Causal Representation Learning for Instantaneous and Temporal Effects in Interactive Systems
Authors: Phillip Lippe
Sara Magliacane
Sindy Löwe
Yuki M Asano
Taco Cohen
Efstratios Gavves
Venue: ICLR 2023 poster (2023)
Keywords: Representation Learning
Causality
Causal Representation Learning
Causal Discovery
Disentanglement

Abstract: Causal representation learning is the task of identifying the underlying causal variables and their relations from high-dimensional observations, such as images. Recent work has shown that one can reconstruct the causal variables from temporal sequences of observations under the assumption that there are no instantaneous causal relations between them. In practical applications, however, our measurement or frame rate might be slower than many of the causal effects. This effectively creates ``instantaneous'' effects and invalidates previous identifiability results. To address this issue, we propose iCITRIS, a causal representation learning method that allows for instantaneous effects in intervened temporal sequences when intervention targets can be observed, e.g., as actions of an agent. iCITRIS identifies the potentially multidimensional causal variables from temporal observations, while simultaneously using a differentiable causal discovery method to learn their causal graph. In experiments on three datasets of interactive systems, iCITRIS accurately identifies the causal variables and their causal graph.

Published as a conference paper at ICLR 2023
CAUSAL REPRESENTATION LEARNING FOR INSTANTANEOUS AND TEMPORAL EFFECTS IN INTERACTIVE SYSTEMS
Phillip Lippe
Sara Magliacane
Sindy Löwe
QUVA Lab
MIT-IBM Watson Al Lab University Amsterdam
UvA-Bosch Delta Lab University of Amsterdam
University of Amsterdam p. w.Lippe@uva..n.
Yuki M. Asano
Taco Cohen
Efstratios Gavves
QUVA Lab University of Amsterdam
Qualcomm Al Research" Amsterdam
QUVA Lab University of Amsterdam
ABSTRACT
Causal representation learning is the task of identifying the underlying causal vari- ables and their relations from high-dimensional observations, such as images. Re- cent work has shown that sequences of observations under the assumption that there are no instantaneous causal relations between them. In practical applications, however, our measure- ment or frame rate might be slower than many of the causal effects. This effectively creates "instantaneous" effects and invalidates previous identifiability results. To address this issue, we propose iCITRIS, a causal representation learning method that allows for instantaneous effects in intervened temporal sequences when inter- vention targets can be observed, e.g., as actions of an agent. iCTTRIS identifies the potentially multidimensional causal variables from temporal observations, while simultaneously using differentiable causal discovery method to learn their causal graph. In experiments on three datasets of interactive systems, iCITRIS accurately identifies the causal variables and their causal graph.
reconstruct the causal variables from temporal
INTRODUCTION
Recently, there has been growing interest in causal representation learning (Schölkopf al., 2021), which aims at learning representations of causal variables in an underlying system from high- dimensional observations like images. Several works have considered identifying causal variables from time series data, assuming that the variables are independent of each other conditioned on the previous time step (Gresele al., 2021; Khemakhem al., 2020a; Lachapelle al., 2022a;b; Lippe et al., 2022b; Yao et al., 2022a:b). This assumes that within each discrete, measured time step. intervening on one causal variable does not affect any other variable instantaneously. However, in real-world systems, this assumption is often violated, as there might be causal effects that act faster than the measurement or frame rate (Faes al., 2010: Hyvärinen al., 2008; Moneta al., 2006; Nuzzi et al., 2021). Consider the example of a light switch and a light bulb. When flipping the switch, there an almost immediate effect on the light by turning it on or off, changing the appearance the whole room instantaneously. In this case, an intervention on a variable (e.g. the switch) also affects other variables (e.g., the bulb) in the same time step, violating the assumption that each variable is independent of the others in the same time step, conditioned on the previous time step. In biology, some protein-protein interactions also occur instatiisttnttaeouusly (Acuner Ozbabacan et al., 201 1).
To overcome this limitation, consider the task of identifying causal variables and their causal graphs from temporal sequences, even in case of instantaneous cause-effect relations. This task contains two main challenges: identifying the causal variables from observations, and learning the causal relations between those variables. We show that, as opposed to temporal sequences without instantaneous
"Qualcomm AI Research initiative of Qualcomm Technologies, Inc.

---

Published as a conference paper at ICLR 2023
effects, neither of these two tasks can be completed without the other: without knowing the variables, we cannot identify the graph: but without knowing the graph, we cannot identify the causal variables, since they are not conditionally independent. In particular, in contrast to causal relations across time steps, the orientations of instantaneous edges are not determined by the temporal ordering, hence requiring to jointly solve the tasks of causal representation learning and causal discovery.
As a starting point, we consider the setting of CITRIS (Causal Identifiability from Temporal Intervened Sequences; Lippe et al. (2022b)). In CITRIS potentially multidimensional causal variables interact over time, and interventions with known targets may have been performed. While in that work all causal relations were assumed be temporal, i.e., from variables in one time step to variables in the next time step. we generalize this setting to include instantaneous causal effects. In particular, we show that in general, causal variables are not identifiable if we do not have access to partially- perfect interventions, i.e., interventions that remove the instantaneous parents. If such interventions are available, we prove that we can identify the minimal causal variables (Lippe et al., 2022b), i.e., the parts of the causal variables that are affected by the interventions, and their temporal and instantaneous causal graph. Our results generalize the identifiability results of Lippe et al. (2022b), since if there instantaneous causal relations, any intervention is partially-perfec by definition. As a practical implementation, we propose instantaneous CITRIS (iCITRIS). iCITRIS maps high-dimensional observations, e.g., images, to lower-dimensional latent space on which it learns an instantaneous causal graph by integrating differentiable causal discovery methods into its prior (Lippe et al., 2022a; Zheng et al., 2018). In experiments on three different video datasets, iCITRIS accurately identifies the causal variables as well as their instantaneous and temporal causal graph. Our contributions are:
We show that causal variables in temporal sequences with instantancous effects are not identifiable without interventions that remove instantaneous parents.
We prove that when having access to such interventions with known targets, the minimal causal variables can be identified along with their causal graph under mild assumptions.
We propose iCITRIS, a causal representation learning method that identifies minimal causal variables and their causal graph even in the case of instantancous causal effects.
Related Work We provide an extended discussion on related work in Appendix C. Early works on causal representation learning focused on identifying independent factors of variations (Klindt et al., 2021; Kumar et al., 2018; Locatello et al., 2019; 2020b: Träuble et al., 2021). in settings similar to Independent Component Analysis (ICA) (Comon, 1994; Hyvärinen et al., 2001; 2019). In particular, Lachapelle et al. (2022a;b); Yao et al. (2022a:b) discuss the identifiability of causal variables from temporal sequences. Yet, in all of these ICA-based setups, causal variables are required to be conditionally independent. For causally-deeendent variables, Yang et al. (2021) learn causal variables from labeled images in supervised manner. Ahuja et al. (2022); Brehmer et al. (2022) identify causal variables with unknown causal relations from pairs of observations that only differ in a subset of causal factors influenced by intervention, i.e., having counterfactual observations. As discussed by Pearl (2009), however, knowing counterfactuals is not realistic in most scenarios. Instead, CITRIS (Lippe al., 2022b) focuses on temporal sequences, in which also the variables that are not intervened upon can still continue evolving over time. On the other hand, in this setting the intervention targets need be known. Moreover, within time step, the causal variables are assumed to be independent conditioned on the variables of the previous time step, hence not allowing for instantancous effects. To the best of our knowledge, iCITRIS the first method to identify causal variables and their causal graph from temporal, intervened sequences even for potentially instantaneous causal effects, without requiring counterfactuals or data labeled with the true causal variables.
2 RELEVANT BACKGROUND AND DEFINITIONS
In this work, we start from the setting of Temporal Intervened Sequences (TRIS) (Lippe et al., 2022b). For clarity, we provide brief overview of TRIS and discuss previous identifiability results, before extending and generalizing the theory to instantaneous effects.
2.1 TEMPORAL INTERVENED SEQUENCES
Temporal intervened sequences (TRIS) (Lippe et al., 2022b) are a latent temporal causal process S with K causal variables (C), C2 ..., Cx yL (e.g., the light switch and bulb), representing a dynamic
2

---

Published as a conference paper at ICLR 2023
Bayesian network (DBN) (Dean et al., 1989; Murphy, 2002). Each causal variable C, is instantiated at each time step t, denoted by Cj and its causal parents pa(C)) are a subset of the variables at time t-
To represent interventions, the causal graph is augmented with binary intervention variables I' e {0, 1)x where 1 = 1 refers to the causal variable C having been intervened at time step t. This setting can model soft interventions (Eberhardt, 2007), i.e., interventions that change the conditional distribution p(C (p(())) I 1) & p(C Ipa(pp((C))) ",, I = 0) (e.g. flipping the light switch). This trivially includes also perfect interventions, do(C; c+) (Pearl, 2009), which cause the target variable to be independent of its parents. To allow for arbitrary sets of interventions, the intervention variables are considered to be confounded by an unobserved variable R'. While the intervention variables I' are assumed to be observed, the actual values of the intervened variables, e.g., the state of the light switch, are not. The graph and its parameters are assumed to be time-invariant (i.e., repeat across time steps), causally sufficient (i.e., no latent confounders besides the variables mentioned before), and faithful (i.e., no additional independences w.r.t. the ones encoded in the graph).
In this setting, causal variables can be scalar or span over multiple dimensions, i.e., C, E D.M, with the dimensionality M; 2 1 and ID; being the domain. For example, a 3d position can be represented as C. E 3 The causal variable space is defined as C = D x D x... x D
Instead of observing the causal variables directly, we measure a niig-diimnnsional observation X e.g., an image, representing a noisy, entangled view of all causal variables C = (Ci, C2. ..., Ck) at time step L. The observation function is defined as h(C C2 ..., Ck. E*) = X', where E E is any noise on the observation X that is independent of C' (e.g. pixel or color shifts), and h : C x E + X is a function from the causal variable space C' and the space of the observation noise E = Rh to the observation space X' C IR To allow unique identification of causal variables from observations, the observation function h is assumed to be bijective, i.e., there exist a unique inverse of for all X E X.
2.2 MINIMAL CAUSAL VARIABLES AND IDENTIFIABILITY CLASS
Multidimensional causal variables are not always fully identifiable in TRIS, when interventions only
affect a subset of the variables' dimensions (Lippe et al., 2022b), e.g., only x in a 3d position [r, y, z]. To account for these interventions, each causal variable Of can be split into an interventiondepeedeen part syar (CT), e-g., [zr). and an intervention- intiiion-ddeeeeeent part (C,),, e.g., [v, 2), where s;())) =
(s) (CI), s. (CP)) )) an invertible function. Under this split, the distribution of C₁ becomes:
p (C) pa (CC)) 11) = =p (x) (6)"((C) (Ca))pa( (aCC) T2) P (C) (pa()))
(1)
With this setup, Lippe et al. (2022b) define a minimal causal variable as follows:
Definition 2.1. The minimal causal variable of a causal variable C w.n.. its intervention variable I is the intervention terpentton-deeeneentt part star(C) (CT) of the split s s((C)) (sTa (C)), sqtx (C)),, such that
the splut maximizes the information content H (stay (Ci)pa(()) in terms of the limiting density of
discrete points (LDDP) (Jaynes, 1957; 1968).
To identify the minimal causal variables from data triplets fa², (4, 111 ](11), ( CITRIS approximates the observation function h by learning an invertible map go X + Z, with Z E M being latent space with M dimensions. For this latent space, CITRIS learns an assignment function eb [[..M]] + [O...].. mapping each dimension of to a causal variable C,.... Ck. The index 0 is used for the observation noise or cnnrreettioninddeennee
variables. We denote the set of latents assigned to the causal variable C, with 2 {zylí []..MM] (1) = i). On this latent space, CITRIS models a prior Po.vo (= (=(+1|=², /141 where each group of latent variables, z++1 is conditioned on the previous time step 2 and its intervention variable r but independent among each other within the same time step. The causal graph can be found by pruning the temporal dependencies in this prior.
Under this model, Lippe et al. (2022b) consider causal system S to be identified by model M if its minimal causal variables are identified up to an invertible transformation, and M recovers the true causal graph in S. CITRIS is shown identify S if it maximizes the information content of Zyoo under the constraint of maximizing the likelihood of data points {X', x _ and no intervention variable I is a deterministic function of any other intervention variable. However, if causal effects occur faster than the observation rate, an intervention influences also other variables in the same time step in addition to its target, leading CITRIS to potentially identify incorrect variables. In this paper, we generalize this identifiability result to systems where instantaneous causal relations may exist.
3

---

Published as a conference paper at ICLR 2023
3 IDENTIFYING CAUSAL VARIABLES WITH INSTANTANEOUT EFFECTS
In this section, we generalize Temporal Intervened Sequences (TRIS) to settings where causal relations can be potentially instantaneous. First, we discuss the challenges arising from instantaneous effects, and then present solutions to overcome these challenges. Finally, we derive our identifiability results.
3.1 ITRIS AND CHALLENGES OF INSTANTANEOUS EFFECTS
We extend TRIS to a setting we call instantancous Temporal Intervened Sequences (iTRIS) which allows for instantancous causal effects. In iTRIS, causal variables within the same time step cause each other, as long as the graph remains acyelic. This means that, for example, C can cause C for * long there is no directed path C2+1 .... C++l Figure 1 summarizes this setting.
While the addition of instantancous effects may seem like a small change, it violates the key assumption of most previous works (Khemakhem et al., 2020a; Lachapelle al., 2022a;b; Lippe et al., 2022b; Yao et al., 2022a;b), namely that causal variables within a time step are independent conditioned some external variable. As consequence, we have to differentiate between causal models in a much larger function space than before, making identifiability a considerably harder task.
To formalize this intuition, consider the follow- ing example. Assume we have two latent causal variables Cj and C2, and, for simplicity, no tem- poral relations. The causal variables C1 and C2 do not cause each other, and we have an arbi- trary observation function hCC, C2) X and distributions pl
Cousal vuriables
X'
Temporal relations
cousal relations
Observation
this example, if our method allows for instantaneous effects, we cannot identify the causal variables or their graph from Pa (X) alone, since there mul- tiple representations that model (X) equally well. For instance, the representation Õ₁ C1, C,,C==C, = + C2 with the causal graph Ć₁ O2 model the same observation distribution since Paz (a(C2(C)) (C C2 C1) = (C2) and hence (1 under soft interven-
C++1
Intervemion variables
Latimi confounding
R
An example causal graph in iTRIS. A la- tent causal variable C+++1 can have as potential par- ents a subset of the causal variables at the previous time step C = (Ci..... Ckx), instantancous par- ents C++1 # j. and its intervention variable I!+1 All causal variables Cr-1 and observation noise E++1 cause the observation h(C++1, E++1) X R is latent confounder allowing for de- pendencies between intervention variables.
Figure 1:
= p(C))p(C2) =
(X). This happens tions, because the causal graph can remain un- changed. However, if we have interventions on C that remove its instantaneous parents, then the learned representation of C, must inde- pendent of any instantaneous effects. This elim- inates Č₁, C22 since C₂ A TT =1 We refer to these interventions as partially-perfect:
Definition 3.1. A partially-perfer intervention on a causal variable C; is soft intervention that removes all parents the instantaneous graph: pl p(C))pp(( (CC))pa(C)).T I; =) Pr (C (Cm where patamp(C (C)) = (C E [L..K] k.. pa(C))) and p1 the cost-interventional distribution.
As an example, consider an intervention that sets C -/-1+ e, where e is noise. While this intervention breaks instantaneous relations, the target still depends on the previous time step, making Using the partially perfect interventions, we can prove the following:
it
terffellyperfeett
Lemma 3.2. In ITRIS, causal variable C₁ cannot be identified up to an invertible transformation T, that independent of all other causal variables, if C; can have instantaneous parents and no partially-perfer interventions C; are provided.
We provide the proof for this lemma and example with temporal relations in Appendix D.2.1. In the following, we assume that all interventions in iTRIS are cartially-perfeee This subsumes the soft intervention setup in TRIS by definition, since the instantaneous graph is empty in this case.
4

---

Published as a conference paper at ICLR 2023
3.2 IDENTIFYING THE MINIMAL CAUSAL VARIABLES IN ITRIS
Using partially-perfect interventions, we introduce our identifiability results in iTRIS. For simplicity of the theoretical analysis, we follow Lippe et al. (2022b) and consider a continuous domain for causal variables, i.e., D = IR, and that distributions have full support. In Section 5, we empirically extend the results to variables with categorical and circular domains, and distributions with limited support. Further, we assume that the temporal dependencies and interventions break all symmetries in the variables' distributions such that distributional implies functional independence. For a Gaussian, this entails that the mean cannot be a constant (see Appendix D.2 for details on the assumptions).
Similar to CITRIS, we learn an invertible mapping, go : X + 2, and an assignment function wh [1. - []..K]] ] to align latent to causal variables. Differently, however, we also learn directed, acyclic graph G' on the K + latent variable groups Zvjo: .... Faxx to model the instantaneous causal relations. The graph G induces a parent structure denoted by 2Fp-r = {z₁ i E [I..M], v(i) E (1) where pag (0) 0, i.e., the variables in 71%o have no instantaneous parents. Meanwhile, the temporal causal graph between C* and C+++1 is implicitly learned by conditioning 2*+1 on all latents of the previous time step, z¹, and can be pruned after training. This results in the following prior:
K
Pa.v. (14t = PP (2)
IPo (2 2411 )
1=1
With this setup, we can now formally define our identifiability class that contains the one in CITRIS:
Definition 3.3. A model M = (0, Φ, vb G) identifies a causal system S = (C, iff for each causal variable C, E [1..K]. the following two conditions hold:
(I) each minimal causal variable is identified up to an invertible transformation Tú, i.c., for all
observations x E X sqaa (c:) = (z where c, = [h- (z)); the value of the true causal variable and zv, = [go(()) is the value of the estimated minimal causal variable, and (2) the estimated parents pal are the same as the true parents of the minimal causal variable aaar (Ct), i.e., the estimated parent set contains z . E [1..K],T T E of sf yaar (C)), and it contains 2/o iff there exist E [1..K] for which sitx is a parent of star (C)).
{1-1,1) 1 iff 83 (CJ) is a parent
For the light switch example, this means that the latent variable Zg, must model the switch's 's state, Zeb the bulb's ' state, and the instantancous graph is 24, - 24hz. Compared to identifiability class explicitly aligns the latent variables with the causal variables and, thus, we do not rely on permutation equivalence class. To identify S from observations, we consider dataset of triplets {a', _ [[+] } with observations x", -++1 E X' and intervention variables μ[+1, This dataset could be created interactively or, for example, recorded by an expert. With this, the objective becomes:
results, this
Po. (2++2 1x", 1 |eet Joe (a++t)) Po.v.G 11,
where the Jacobian of go. |det Joo (x(+1)| is introduced due to the change of variables of x to 2. dim(X') > dim(C x E), we consider that ge contains an arbitrary, fixed map from X' to the lower dimensionality. Under the assumption that go. Pe are universal function approximators and the dataset is unlimited in size, we derive the following identifiability result:
Theorem 3.4. In iTRIS, model M = (0",0" ",,"""C. GO) identifies causal system S (C, h) (Definition 3.3) if M under the constraint of maximizing the likelihood Po.0,a(X't111> (xx
11t1);
(I) maximizes the information content H22+++12² ) in terms of the LDDP (Jaynes, 1957; 1968),
(2) minimizes the number of edges in G and
(3) no intervention variables T;, 13 are deterministically related, i.e., Vj #i: -(I, VI 7 ((T)).
Intuitively, this theorem shows that we can identify the minimal causal variables, even when instanta- neous effects are present, under the same constraints as CITRIS. The proof in Appendix D follows three main steps. First, we show that the true observation function constitutes a global optimum of Equation (3), but is not necessarily unique. Second, we derive that any global optimum must iden- tify the minimal causal variables. Finally, we show that optimizing the data likelihood identifies the complete causal graph, i.e., instantaneous and temporal, between the minimal causal variables.
5

---

Published as a conference paper at ICLR 2023
4 CAUSAL REPRESENTATION LEARNING WITH INSTANTANEOUS EFFECTS
Based on our theoretical results, we propose iCITRIS, a generalization of CITRIS (Lippe et al., 2022b). We first review the original CITRIS architecture, and then describe extensions in iCITRIS.
4.1 BASELINE: CITRIS
CITRIS is build upon variational autoencoder (VAE) (Kingma et al., 2014), where the (convolutional) encoder and decoder approximate the invertible map ge. To promote the identification of the causal variables in latent space, the prior of the VAE follows Equation (2), excluding the instantaneous parents. All latent distributions Ps are usually implemented as conditional Gaussians, with the mean and std predicted by small MLP per latent variable. Finally, the VAE is trained via maximum likelihood on p( ( x [X', I(+1). Alternatively, CITRIS can also be trained on the representations of pretrained autoencoder, where the map ge is replaced by a normalizing flow (Rezende et al., 2015). We follow the same setup in iCITRIS, but extend CITRIS' prior to instantancous effects.
4.2 LEARNING THE INSTANTANEOUS CAUSAL GRAPH
To learn the instantaneous causal graph simultaneously with the causal representation, incorporate recent differentiable score-based causal discovery methods in iCITRIS. Given a distribution over graphs p(G), the conditional distribution over the latent variables = _t+1 of Equation (2) becomes:
Pa.va 12 1++1 Pe (2*t) (EGD(()
II Po
(2 '2 2 (111)))
(4)
where the parent sets, Z0re depend on the graph structure G. The goal is to jointly optimize Ps and p(G) under maximizing the likelihood objective of Equation (3), such that p(G) is peaked at the correct causal graph. To this end, experiment with two causal discovery methods that allow for continuous optimization: NOTEARS (Zheng et al., 2018), and ENCO (Lippe et al., 2022a).
NOTEARS (Zheng al., 2018) casts structure learning as continuous optimization problem by providing continuous constraint on the adjacency matrix to enforce acyclicity, Following Ng et al. (2022), we model the adjacency matrix with independent edge likelihoods, and differentially sample from it using the Gumbel-Softmax trick (Jang et al., 2017). We use these samples as graphs in the prior pa (= (='+) lz+2) to mask the parents the individual causal variables, and obtain gradients for the graph through the maximum likelihood objective of the prior. In order to promote acyclicity, we use the constraint as regularizer, and exponentially increase its weight over training.
ENCO (Lippe et al., 2022a), on the other hand, uses interventional data and two separate parameter sets: one for the orientation per edge, and one for the existence per edge. By only using interventions to update the orientation parameters, ENCO converges to the true, acyclic graph under single-target interventions in the sample limit. Yet, we found it to also work well under multi-targe interventions in iCITRIS. ENCO uses low-variance, unbiased gradients based on REINFORCE (Williams, 1992), potentially providing a more stable optimization than NOTEARS. For efficiency, we merge the two learning stages of ENCO and update both the graph and distribution parameters at each iteration.
4.3 STABILIZING THE OPTIMIZATION PROCESS
Simultaneously identifying the causal variables and their graph leads to chicken-and -nd--eg situation: without knowing the variables, we cannot identify the graph; but without knowing the graph, we cannot identify the causal variables. This can cause the optimization to be unstable and to converge to local minima with incorrect graphs. To stabilize it, we propose the two following approaches.
Graph learning scheduler During the first training iterations, the assignment of latent to causal variables is almost random, since the gradients for the graph parameters are very noisy and uninfor- mative. Thus, we use learning rate schedule for the graph learning parameters such that the graph parameters are frozen for the first couple of epochs. During those training iterations, the model learns to fit the latent variables to the intervention variables under arbitrary graph, leading an initial, rough assignment of latent to causal variables. Then, we warm up the learning rate to slowly start the graph learning process while continuing to separate the causal variables in latent space.

---

Published as a conference paper at ICLR 2023
Mutual information estimator If the provided interventions are fully perfect, L.e, they remove temporal dependencies as well, we can exploit this independence by masking out the temporal parents in the prior distribution under interventions. Furthermore, with perfect interventions, also enforce the mutual information (Mf) (Kullback, 1997) between parents and children under interventions to he zero as additional regularization. Following work on neural MI estimation (Belghazi et al., 2018; Hjelm et al., 2019; van den Oord et al., 20) we train a network to distinguish between samples from the joint distribution p(Z 04 3 2 4 1) and the product of their marginals, p( 14
1)p(< 2 ! 1). While the MI estimator optimizes its accuracy, the latents are optimized to do the opposite, effectively forcing z/e, and its parents to be independent under interventions.
5 EXPERIMENTS
We evaluate iCITRIS on three video datasets with varying difficulties and compare it to common causal representation methods. We include further dataset details in Appendix E, discuss hyperpa- rameters in Appendix F, and provide the code at https: /ttl:://ggthttt com/ wwt/po/m//cc.ceeecccceeeeeecceee httpppphpppppp/////////////
5.1 EXPERIMENTAL SETTINGS
Baselines Since iCITRIS is, to the best of our knowledge, the first method to identify causal variables with instantaneous effects in this setting, we compare independent causal variables. Firstly, we use CITRIS (Lippe et al., 2022b) and the Identifiable VAE (iVAE) (Khemakhem al., 2020a), which both use the previous time step and intervention targets to model conditionally independent variables. Further, to compare to model with dependencies among latent variables, we evaluate the iVAE with an autoregressive prior, which we denote with iVAE-AR. All methods share the general model setup, e.g., the encoder network architecture, where possible.
methods for identifying conditionally
Evaluation metrics To evaluate the identification of the causal variables, we follow Lippe et al. (2022b) and report the R2 scores for correlations. In particular, R2 is the score between the true causal variable C; and the latents that have been assigned to the causal variable Cj by the learned model, i.c., Zn//-- We denote the average correlation of the predicted variable its true value with R2 diag- 1/K) E, Ri (optimal 1), and the maximum correlation besides its true variable with R sep 1/K Σ, maxi4i RR, (optimal 0). Furthermore, to investigate the modeling of the temporal and instantaneous relations between the causal variables, we perform causal discovery as a post- processing step on the latent representations since the baselines do not explicitly learn the graph, and report the Structural Hamming Distance (SHD) between the predicted and true causal graph.
5.2 2D COLORED CELLS WITH CAUSAL EFFECTS: VORONOI BENCHMARK
We first conduct experiments on synthetically generated causal graphs with various instantaneous structures to investigate the difficulty and challenges of the task. We consider three instantancous graph structures: random has a randomly sampled, acyelic graph structure with a probability of 0.5 of two variables being connected by direct edge, and cha i n and full 1 represent the minimally- and maximally-comnected DAGs respectively. For each graph, we sample temporal edges with an edge probability of 0.25 matching the density of the instantaneous causal graph. Based on these graphs, we create the variable's observational distributions as Gaussians parameterized by randomly initialized neural networks, and provide for simplicity single-target, perfect interventions for all variables. The causal variables are mapped to image space X' by firstly applying randomly initialized, two-layer normalizing flow, and afterwards plotting them as colors in a 32x32 pixels image of fixed Voronoi diagram as an irregular structure. Thus, the representation learning models need to distinguish between the entanglement by the random normalizing flow and the underlying causal graphs to identify the causal variables, while also performing causal discovery to find the correct causal graph.
In Figure 2, we show the results of all models on graphs of 4, 6, and 9 variables. For the random and chain graphs, iCITRIS-ENT ENCO identifies the causal variables and their causal graph with only minor errors, even for the largest graphs of 9 variables. Even on the challenging full graph, iCITRIS- ENCO considerably outperforms the other models. In contrast, iCITRIS-NOTEARE struggles with the edge orientations and converges to edge probabilities noticeably lower than 1.0, with which the variables cannot be perfectly identified anymore, especially for increasing graph sizes. Meanwhile,
7

---

Published as a conference paper at ICLR 2023
Models
Error type
(ouns)
(oum)
0.6
0.6
06
0.4
0.4
cror
04
random
chdli Graph etructure
Edl
randon
chain Graph structure
Mil
(0 roddo
chain Graph structure
full
(a) R² error- 4 variables
(b) R² error variables
(c) R error 9 variables
Models
Error type
CITRRBEENCO (ours)
NOTRRIS-NTTEARBBO (ours)
CITReB
NAE
NAE-AR
Vnstrrssneuus edge
Terppons edgys
10.0
30
60
SHD
40
10
2.5
6.0
obait
random
nandom
ctan
Graph aisucture
Graph structuee
Graph stucturo
(d) SHD 4 variables
(e) SHD 6 variables
(f) SHD 9 variables
Figure 2: Results on the Voronoi benchmark over three graph structures and sizes with five seeds (error bars in Appendix G.I). For all metrics, lower is better. Top row (a-c): Plotting the R2 correlation error as the average distance between predicted and true causal variables (1-"R² diag", solid bars), plus the maximum correlation any other variable (" I2 sep". striped bars). iCITRIS-ENCE performs well across graph structures and sizes. Bottom row (d-f): The SHD between predicted and ground truth causal graph, divided into instantaneous (solid bars) and temporal (striped bars) edges. iCITRIS- ENCO obtains the lowest error across graphs, with close to zero for the graphs r random and chain.
CITRIS and iVAE find K independent dimensions, conditioned on the previous time step and the intervention targets, instead of the true causal variables. which leads to sparse instantancous, but wrongly dense temporal graphs. Finally, the autoregressive baseline, iVAE-AR, naturally entangles all dimensions in the latent space, on which the true causal graph cannot be recovered anymore. This underlines the non-tric triviality of identifying rsstantaneously-rlattee iCITRIS identifies the causal variables and graph well across graph structures and sizes, with ENCO outperforming NOTEARS due to more stable optimization, especially for larger, complex graphs.
causal variables. In conclusion,
5.3
3D OBJECT RENDERINGS: INSTANTANEOUS TEMPORAL CAUSAL3DIDENT
As visually challenging dataset, we use the Temporal Causal3DIdent dataset (Lippe et al., 2022b; Kügelgen et al., 2021) which contains 3D renderings (64 x 64 pixels) of different object shapes under varying positions, rotations, and lights. The dataset has seven causal variables, in- cluding categorical and circular variables, going beyond iCITRIS's theoretical setting. To intro- duce instantaneous effects, we replace all temporal relations with instantaneous edges, except those on the same variable (C + C)+)). For instance, change in the rotation leads to an instanta- neous change in the position of the object, which again influences the spotlight. Overall, we ob- tain an instantaneous graph of eight edges between the seven multidimensional causal variables. We provide partially-perfee interventions that remove instantaneous parents, but leave the ex- isting temporal dependencies unchanged. Since the dataset is visually complex, we use the nor- malizing flow variant of iCITRIS and CITRIS applicd on pretrained autoencoder.
Table 1: Results the Instantaneous Temporal Causal3DIdent dataset over three seeds (standard deviations in Table 9). iCITRIS-ENCO performs best in identifying the variables and their graph.
Table shows that ECITRIS-ENCE identifies the causal variables well and recovers most in- stantaneous relations, with up two errors average. The temporal graph had more false pos- itive edges due to minor correlations. iCITRIS-
Model
R (ding / sop () SHI (instant temp
iCITRIS-ENT ENCO NOTEISS
0.96/ 0.07
0.95/0.10
/
CITRIS IVAE IVAE-RR
0.79/0.22 0.221
6.00 15.00
0.74/
10.67 / 12
8

---

Published as a conference paper at ICLR 2023
NOTEARS incorrectly orients several edges during training, underlining the benefit of ENCO as the graph learning method in iCITRIS. The baselines have a significantly higher entanglement of the causal variables and struggle with finding the true causal graph. Further, in Appendix G.2, apply iCTTRIS to the original Temppora Causal3DIdent dataset, which contains only temporal causal relations and no instantaneous effects. In this setting, iCITRIS performs on par with CITRIS, verify- ing that iCITRIS generalizes CITRIS across datasets. In summary, causal variables along with their instantancous graph well, even in visually challenging dataset.
can identify the
5.4 REAL GAME DYNAMICS: CAUSAL PINBALL
Finally, consider simplified version of the game Pinball, which naturally has instantaneous causal effects: if the paddles are activated when the ball is close, the ball is accelerated immediately. Similarly, when the ball hits a bumper, its light turns on and the score increases immediately. This results in instantaneous effects, especially under common frame rates. In this envi- ronment, we consider five causal variables: the position of the left paddle, the right paddle, the ball (position and velocity), the state of the bumpers, and the score. Interventions again remove instantaneous, but keep temporal parents. Pinball closer real- environment than the other two datasets and has two characteristic differences: (1) many aspects of the environment are deterministic, e.g., the ball movement, and (2) the instantaneous effects are sparse, e.g., the paddles do not influence the ball if it is far away of them. Such a setting violates several assumptions like faithfulness, the full support and potential symmetries in the observational and interventional distributions, questioning whether iCITRIS empirically works here.
Figure 3: Example of Causal Pinball.
The results in Table 2 suggest that iCITRIS still works well on this environment. Besides identi- fying the causal variables well, iCITRIS ENCO identifies the instantaneous causal graph with minor errors. In contrast, CITRIS entangles the variables much stronger, while iVAE has diffi- culties identifying all variables in the environ- ment. This shows that iCITRIS can be applied in challenging environments beyond our theoret-
Table 2: Results on the Causal Pinball dataset three seeds (see Table 12 for standard deviations).
Model
R (diag / sep U)
SHD (instant L/ temp
ICITRIS-ENCO NOTRRSS
0.99/00 /0.12
0.67/3.000
0.98 / 0.
3.53 / 1.67
CITRIS IVAE IVAE-RR
0.39
0.44 0.05
4.33 / 4.67
0..15
8.00//3.01
ical limitations, even with deterministic causal effects, while maintaining strong empirical results.
CONCLUSION AND DISCUSSION
We propose iCITRIS, a causal representation learning framework for temporal intervened sequences with potentially instantaneous effects. From such sequences, iCITRIS identifies the minimal causal variables while jointly learning the instantaneous and temporal causal graph. In experiments, iCITRIS accurately recovers the causal variables and their graph in three video datasets.
Since instantaneous effects are common in real-world settings (Hyvärinen et al., 2008; Nuzzi et al., 2021), we believe that iCITRIS contributes an important step towards practical causal representation learning methods. Still, as with most other theoretical results, our identifiability theorem is limited by the assumptions it takes. The two most crucial assumptions in iCITRIS are having a dataset, potentially recorded by an expert, that has (1) non-detemiinittcally related, known intervention targets and (2) partially-perfees interventions, i.e., interventions that can remove instantaneous parents. Without the first assumption, causal variables may become entangled in the latent space, and without the latter, instantaneous causal relations may be predicted where none truly exist. However, as demonstrated in experiments on Causal3DIdent and Causal Pinball, iCITRIS still achieves strong empirical performance in settings that violate other assumptions. For instance, in these experiments, the distributions had limited support and some variables had circular categorical domains.
To extend iCITRIS to even more settings, future work includes investigating a setup where interven- tions are not directly available, but can be performed by sequences of actions, and targets must be learned in an unsupervised manner. Further, iCITRIS limited to acyclic graphs, while for instanta- neous effects cycles could occur under low frame rates, which is also interesting future direction.

---

Published as a conference paper at ICLR 2023
AUTHOR CONTRIBUTIONS
Lippe conceived the idea, derived the theoretical results, implemented the models and datasets, and wrote the paper. S. Magliacane, S. Löwe, Y. M. Asano, T. Cohen, E. Gavves advised during the project and helped in writing the paper.
ACKNOWLEDGMENTS
We thank Johann Brehmer and Pim de Haan for valuable discussions throughout the project. We also thank SURFsara for the support in using the Lisa Compute Cluster. This work is financially supported by Qualcomm Technologies Inc., the University of Amsterdam and the allowance Top consortia for Knowledge and Innovation (TKIs) from the Netherlands Ministry of Economic Affairs and Climate Policy.
REFERENCES
Saliha Ece Acuner Ozbabacan, Hatice Billur Engin, Attila Gursoy, and Ozlem Keskin. Transient protein-protein interactions. Protein Engineering, Design and Selection, 24(9):635-648, 06 2011.
Kartik Ahuja, Jason Hartford, and Yoshua Bengio. Weakly Supervised Representation Learning with Sparse Perturbations. In Advances in Neural Information Processing Systems 35: Annual Conference on Neural Information Processing Systems 2022, NeurlPS 2022 (to appear), 2022.
Mohamed Ishmael Belghazi, Aristide Baratin, Sai Rajeshwar, Sherjil Ozair, Yoshua Bengio, Aaron Courville, and Devon Hjelm. Mutual Information Neural Estimation. Jennifer Dy and Andreas Krause (eds.), Proceedings of the 35th International Conference Machine Learning, volume 80 of Proceedings of Machine Learning Research, pp. 531-540. PMLR, 10-15 Jul 2018
Blender Online Community. Blender- - 3D modelling and rendering package. Blender Foundation, Stichting Blender Foundation, Amsterdam, 2021.
Johann Brehmer, Pim de Haan, Phillip Lippe, and Taco Cohen. Weakly supervised causal representa- tion learning. In Advances in Neural Information Processing Systems 35: Annual Conference on Neural Information Processing Systems 2022, NeurIPS 2022 (to appear), 2022.
Philippe Brouillard, Sébastien Lachapelle, Alexandre Lacoste, Simon Lacoste-Julien and Alexan- dre Drouin. Differentiable Causal Discovery from Interventional Data. In Hugo Larochelle, Marc' Aurelio Ranzato, Raia Hadsell, Maria-Florina Balcan, and Hsuan-Tien Lin (eds.), Advances in Neural Information Processing Systems 33: Annual Conference on Neural Information Process- ing Systems 2020, NeurlPS 2020, December 6-12, 2020, virtual, 2020.
Pierre Comon. Independent component analysis, A new concept? Signal Processing, 36(3):287-314,
April 1994.
Thomas M. Cover and Joy A. Thomas. Elements of Information Theory. John Wiley and Sons, Ltd, 2005. ISBN 9780471748823. doi:
ffmater
Keenan Crane. Keenan's 3D Model Repository. https: / /tww://wwwecc//eco/////////////o cs. emu edu/-kmerane/ Projects/Modes Reposi tory/ 2021.
Brian Curless and Mare Levoy. A Volumetric Method for Building Complex Models from Range Images. Proceedings of the 23rd Annual Conference on Computer Graphics and Interactive Techniques, SIGGRAPH '96, pp. 303-312, New York, NY, USA, 1996. Association for Computing Machinery. ISBN 0897917464. doi: 10.1145/237170.257200
Pim De Haan, Dinesh Jayaraman, and Sergey Levine. Causal confusion in imitation learning. Advances in Neural Infommaionn Processing Systems, 32, 2019.
Thomas Dean and Keiji Kanazawa. A model for reasoning about persistence and causation. Computa- 16990...999.
tional Intelligence, 5(2):142-150, 1989. doi:
Frederick Eberhardt. Causation and intervention. Unpublished doctoral dissertation, Camegie Mellon University, pp. 93, 2007.
10

---

Published as a conference paper at ICLR 2023
Martin Engelcke, Adam R Kosiorek, Oiwi Parker Jones, and Ingmar Posner. GENESIS: Generative scene inference and sampling with object-centric latent representations. International Conference on Learning Representations (ICLR), 2020.
Luca Faes and Giandomenico Nollo, Extended causal modeling to assess partial directed coherence in multiple time series with significant instantancous interactions. Biological cybernetics, 103(5): 387-400, 2010.
William Falcon and The PyTorch Lightning tcam. PyTorch Lightning, 2019.
Luigi Gresele, Julius von Kügelgen, Vincent Stimper, Bernhard Schölkopf, and Michel Besserve. Independent mechanism analysis, a new concept? In A. Beygelzimer, Y. Dauphin, P. Liang, and Wortman Vaughan (eds.), Advances in Neural Information Processing Systems, 2021.
Kaiming He, Xiangyu Zhang, Shaoqing Ren, and Jian Sun. Deep residual learning for image recognition. In Proceedings of the IEEE conference on computer vision and pattern recognition, pp. 770-778, 2016.
Devon Hjelm, Alex Fedorov, Samuel Lavoie-Marchildeee Karan Grewal, Phil Bachman, Adam Trischler, and Yoshua Bengio. Learning deep representations by mutual information estimation and maximization. In International Conference on Learning Representations, 2019.
John D. Hunter. Matplotlib: A 2D graphics environment. Computing in Science Engineering, 9()): 90-95, 2007. doi: 10. 10110//C..22000.55..
Antti Hyttinen, Frederick Eberhardt, and Patrik O Hoyer. Experiment selection for causal discovery. Journal of Machine Learning Research, 14:3041-3071, 2013.
Aapo Hyvärinen and Hiroshi Morioka. Unsupervised Feature Extraction by Time-Contrastive Learning and Nonlinear ICA. In Proceedings of the 30th International Conference on Neural Information Processing Systems, NIPS' 16, pp. 3772-37888 Red Hook, NY, USA, 2016. Curran Associates Inc. ISBN 9781510838819.
Aapo Hyvärinen, Juha Karhunen, and Erkki Oja. Independent Component Analysis. John Wiley & Sons, June 2001.
Aapo Hyvärinen. Shohei Shimizu, and Patrik O. Hoyer. Causal Modelling Combining Instantaneous and Lagged Effects: An Identifiable Model Based on Non- -ousaiausia In Proceedings of the 25th International Conference on Machine Learning, ICML '08, pp. 424 434, New York, NY, USA, 2008. Association for Computing Machinery, ISBN 9781 605582054. 1390210. URL https:ttp/// do wwwooo/////0 10000000000 1141114544 3955/3311155 1390210.
doi: 10.1145/1390156.
Aapo Hyvärinen, Hiroaki Sasaki, and Richard Turner Nonlinear ICA Using Auxiliary Variables and Generalized Contrastive Learning. In Kamalika Chaudhuri and Masashi Sugiyama (eds.), Proceedings of the Twenty-Secone International Conference Artificial Intelligence and Statistics, volume 89 of Proceedings of Machine Learning Research, pp. 859-868. PMLR, 2019.
Sergey Ioffe and Christian Szegedy. Batch Normalization: Accelerating Deep Network Training by Reducing Internal Covariate Shift. In Francis Bach and David Blei (eds.), Proceedings of the 32nd International Conference on Machine Learning, volume 37 of Proceedings of Machine Learning Research, pp. 448-456, Lille, France, 07-09 Jul 2015. PMLR.
Eric Jang, Shixiang Gu, and Ben Poole. Categorical Reparameterization with Gumbel-Softmax, In 5th International Conference on Learming Representations, ICLR 2017, Toulon, France, April 24- 26, 2017, Conference Truck Proceedings, 2017.
Edward T. Jaynes. Information Theory and Statistical Mechanics. Phys. Rev. 106:620-630, May 1957. doi: 10. 101.100yyeee..1066. 106.620.
Edwin T. Jaynes. Prior Probabilities. IEEE Transactions on Systems Science and Cybernetics, 4(3): 227-241, 1968. doi: 10.1109/TSSC.1100 1968.300119

---

Published as a conference paper at ICLR 2023
Ilyes Khemakhem, Diederik Kingma, Ricardo Monti, and Aapo Hyvarinen. Variational Autoencoders and Nonlinear ICA: A Unifying Framework. In Silvia Chiappa and Roberto Calandra (eds.), Proceedings of the Twenty Third International Conference Artificial Intelligence and Statistics, volume 108 of Proceedings of Machine Learning Research, pp. 2207-2217. PMLR, 2020a.
Ilyes Khemakhem, Ricardo Monti, Diederik Kingma, and Aapo Hyvarinen. ICE-BeeNs Identifiable Conditional Energy-Basse -ased Deep Models Based on Nonlinear ICA. In H. Larochelle, M. Ranzato, R. Hadsell, M. Balcan, and H. Lin (eds.), Advances in Neural Information Processing Systems, Curran Associates, Inc., 2020b.
volume 33, pp. 12768-12777
Diederik P. Kingma and Jimmy Ba. Adam: A Method for Stochastic Optimization. In Yoshua Bengio and Yann LeCun (eds.), 3rd International Conference on Learning Reprexentations, ICLR 2015, San Diego, CA, USA, May 7-9, 2015, Conference Track Proceedings, 2015.
Diederik P. Kingma and Max Welling. Auto-Encoding Variational Bayes. In Yoshua Bengio and Yann LeCun (eds.), 2nd International Conference on Learning Representations, ICLR 2014, Banff AB, Canada, April 14-16, 2014, Conference Track Proceedings, 2014.
Durk P Kingma and Prafulla Dhariwal. Glow: Generative Flow with Invertible 1x1 Convolutions. and R. Garnett (eds.),
In S. Bengio, H. Wallach, H. Larochelle, K. Grauman, N. Advances in Neural Information Processing Systems, volume 31. Curran Associates, Inc., 2018.
Durk P Kingma, Tim Salimans, Rafal Jozefowicz, Xi Chen, Ilya Sutskever, and Max Welling. Improved variational inference with inverse autoregressive flow. Advances in neural information processing systems, 29, 2016.
Thomas Kipf, Gamaleldin F Elsayed, Aravindh Mahendran, Austin Stone, Sara Sabour, Georg Heigold, Rico Jonschkowski, Alexey Dosovitskiy, and Klaus Greff. Conditional Object-Centric Learning from Video. International Conference Learning Representations (ICLR), 2022.
David Klindt, Lukas Schott, Yash Sharma, Ivan Ustyuzhaninov. Wieland Brendel, Matthias Bethge, and Dylan Paiton. Towards Nonlinear Disentanglement in Natural Data with Temporal Sparse Coding. In International Conference on Learning Representations (ICLR), 2021.
Venkat Krishnamurthy and Mare Levoy. Fitting Smooth Surfaces to Dense Polygon Meshes. In Proceedings of the 23rd Annual Conference Computer Graphics and Interactive Techniques, SIGGRAPH '96, pp. 313-324, New York, NY, USA, 1996. Association for Computing Machinery. ISBN 0897917464. doi: 10. 10.1145/37770.233277.
Solomon Kullback. Information theory and statistics. Courier Corporation, 1997.
Abhishek Kumar, Prasanna Sattigeri, and Avinash Balakrishnan. Variational Inference of Disentan- gled Latent Concepts from Unlabeled Observations. In 6th International Conference on Learning Representations, ICLR 2018, Vancouver, BC, Canada, April 30 May 3, 2018, Conference Track Proceedings, 2018
Sébastien Lachapelle and Simon Lacoste-Julien. Partial Disentanglement via Mechanism Sparsity. arXiv preprint arXiv:2207.077:: 2022a.
Sebastien Lachapelle, Pau Rodriguez, Rémi Le, Yash Sharma, Katie E Everett, Alexandre Lacoste, and Simon Lacoste-Julien. Disentanglement via Mechanism Sparsity Regularization: A New Principle for Nonlinear ICA. In First Conference on Causal Learning and Reasoning, 2022b.
Phillip Lippe. Taco Cohen, and Efstratios Gavves. Efficient Neural Causal Discovery without Acyclicity Constraints. In International Conference on Learning Representations, 2022a.
Phillip Lippe, Sara Magliacane, Sindy Löwe, Yuki M. Asano, Taco Cohen, and Efstratios Gavves. CITRIS: Causal Identifiability from Temporal Intervened Sequences. In Proceedings of the 39th International Conference on Machine Learning, ICML 2022, 2022b.
Francesco Locatello, Stefan Bauer, Mario Lucic, Gunnar Raetsch, Sylvain Gelly, Bernhard Schölkopf, and Olivier Bachem. Challenging Common Assumptions in the Unsupervised Learning of Disen- tangled Representations. In Kamalika Chaudhuri and Ruslan Salakhutdinov (eds.), Proceedings of the 36th International Conference on Machine Learning. volume 97 of Proceedings of Machine Learning Research, pp. 4124 PMLR, 09-15 Jun 2019.
12

---

Published as a conference paper at ICLR 2023
Francesco Locatello, Ben Poole, Gunnar Rätsch, Bernhard Schölkopf, Olivier Bachem, and Michael Tschannen.
Disentanglement Without Compromises. In Proceedings of the 37th International Conference Machine Learning. ICML 2020, 13-18 July 2020. Virtual Event, volume 119 of Proceedings of Machine Learning Research, pp. 6348-668866 -448..39. PMLR, 2020a.
Francesco Locatello, Michael Tschannen, Stefan Bauer, Gunnar Rätsch. Bernhard Schölkopf, and Olivier Bachem. Disentangling Factors of Variations Using Few Labels, In International Confer- ence on Learning Representations, 2020b.
Francesco Locatello, Dirk Weissenborn, Thomas Unterthiner, Aravindh Mahendran, Georg Heigold, Jakob Uszkoreit, Alexey Dosovitskiy, and Thomas Kipf. Object-Centric Learning with Slot Attention. In H. Larochelle, M. Ranzato, R. Hadsell, M.F. Balcan, and H. Lin (eds.), Advances in Neural Information Processing Systems, volume 33, pp. 11525-15 1322511555 11538. 2020c.
Curran Associates, Inc.,
Alessio Moneta and Peter Spirtes. Graphical models for the identification of causal structures in multivariate time series models. In 9th Joint International Conference on Information Sciences (JCIS-06), pp. 613-616. Atlantis press, 2006.
Ricardo Pio Monti, Kun Zhang, and Aapo Hyvärinen. Causal Discovery with General Non-Linear Relationships using Non-Linear ICA. In Amir Globerson and Ricardo Silva (eds.). Proceedings of the Thirty-Fifth Conference on Uncertainty Artificial Intelligence, UAI 2019, Tel Aviv, Israel, July 22-25, 2019, volume 115 of Proceedings of Machine Learning Research, pp. 186-195, AUAI Press, 2019.
Kevin Murphy. Dynamic Bayesian Networks: Representation, Inference and Learning. UC Berkeley, Computer Science Division, 2002.
Martin Edward Newell. The Utilization of Procedure Models in Digital Image Synthesis. PhD thesis, The University of Utah, 1975. AA17529894.
Ignavier Ng. Shengyu Zhu, Zhuangyan Fang, Haoyang Li, Zhitang Chen, and Jun Wang. Masked Gradient-Based Causal Structure Learning. In Proceedings of the 2022 SIAM International Conference on Data Mining (SDM), pp. 424-432, 2022.
Didrik Nielsen, Priyank Jaini, Emiel Hoogeboom, Ole Winther, and Max Welling. Survae flows: Surjections to bridge the gap between vaes and flows. Advances in Neural Information Processing Systems, 13:12685-12666 2020.
D Nuzzi, Stramaglia, M Javorka, Daniele Marinazzo, A Porta, and Luca Faes. Extending the spectral decomposition of granger causality to include instantaneous influences: application the control mechanisms of heart rate variability. Philosophical Transactions of the Royal Society A, 379(2212):2020023, 2021.
Adam Paszke, Sam Gross, Francisco Massa, Adam Lerer, James Bradbury, Gregory Chanan, Trevor Killeen, Zeming Lin, Natalia Gimelshein, Luca Antiga, Alban Desmaison, Andreas Köpf, Edward Yang. Zachary Vetito, Martin Raison, Alykhan Tejani, Sasank Chilamkurthy, Benoit Steiner, Lu Fang, Junjie Bai, and Soumith Chintala. PyTrcch: An Imperative Style, Righ-Performance Deep Learning Library. In Hanna M. Wallach, Hugo Larochelle, Alina Beygelzimer, Florence d' d'Alcéé-Buc, Emily B. Fox, and Roman Garnett (eds.), Advances in Neural Information Processing Systems 32: Annual Conference on Neural Information Processing Systems 2019, NeurIPS 2019, December 8-14, 2019, Vancouver, BC, Canada, pp. 8024-8035, 2019.
Judea Pearl. Causality: Models, Reasoning and Inference. Cambridge University Press, USA, 2nd edition, 2009. ISBN 052189560X
Emil Praun, Adam Finkelstein, and Hugues Hoppe. Lapped Textures. In Proceedings of ACM SIGGRAPH 2000, pp. 465-470, July 2000.
Prajit Ramachandran, Barret Zoph, and Quoe V Le. Searching for activation functions. arXiv preprint arXiv: 2017.
13

---

Published as a conference paper at ICLR 2023
Patrik Reizinger, Luigi Gresele, Jack Brady, Julius von Kügelgen, Dominik Zietlow, Bernhard Schölkopf, Georg Martius, Wieland Brendel, and Michel Besserve. Embrace the Gap: VAEs Perform Independent Mechanism Analysis. In Advances in Neural Information Processing Systems 35: Annual Conference on Neural Information Processing Systems 2022, NeurlPS 2022 (to appear), 2022.
Danilo Jimenez Rezende and Shakir Mohamed. Variational Inference with Normalizing Flows. In Francis R. Bach and David M. Blei (eds.), Proceedings of the 32nd International Conference on Machine Learning, ICMI. 2015, Lille, France, 6-11 July 2015, volume 37 of JMLR Workshop and Conference Proceedings. pp. 1530-1538. JMLR.org, 2015.
Szymon Rusinkiewicz, Doug DeCarlo, Adam Finkelstein, and Anothony Santella. Suggestive cs princeton. edu/proo/ http://ooo/suooono////om cemooonmmoo /m////mooelll/eemmem 2021.
Contour Gallery. https: /gfx. /ggpx
Bernhard Schölkopf, Francesco Locatello, Stefan Bauer, Nan Rosemary Ke, Nal Kalchbrenner, Anirudh Goyal, and Yoshua Bengio. Toward causal representation learning. Proceedings of the IEEE, 109(5):612-66) 634,,22666662266 2021.
Maximilian Seitzer, Bernhard Schölkopf, and Georg Martius. Causal influence detection for improv- ing efficiency in reinforcement learning. Advances in Neural Information Processing Systems, 34, 2021.
Shohei Shimizu, Patrik O. Hoyer, Aapo Hyvärinen, and Antti Kerminen. linear non- non-gaussian acyclic model for causal discovery. J. Mach. Learn. Res., 7:2003-2030, dec 2006. ISSN 1532-4435.
Peter Sorrenson, Carsten Rother, and Ullrich Köthe. Disentanglement by Nonlinear ICA with General Incompressible-fov Networks (GIN). In International Conference on Learning Representations, 2020.
Frederik Träuble, Elliot Creager, Niki Kilbertus, Francesco Locatello, Andrea Dittadi, Anirudh Goyal, Bernhard Schölkopf, and Stefan Bauer. On Disentangled Representations Learned from Correlated Data. In Marina Meila and Tong Zhang (eds.), Proceedings of the 38th International Conference on Machine Learning, volume 139 of Proceedings of Machine Learning Research, pp. 10401- 10412. PMLR, 18-24 Jul 2021.
Greg Turk and Mare Levoy. Zippered Polygon Meshes from Range Images. In Proceedings of the 21 Annual Conference on Computer Graphics and Interactive Techniques, SIGGRAPH '94, pp. 311-318, New York, NY, USA, 1994. Association for Computing Machinery. ISBN 0897916677.. doi: 10.1145/12211.119224..
Aäron van den Oord, Yazhe Li, and Oriol Vinyals. Representation Learning with Contrastive Predictive Coding. CoRR, abs/1807 00744,, 2018.
Julius Kügelgen, Yash Sharma, Luigi Gresele, Wieland Brendel, Bernhard Schölkopf. Michel Besserve, and Francesco Locatello. Self-Supervised Learning with Data Augmentations Provably Isolates Content from Style. In Thirty-Fifth Conference on Neural Information Processing Systems, 2021.
Ronald J. Williams. Simple statistical gralicnt-following algorithms for connectionist reinforcement learning. In Machine Learning, pp. 229-2566 1992.
Mengyue Yang, Furui Liu, Zhitang Chen, Xinwei Shen, Jianye Hao, and Jun Wang. Causal disentangled representation learning via neural structural causal models. In Proceedings of the IEEF/CVF Conference on Computer Vision and Pattern Recognition, pp. 9593-9602, 2021.
Weiran Yao, Guangyi Chen, and Kun Zhang. Learning Latent Causal Dynamics. arXiv preprint arXiv.2202.04822, 2022a.
Weiran Yao, Yuewen Sun, Alex Ho, Changyin Sun, and Kun Zhang. Learning Temporally Causal Latent Processes from General Temporal Data. In International Conference Learning Represen- tations, 2022b.
Junzhe Zhang, Daniel Kumor, and Elias Bareinboim. Causal imitation learning with unobserved confounders. Advances in neural information processing systems, 33:12263-12274 2020.
14

---

Published as a conference paper at ICLR 2023
Xun Zheng, Bryon Aragam, Pradeep Ravikumar, and Eric P. Xing. DAGs with NO TEARS: Continuous Optimization for Structure Learning. In Samy Bengio, Hanna M. Wallach, Hugo Larochelle, Kristen Grauman, Nicolò Cesa-Bianchi, and Roman Gamett (eds.), Advances in Neural Information Processing Systems 31: Annual Conference on Neural Information Processing Systems 2018, NeurlPS 2018, December 3-8, 2018, Montréal, Canada, pp. 9492-9503, 2018.
Roland S. Zimmermann, Yash Sharma, Steffen Schneider, Matthias Bethge, and Wieland Brendel. Contrastive Learning Inverts the Data Generating Process. In Proceedings of the 38th International Conference on Machine Learning, ICMI. 2021, 18-24 July 2021, Virtual Event, volume 139 of Proceedings of Machine Learning Research, pp. 12979-129999 PMLR, 2021.
15

---

Published as a conference paper at ICLR 2023
SUPPLEMENTARY MATERIAL CAUSAL REPRESENTATION LEARNING FOR INSTANTANEOUS AND TEMPORAL EFFECTS
TABLE OF CONTENTS
Broader Impact
17
Reproducibility statement
17
C Expanded Related Work
17
D Proofs
19
D.I Preliminaries
19
D.2 Assumptions for Identifiability
20
D.3 Theorem 3.4 Proof outline
25
D.4 Theorem 3.4 Proof Step 1: The true model a global optimum of the likelihood objective
25
D.5 Theorem 3.4 Proof Step 2: Characterizing the disentanglement class
28
D.6 Theorem 3.4 Proof Step 3: Identifiability of the causal graph
31
D.7
Theorem 3.4 Proof Step Final identifiability result
34
Datasets
35
E.1 Voronoi benchmark
35
E.2
Instantaneous Temporal Causal3DIdent
37
E.3
Causal Pinball
38
Experimental details
40
F.1
iCITRIS Model details
40
F.2
Hyperparameters
42
F3
Evaluation metrics
43
Additional Experimental Results and Ablation Studies
46
G.1
Voronoi Benchmark
46
G.2
Instantaneous Temporal Causal3DIdem
49
G.3
Causal Pinball
50
16

---

Published as a conference paper at ICLR 2023
A BROADER IMPACT
The importance of causal reasoning for machine learning applications, especially reinforcement learning and latent dynamics understanding, has been emphasized by several previous works (De Haan et al., 2019; Lachapelle et al., 2022b; Pearl, 2009; Schölkopf et al., 2021; Seitzer et al., 2021; Zhang al., 2020). Thereby, starting from low-level information like pixels constitutes a considerable challenge, since we aim at reasoning about objects and abstract concepts instead of low-level pixels. We believe that this work contributes an important step towards tackling this challenge since goes beyond previous work by considering instantaneous effects, a common property in real- systems (Faes al., 2010; Hyvärinen et al., 2008; Moneta al., 2006; Nuzzi al., 2021). Besides providing theoretical identifiability results, we also propose practical algorithm with which one can learn the causal variables and their graph from high-level observations. Furthermore, we envision a reinforcement learning setting as future application, where robotic system may be able to interact with an environment. However, the main assumption that prevent us from doing this so far, is the availability of interventions with known targets. In many systems, one might not be able directly perform such interventions, but rather require several steps of low-level actions. For instance, instead of being provided the intervention targets, future work could consider a robotic setup where one can control a robot arm which can perform several interactions (e.g.. flipping switch), and we believe in the
that our work can constitute the starting point for such extension. Moreover, as we have experiments on the Causal3DDdent dataset and the Causal Pinball environment, not all assumptions must be strictly fulfilled to identify the variables empirically. Moving towards this empirical goal, recent advances in unsupervised object-centric learning (Engelcke et al., 2020; Kipf et al., 2022; Locatello et al., 2020c) have shown that objects, which can often be considered as groups of causal variables like position and velocity, can be identified from high-dimensional data without labels. possible combination of such object-centric approaches with our causal representation learning method can relax further assumptions by using the objects prior disentanglement of information, opening up further possible applications of iCITRIS. Thus, we believe that this work can form the basis of several future works in this direction.
Since the possible applications of causal representation learning and specifically iCITRIS are fairly wide-ranging, there might be potential impacts we cannot forecast at the current time. This includes misuses of the method for unethical purposes. For instance, an incorrect application of the method can be used justify false causal relations, such referencing gender and race as causes for other characteristics of a person. Hence, the obligation to use this method in a correct way within ethical boundaries lies on the user, and the outputs of the method should always be critically evaluated. We will emphasize this responsibility of the user in the public license of our code.
B REPRODUCIBILITY STATEMENT
For reproducibility. the code for all models used in this paper is publicly available at https: / //gith (ggtthu... com/ httppphhppppccccc CITRIS. Further, we provide the code for generating the Voronoi benchmark, the Instantaneous Temporal Causal3DIdent dataset, and the Causal Pinball environment. More details on the datasets and visualizations are outlined in Appendix E.
Moreover, for all experiments of Section 5, we have included a detailed overview of the hyperparam- eters in F.2 and additional implementation details of the evaluation metries and model architecture components in Appendix F.I. All experiments have heen repeated for at least 3 seeds (5 seeds for the Voronoi benchmark) to obtain stable, reproducible results. We provide an overview of the standard deviations, as well as additional results and ablation studies in Appendix G.
Finally, all experiments this paper were performed on a single NVIDIA TitanRTX GPU with 6-core CPU. The overall computation time of all experiments together in this paper correspond to approximately 80 GPU days (excluding hyperparameter search and trials during the research).
C EXPANDED RELATED WORK
Early works on causal representation learning focused identifying independent factors of variations (Klindt et al., 2021; Kumar et al., 2018; Locatello et al., 2019; 2020b: Träuble et al., 2021). A related line of work, Independent Component Analysis (ICA) (Comon, 1994; Hyvärinen et al., 2001), tries
17

---

Published as a conference paper at ICLR 2023
to recover independent latent variables that were transformed by some invertible transformation. ICA was extended to non-linear transformations by exploiting auxiliary variables that make latents mutually conditionally independent (Hyvärinen et al., 2016; Hyvärinen al., 2019), combined with deep learning methods like VAEs (Khemakhem et al., 2020a;b; Reizinger et al., 2022; Sorrenson al., 2020; Zimmermann et al., 2021) and applied causality (Gresele al., 2021; Monti et al., 2019; Shimizu et al., 2006). In particular, Lachapelle et al. (2022a:b); Yao et al. (2022a:b) discuss the identifiability of causal variables from temporal sequences. As forms of interventions, Lachapelle et al. (2022a;b) consider external actions, while Yao et al. (2022a;b) use non-stationary noise. Yet, in all of these ICA-based setups, causal variables are required to be conditionally independent. Alternatvely, Yang et al. (2021) learn causal variables from labeled images in a supervised manner.
Given a known causal structure, von Kügelgen al. (2021) demonstrate that common contrastive learning methods can block-identify causal variables that remain unchanged under augmentations. Locatello et (2020a) identify independent latent causal factors from pairs of observations that only differ in subset of causal factors. Brehmer et al. (2022) extend this setup variables that are causally related with access to single-target interventions. Similarly, Ahuja et al. (2022) extend the setup of Locatello al. (2020a) to variables with interdependencies, relying interventions that only affect their target. All these methods require pairs counterfactual observations, where only subset of variables is changed by the intervention, while the rest are frozen, i.e., they keep the same values before and after an intervention. As discussed by Pearl (2009), however, knowing counterfactuals is not realistic most scenarios. Instead, CITRIS (Lippe al., 2022b) focuses on temporal sequences, in which also the variables that are not intervened upon at given time step can still continue evolving over time. On the other hand, in this setting the intervention targets need be known. Moreover, within a time step, the causal variables are assumed to be independent conditioned on the variables of the previous time step, hence not allowing for instantaneous effects. To the best of our knowledge, iCITRIS is the first method to identify causal variables and their causal graph from temporal, intervened sequences even for potentially instantaneous causal effects, without requiring counterfactuals or data labeled with the true causal variables.
18

---

Published as a conference paper at ICLR 2023
D PROOFS
In this section, we provide the proof for the identifiability theorem 3.4 in Section 3 and Lemma 3.2. The section is structured into three main parts. First, in Appendix D.I, we give an overview of the notation and elements that are used in the proof. Next, we discuss the assumptions needed for Theorem 3.4, with focus on why they are needed and what violation of these assumptions can cause. Additionally, we provide a proof of Lemma 3.2 in this subsection. Finally, we provide the proof of Theorem 3.4, structured into multiple subsections as different main steps of the proof. detailed overview of the proof is provided in Appendix D.3.
D.1 PRELIMINARIES
Throughout the proof, we will the same notation as used in the main paper, and try to align it as much possible with Lippe et al. (2022b). As summary, we review here an adapted version of the notation and preliminaries of the proof for CITRIS:
We denote the K causal factors in the latent causal dynamical system Ci...., Cx:
The dimensions and space of causal variable is denoted C. E D. with M, 1. In the remainder of the proof, we consider D; to be IR, i.e., C, being a continuous variable;
We group all causal factors in single variable C (Ci, .C.,.... Cx) C, where C is the causal factor space C W.M: Da ... Mx:
The data we base our identifiability on is generated by latent Dynamic Bayesian network with variables (C1, C2 ..., C+ ]
We assume to know each time step the binary intervention variables 1' {0, 1) |] where 1. = 1 refers to an intervention on the causal factor C₁ As special case 1 = 0 for all f; For each causal factor C, there exists minimal causal split s (Ct), s!me (C,) such that ((a((C)) represents only the variable/manipulabll part of C, while s!"v (C²) represents the invariable part of C₁;
At each time step, we can access observations x², x E X C R.N;
There exist a bijective mapping between observations and causal/noise space, denoted by h C E + X, where E is the space of the noise variable. The bijective map implies that the observations, X. live in a peweerdimensional manifold of size dim x E) in RVV For example, in Causal Pinball, we have a limited set of images that means that there exists an inverse to the observation function, h 11 such that h(h h(h-=(()) = X for all X E X, and h (n([C; F))) [C; F?] for all C E C, E
occur. Formally, this
The noise Eº E & at time step f subsumes all randomness besides the causal model which influences the observations. For example, this could be brightness shifts in Causal3D, or color shifts in the Causal Pinball environment since in these setups, no causal factor is encoded in brightmess and color respectively. While this setting is quite general, we still require that the values of the causal factors must be identifiable from single observations. Hence, the joint dimensionality of the observation noise and causal model limited to the image size.
For any model learning a latent space, we denote the vector of latent variables by 2' E Z S M where Z is the latent space of dimension AM = dim(E) + dim(C). In practice we usually overestimate M, i.e., M > dim + dim(C);
In iCITRIS, we learn the inverse of the observation function as ge X - Z. If dim(X) > dim(C x E), we consider that contains an arbitrary, fixed map from X' to the lower dimensionality. This map can, in theory, be trivially found by ensuring invertibility of all X X' while minimizing the number of dimensions. An alternative interpretation is that ge is a deterministic variational autoencoder with zero reconstruction loss. In this limit, Nielsen et al. (2020) showed that the encoder-decoder function as an invertible normalizing flow, which is what we base our analysis on as well. In practice, we train deterministic autoencoder with a latent dimension greater than dim(E) + dim(C), and work on this larger dimensionality; In iCITRIS, we learn an assignment from latent dimensions to causal factors, denoted by
1b []..M] + []:
'The latent variables assigned to each causal factor C2 by &b are denoted as Zp: = {(ッíØ e [1..M]] +b))) {} (go(x*), 22')) [1..M], vk() i):
The remaining latent variables that are not assigned to any causal factor are denoted as 2
In iCITRIS, we learn directed, acyclic graph G (V, F)) where V ( E [].
19

---

Published as a conference paper at ICLR 2023
and the edges represent directed causal relations;
The graph G induces a parent structure which we denote by ":00" = {zylı E [1..M] e)) E
(ag where pag(0) = 0, i.e., the variables in Zvo having no instantaneous parents; The parents of causal variable within the same time step t are denoted by pa"t (C(+))) and the parents of the previous time step by pa' pa
As a special case, we denote the function go with the parameters that precisely model the in- verse of the true observation function, h 1- as the disentanglement function X + C & with C = D. x x D and M, being the number of latent dimensions assigned to the causal factor C₁ by vi" We denote the output of 8* for an observation as (x)
(C1. C2 ..., E). The representation of 8 learnable function is denoted by 9ø and s=: In the following proof, we will use entropy variable. be invariant to possible invertible transformations, egg., scaling by 2, we use the notion of the limiting density of discrete points (LDDP) (Jaynes, 1957; 1968). In contrast to differential entropy, LDDP introduces an invariant measure m(X), which can be seen as a reference distribution we measure the entropy of p(X) to. The entropy is thereby defined as:
measure of information content in random
H(X) = - p()) p(X) log
H(X)=
((X) m(x) dr
(S)
In the following proof, we will consider entropy measures over latent and causal variables. For the latent variables, we consider m(X) but fixed distribution X' (e.g. random Gaussian
x) to be the push-forward distribution of an arbitrary,
= RT) through gs. For the causal variables, we consider it to be the push-forward through h-1 For more details on LDDP, see Lippe et al. (2022b, Appendix A.1.2) and Jaynes (1957; 1968).
D.2 ASSUMPTIONS FOR IDENTIFABILITY
In this section, we provide a detailed discussion of the assumptions of iCITRIS to enable the identification of an underlying causal graph with instantaneous effects. We thereby focus on why these assumptions are necessary, and how a violation of those can lead to scenarios where the causal variables and graph not identifiable.
D.2.1 ASSUMPTION THE INTERVENTIONS ON THE CAUSAL VARIABLES REMOVE INSTANTANEOUS PARENTS
iCITRIS requires interventions on the causal variables that remove instantaneous parents, in order to separate the variables in latent space, as stated in Lemma 3,2 and copied here for completeness:
Lemma D.1. In iTRIS, a causal variable C₁ cannot be identified up to an invertible transformation T. that independent of all other causal variables, if C2 can have instantaneous parents and no partially-perfect interventions on C; are provided.
Proof. To prove this Lemma, consider a causal variable C; that has C, as an instantaneous parent. The conditional distribution of C₁ as defined in iTRIS, can be written as (CC+ S, 1
where S C CC++ (C(+1 C J.te,, i.e., any additional parent without introducing cycles. We do not put any constraints on the distribution and also on the provided interventions, except that we do not have the knowledge whether under / 1, Ch becomes independent of Ch+1 or not. This implies that one must consider the most general form of interventions for C, i.e., modeling the distribution p. (C |Ct+ S, I 1) under interventions with possible unknown independences. To keep this result general, we consider an arbitrary observation function h(C E²) = X
Under this setting, it sufficient to show that there exist another representation Ó that cannot be distinguished from C' solely based observation triples (X', X*+1 /++1), and that there exist invertible function f such that f f(Ć;) = C² for any t. Note that we exclude a permutation of variables, since the intervention targets 1++ align the two representations.
As an alternative representation, consider Ć = (CI,..., C;-1, Ci Cj, Cit1. the number of causal variables. Then, the distribution of p(C) only differs in the conditional of p, as
..., CR) with K being
20

---

Published as a conference paper at ICLR 2023
follows:
Di - ,5, =i(C + 1f
(6)
Because D; is conditioned on 1 there exist invertible, pllume-preserving transformation from C + Cft1 to C++1, Le., w(dC)+)) = c c c Cj+1. Hence, follows that:
p. + cft ,,,1T1) = p.
(8)
and overall that p(O) = (C)) Furthermore, there exist a function å that maps Ć to the same observations as h does for C:
h(C* E-) h(C C. Ci.,...,,,,,,E =(C",,
(9)
Therefore, both representations, C and C, can model the same data generation process for {X', x , 1]+ and are indistinguishable from these observations alone. Finally, it is apparent that there exist no invertible transformation from C to C that is independent of Cy. Thus, the causal variable C₁ is not identifiable up to invertible, componentwise transformations.
As an example of how this effects standard identification problem, consider two random, causal variables C₁, O2 with the causal graph Cf - C++1 C2 - C+++1 The two causal variables C₁ C2 have therefore no instantaneous relations. Further, consider the (soft-interventionaa)) distributions Pi (C IC, I++1 and p2 (2 [C2, 1 whose form can be arbitrary, but for this example, choose them to be Gaussian with constant variance:
[M(C (1)
if
(10)
(W (C),
if =
if
p2(Ct+1|C 1241)
(11)
if 1++1=
where P1, (i,. M2, f2, 01. 01 02 Ø2 are arbitrary, potentially non-linear functions of C₁ and C2 respectively. Further, to consider the simplest case, suppose that the observation X a time step are the causal variables themselves, X === [C], C%), and we observe data points of all intervention settings, i.e., I!+1 ~ Bernoull(gg) with 0 < <.11.
Under this setup, the true generative model follows the distribution:
p(X+11 X 1+ p(C Ct C2 (1t1)) p(C C2 111, 1142) MCC 14,, 1241) = pI(CttIIC, (())
(12)
(13)
1C +
(14)
where C++1 need at least one other representation which can achieve the same likelihood as the true generative model under all intervention settings /++1, For this, consider the following distribution:
CX
J++ 1 To show that the causal variables not uniquely identifiable,
Ix',
p(C cCt I ++1 11+1)
(15)
IC I+1) p( I0 C. Ctt , ,1t1)
(16)
= =P(CCtyCH ptt) (2 + Ct Gt+1, 1111)
(17)
-pi(Ctticc p(1) in ( Ca Ott1 1111
(18)
with Gt+1 = C = C1+1 + C3+l, Note the additional dependency of C on Ö++1, which is possible in the space of possible causal models with additional instantaneous causal edge
21

---

Published as a conference paper at ICLR 2023
2
2
-2
(a) Observational distribution p(X'|x"-11 C,I 0, 0)
(b) Interventional distribution
(c) Interventional distribution
p(X'|x-= C, 1,
C, = = 1)
Figure 4: Example distribution for showcasing the necessity of partially-perfect interventions for disentangling causal variables with instantaneous effects. Suppose we are given two-dimensional observations X", for which the observational and interventional distributions are plotted in (a)-(c). The central plot of each subfigure shows 2D histogram, and the subplots above and on the right show the 1D marginal histograms. For simplicity, we keep the previous time step, Xt-1 constant here. From the interventional distribution, one might suggest that we have the latent causal graph C + C2 since under _ 1, the distribution of both observational distributions change, while _ =1 keeps X2 unchanged. However, the data has been actually generated from two independent causal variables, which have been entangled by having X [C], Ci C%. We cannot distinguish between these two latent models from interventions that do not reliably break instantaneous causal effects, showing the need for partially-penn -prffect interventions.
Ctt1 + O221. The new distribution ba is identical to the true distribution, since:
D2(CT-1 c Gt,, ,1,,, = 0) = N( + C + o2(C))
(19)
1 (Cf + +O2+1 (ct+1 + p2(Cg))))
cxp
V2 (C²)
2
o2()2)²
(00)
1 (Cht1 (2()))
cxp 2
(21)
2
0²(2
= N(( (2((C) 02 (C()))
(22)
P2(Ct+1 C2 [+] 0)
(23)
Similarly, one can show that (2 + C3+ C7+1 1 = 1) p 15 = 1).
Hence, the alternative representation Gi+1, C+++ can model the distribution 1 as well as the true causal model. In conclusion, from the samples alone, we cannot distinguish between the two representation C1 C2 and Ć1, C2. and the model is therefore not identifiable up to invertible transformations.
An alternative example with a non-trivial observation function visualized in Figure 4, which further underlines the problem.
This shows that with soft interventions, one cannot distinguish between causal relations introduced by the observation function and those that are in the true causal model. Prrtiallly)Perfed interventions, however, provide an opportunity to do so since if we had known that the intervention on C.2 renders independent of C1, the second causal model could not have modeled the correct distribution under In 1. Thus, we can distinguish between the two, allowing us to identify the correct causal model.
Note that under partially-perfi intervention, the intervention -nnepeddnnnndeeeedeett part of causal variable, givv (C.). automatically cannot have any instantancous parents, since otherwise, the intervention does not remove all instantaneous parents and hence is actually not partially-perfes
22

---

Published as a conference paper at ICLR 2023
D.2.2 ASSUMPTION 2: THE INTERVENTION VARIABLES ARE NOT DETERMINISTIC FUNCTION OF EACH OTHER
iCITRIS builds upon interventions to identify the causal variables. The intervention targets are not necessarily independent of each other, but can be confounded. For instance, we could have a setting where we only obtain single-targe interventions, or a certain variable C₁ can only be jointly intervened upon with another variable Cj. this large space of possible experimental settings, we naturally cannot guarantee identifiability all the time. In particular, we require that intervention targets for the different causal variables are unique:
Lemma D.2. All information that is strictly dependent the intervention target 12 i.e. syar the minimal causal variable of Cs cannot be disentangled from another causal variable, Cg with j * i, if their intervention targets are identical: Vt, 1
Proof. Lippe et al. (2022b) have shown that two causal variables C, Cj cannot be disentangled from observational data alone if they follow a Gaussian distribution with equal variance over time. Taking this setup, consider that additionally to observational data, observe samples where both variables have been intervened upon, 1 = 1 1. If the interventional distribution of C2 and C3 are both Gaussian with the same variance, we have the same non-identifiability as in the observational case. Since the entanglement axes can transfer between the two setups, C₁ and Cj cannot be disentangled, and therefore their minimal causal variables.
In other words, if two variables are always jointly intervened or passively observed, we cannot distinguish whether information belongs to causal variable C, or Cj. Since the causal system is stationary, having one time step & for which 1 * I implies that in the sample limit, we will observe samples with I * I in the limit well. Further, when we only observe joint interventions two variables, C,,, Cj. the causal graph among the two variable cannot be identified for arbitrary distributions (Eberhardt, 2007), making the identifiability of the graph and variables impossible.
Following Lippe et al. (2022b), we require that the following independence holds for every causal variable C2 with observed interventions:
C.+1 X C (c(+)),), for any i +
(24)
This also implies that there does not exist variable Cj for which Vt, 7ī = 1 - IJ. As mentioned before, under additional assumptions such that every causal variable has at least one parents, can be relaxed to unique interventions.
D.2.3 ASSUMPTION 3: DISTRIBUTIONS HAVE FULL SUPPORT
Following several previous works (Brehmer et al., 2022; von Kügelgen et al., 2021), we consider for the theoretical results that all distributions have full support. If the observational and interventional distribution do not share the same support, there exist data points for which the intervention targets can be determined from the observation X alone. In such situation, the encoder can change encoding depending on the intervention target, as long as the decoder can yet recover the full observation. This can potentially create representation models that ignore the latent structure, since the intervention targets are already known. Furthermore, when intervention targets are known from seeing causal variables, we potentially introduce new independencies from intervention targets. For instance, if we have the graph C1, C2 - C3 where I3 = 1 only if 1 1, 12 0, we can induce the intervention targets from other causal factors, making C3 essential independent of Iz. To prevent such degenerate solutions, take the assumption that the observational and intervention distributions share the same support. This assumption implies that any data point could come from either the interventional or observational regime, ensuring that the intervention target cannot deterministically be found from an observation X
D.2.4 ASSUMPTION 4: TEMPORAL CONNECTIONS AND INTERVENTIONS BREAK ALL SYMMETRIES IN THE DISTRIBUTIONS
The temporal and interventional dependencies are essential part in iCITRIS to guarantee identifia- bility and disentanglement of the causal variables. Without any of these dependencies, there may exist multiple representations that model the same distribution p( X*) x 7*), while following the
23

---

Published as a conference paper at ICLR 2023
C1
C2
C3
Figure 5: Example instantaneous causal graph between 3 causal variables C1 C2 Cz. Without temporal dependencies, we could encode information of C₁ dependent on C3 without needing an edge in the distribution.
enforced latent structure by iCITRIS. The problem that variables can functional dependent on each other, where these dependencies exploit symmetries, leaving the distribution unchanged.
For instance, consider the instantaneous causal graph of three variables C2 C2, C3 with C1, - C2. as depicted in Figure 5. Suppose that C₁ does not have any temporal parents, and the observational
distribution of it follows Gaussian: p(Ci [ 0) N [ug of) with Pl. of being constants.
Further, suppose that under interventions, only the standard deviation changes, i.e. [ = 1)
NCC (4,, of) with of & of. Then, for any point Cj = C1, there exists a second point, 6j 2/11 C1. which has the same probability for any value of 1 This is because both distributions, p(C) [ 0)
and p( = 1), share a symmetry around the mean f(1-
Now, suppose we have the optimal encoder which maps an observation X of this system to the three causal variables with their ground truth values. Then, there exist an alternative encoder, which flips the observed value of Cj around the mean μl, deterministically conditioned on the remaining variables Cj and Cg. For instance, we could have the following representation
Č₁ C2 Oj for the causal variables:
Cf 12p1 Cj otherwise
if O > 0
C =
(25)
This alternative representation model shares the same likelihood as the optimal encoder in terms of p( (X 7'), since flipping the value of Cf around the mean does not change its probability. Further, despite the flipping, the original observation X can be recovered from this alternative representation (H by the decoder, because the possible conditioning factors, i.e. Cÿ in this case, observable to the decoder. Hence, both representations are equally valid for the causal models. Yet, one cannot recover the value of the true causal variable, C1, from its alternative representation Ci alone, since Ć₁ needs to be known to invert the example condition. This shows that we can have functional dependencies between representations of causal variables while their distributions remain independent. Thus, there exist more than one representation that cannot be distinguished between from having samples of p(X X 7²) alone.
More generally speaking, functional dependencies between variables can be introduced if there exists a transformation that leaves the probability of variable C, unchanged for any possible value of its parents unseen in X i.. its intervention target 1ţ and temporal parents Cr--1 Whether this transformation performed or not can now be conditioned on other variables at time step t. Meanwhile, this transformation does not introduce additional dependencies in the causal graph, since the distribution does not change.
To prevent such transformations from being possible, the temporal parents and intervention targets need to break all symmetries in the distributions. We can specify in the following assumption:
Assumption 4:
For causal variable C: and its causal mechanism p(C++ Ipal+1 (C, ).pp' (c I!+1), there exist no invertible, smooth transformation with T(C +1 ((tt)) = Cftt besides the identity, for which the following holds:
YCt 1041,1t+
(pa"tt(C
(C(++),
aT(C++1C441 act+1
(26)
pa (C,t()))) I1+1)
Intuitively, this means that there does not exist any symmetry that is shared across all possible values of the parents (temporal and interventions) of a causal variable. While this might first sound restricting, this assumption will likely hold in most practical scenarios. For instance, if the distribution a Gaussian, then the assumption holds as long as the mean is not constant since the intervention breaks any parent dependencies are broken by the perfect interventions. The same holds in higher
24

---

Published as a conference paper at ICLR 2023
dimensions, the new symmetries, i.e. rotations, are yet broken if the center point is not constant. Note that these symmetries can be smooth transformations, in contrast to the discontinuous flipping operation on the Gaussian (i.e., either we flip the distribution not, but there is no step in between).
D.2.5 ASSUMPTION 5: CAUSAL GRAPH STRUCTURE REQUIREMENTS
Besides disentangling and identifying the true causal variables, we are also interested in finding the instantaneous causal graph. This requires us perform causal discovery, for which need to take additional assumptions. First, we assume that the causal graph is acyclic, i.., for any causal variable C₁ there does not exist path through the directed causal graph that loops back to it. Note that this excludes different instances over time, meaning that path from C to C is not considered a loop. In real-world setups, there potentially exist instantaneous graphs which are not acyclic, which essentially model feedback loop over multiple variables. However, rely on the graph as a distribution factorization, we assume it to be acyclic, and leave extension to cyclic causal graphs for future work. As the second causal graph assumption, we require that the causal graph is faithful, which means that all independences between causal variables are implications of the graph structure, not the specific parameterization of the distributions (Hyttinen al., 2013; Pearl, 2009). Without faithfulness, the graph might not be fully recoverable. Finally, we assume causal sufficiency, iee. there do not exist any additional latent confounders that introduce dependencies between variables beyond the ones we model. Note that this excludes the potential latent confounder between the intervention targets, and we rather focus on confounders on the causal variables C1...., Cx besides their intervention targets, the previous time step C*, and instantaneous parents C++11
D.3 THEOREM 3.4 PROOF OUTLINE
The goal of this section is to proof Theorem 3.4: the global optimum of iCITRIS will identify the minimal causal variables and their instantancous causal graph. The proof follows a similar structure as Lippe et al. (2022b) used for proofing the identifiability in CITRIS, but requires additional steps to integrate the possible instantancous relations. In summary, will take the following steps:
1. (Appendix D.4) Firstly, we show that the function 8* that finds the true latent variables Ci,... CK and assigns them to the corresponding sets ".,...., 20bxe constitutes a global, but not necessarily unique, optimum for maximizing the likelihood p(X(+(1))))) JL+2).
2. (Appendix D.5) Next, we characterize the class of disentanglement functions A which all represent a global maximum of the likelihood, i.e., get the same score the true function 8". We do this by proving that all functions in A must identify the minimal causal variables.
3. (Appendix D.6) In third step, we show that based on the identification of the minimal causal variables, the causal graph on these learned representations must contain at least the same edges as in the ground truth graph.
4. (Appendix D.7) Finally, we put all parts together and derive Theorem 3.4.
We will make use of Figure 6 summarizing the temporal causal graph, and the notation introduced in Appendix D.1. For the remainder of the proof, we assume for simplicity of exposition that:
The invertible map and the prior pa | I++) are sufficiently complex to approxi- mate any possible function and distribution one might consider in iTRIS. In practice, over- parameterized neural networks can approximate most functions with sufficient accuracy.
The sample size for the provided experimental settings is unlimited. This ensures that dependencies and conditional independencies in the causal graph of Figure 6 transfer to the observed dataset, and no additional relations introduced by sample biases. In practice, large sample size is likely to give an accurate enough description of the true distributions.
D.4 THEOREM 3.4 PROOF STEP 1: THE TRUE MODEL IS A GLOBAL OPTIMUM OF THE LIKELIHOOD OBJECTIVE
We start the identifiability discussion by proving the following Lemma:
25

---

Published as a conference paper at ICLR 2023
Causal variables
C
X
h
Temporal causal relations
E
Instantancous causal relations
Observations
C
C
O
C
X
Intervention variables
RTH
1-11
[
Latent confounding
R++1
Figure 6: An example causal graph in iTRIS. A latent causal factor C++1 can have potential parents the causal factors at the previous time step C = (C. C instantaneous parents Cgt i + j. and its intervention variables 1++ All causal variables C++1 and the noise E++1 cause the observation X++1 R is potential latent confounder between the intervention targets.
Lemma D.3. The true identification function 8* that correctly identifies the true causal factors C++1, .... C1+1 from observations X X using the vi assignment function on the latent variables 7c+1 and the true causal graph G is one of the global maxima of the likelihood of p(X++IXX (xt))
This lemma ensures that the true model is part of the solution space of maximum likelihood objective on p(X++|XX /(+1).
Proof. In order prove this, first rewrite the objective in terms of the true causal factors. This can be done by using the causal graph in Figure 6, which represents the true generative model:
K
p(x, x Ct 00 /(11 (
| ..'..pa"" ( (C++))
(27)
p(X* (C) p(C))) p(1
The context variable R++1 is subsumed in p(I++), since it a confounder between the intervention targets and is independent of all other factors given 71+1
In order to obtain p(X'=I|x' f++1) from p(x', X++1 C. CC+1, I(+1), we need to marginalize out C and C++1, and condition the distribution X and 1++1;
K
[ ( p(x 10tt))
PO |C',, (ctt) (,44
p(x* ++ xx )
(28)
the assumptions with respect to the observation function h we have defined to be bijective, meaning that there exists inverse h 11 that can identify the causal factors C" and noise variable E from X Using the invertible map, we can write p( p(C* (X') = 8,-1(x*)= [C where à is Dirac
delta. We also remove ES from the conditioning set since it is independent of X 1+1 This leads us to:
K
1 )+ = C++1
I1
pad (C) 1.1 ), (,
(29)
26

---

Published as a conference paper at ICLR 2023
We can similar step relate X with C*+1 and E+ However, since model distribution over X we need to respect possible non- non-volume preserving transformations. Hence, we use the change of variables formula with the Jacobian Jn Oh(C*+1 g obtain:
of the observation function / to
K I pa (C+))) 15t11) p(E++1)
p p(x+1xx /+1)= = [Jal
(30)
Since Equation (30) is a derivation of the true generative model p(X", x C² gC-1, r(+)), it con- stitutes a global optimum of the maximum likelihood. Hence, one cannot achieve higher likelihoods by reparameterizing the causal factors or having a different graph, as long as the graph is directed and acyclic.
In the next step, we relate this maximum likelihood solution to iCITRIS, more specifically, the prior of iCITRIS. For this setting, the learnable, invertible map ge is identical the inverse of the observation function, h In terms of the latent variable prior, we have defined our objective of iCITRIS as:
K
pa (=* 1 10,1 = III Ps (=
.+1 1 =
(31)
t=0
Since we know that ge is an invertible function between X' and Z, we know that z' must include all information of X Thus, we can also replace it with 2' = [C E.), giving us:
K
Po (2 1 |C E", E,, 1+11 - II Pa (##"IC* E., E², 2 4411
(32)
i=
Next, we consider the assignment function th". The optimal assignment function v/* assigns sufficient dimensions to each causal factor C1.... Ck. such that we can consider 2+1 Further, the same graph G is used in the latent space as in the ground truth, except that we additionally condition : i 1,... K
z C for = L,.,,... K.
With that, Equation (32) becomes:
K
Po [C E., , 142)
_411 1+1 II Pe -
11 " 10
p( p(= C',, F²)
(33)
where we remove E from the conditioning set for the causal factors, since know that C++1 and E++1 independent of E Now, Zgg must summarize all information of =4+1 which is not modeled in the causal graph. Thus, 20g represents the noise variables:
241 E++1
2
K
1441)
Po (3* C", E", 1', = II
c
IC"
(+1 241 A48
p( 11 E
EE)
Po
(34)
Finally, by using g8. we can replace the distribution on +11 by a distribution on X ++ by the change of variables formula:
Ogg 0g (2
K II
.+1 We can simplify this distribution by using the independencies of the noise term E++1 in the causal graph of Figure 6:
Ps (X++lIC* F., (+))+
CC
p+
1
(35)
p(= = EU+1 IC Ft
Ps C* F., r+41= = Ögā (2 0zt+1
K
)
II Po
IC 1+1
(36)
p(2(+1 E(+1)
With this, Equation (36) represents the exact same distribution Equation (30). Therefore, we have shown that the function 8* that identifies the true latent variables C1...., Cx and assigns them to the corresponding sets E s......, ZΦх constitutes global optimum for maximizing the likelihood. However, this solution not necessarily unique, and additional optima may exist. In the next steps of the proof, will discuss the class of functions and graphs that lead to the same optimum.
n
27

---

Published as a conference paper at ICLR 2023
C
C
s inv (O++1)
pa pat (Ott1)
C++11
St
var
(C++1)
1++1
1
(a) Original causal graph of C;
(b) Minimal causal split graph of C;
Figure 7: The minimal causal variable in terms of causal graph under iTRIS. (a) the original causal graph, C++1 has as potential parents the causal variables of the previous time step C' (eventually
subset), its instantaneous parents (a;+ and the intervention target I (b) The minimal causal variable splits C++1 into an invariable part x; (c++) and variable part s (C++1). The invariable part iinv (Ct+) is independent of the instantaneous parents and the intervention target.
Further, it can be parent of syar (C++)) due to the autoregressive distribution modeling.
D.5 THEOREM 3.4 PROOF STEP 2: CHARACTERIZING THE DISENTANGLEMENT CLASS
In this section, we discuss the identifiability results of the causal variables in iCITRIS. We first describe the minimal causal variables in iTRIS, and how they differ to TRIS in CITRIS (Lippe et al., 2022b). Next, we identify the information that must assigned individual parts of the latent representation. Finally, we discuss the final setup to ensure identification of the variables according Definition 3.3, including the additional variables in 2
D.5.1 MINIMAL CAUSAL VARIABLES
Lippe al. (2022b) introduced the concept of minimal causal variable as an invertible split of a causal variable (c,) (s*)e(;;) s intervention, s var (C;), and part that is independent of it, (',((C,) (see Definiton 2.1). ). In other words, the minimal causal variable is the smallest part of causal variable that strictly depends on the provided intervention.
into one part that is strictly dependent on the
For iCITRIS, we consider the same concept, but adapt it to the setup of iTRIS. First, iTRIS assumes the presence of interventions that render a variable independent of its instantaneous parents. Hence, when given these interventions, we can ensure that givv (C.) does not have any instantaneous parents. Second, the presence of a causal graph in iCITRIS allows dependencies between different parts the latent space. Further, 2 can be the parent of any other set of variables, thus allowing for potential dependencies between slnv (C;) and & gar (C:). Note that those for the same time step, however, must also be cut off by the intervention. Hence, the split s; s;(C) = (s)"r(C3) (Ct), s! (Cf)) must have the following distribution structure:
p (s,(C) ICC pa paatt(aa (Ct*), 1,1+1) (Ct+) (C (C (C(+))) 1 p (s! (C(+))(C4)
(37)
where
p (s) (C+ pa "l +++))
i (x) (C++))))))))))
if 7f =1
(38)
(p (s (C pa + (C+ I
(C(+))
otherwise
Thereby, the minimal causal variable with respect to its intervention variable +1+1 is the split s, which maximizes the information content H (s (C*). These relations are visualized in Figure 7.
1
Causal variables for which the intervention target is constant, i.e., interventions have been observed,
28

---

Published as a conference paper at ICLR 2023
were modeled by s (C,) s (C,) naturally hold anymore since sinv (C,) is restricted to not having any instantaneous parents. However, stated in assumption 1, variable without interventions cannot be an instantancous child of any variable. Hence, for a causal variable C. if 7 = 0 for all f. its minimal causal split is defined as sta glnv (Ci) 8 (C,) = 0, in CITRIS (Lippe al., 2022b).
0 in CITRIS (Lippe al., 2022b). Here, this does not
D.5.2 IDENTIFYING THE MINIMAL CAUSAL VARIABLES
As first step, postulate the following lemma:
Lemma D.4. For all representation functions in the class A", there exist deterministic map from the latent representation Zy, to the minimal causal variable 8 s (C:) for all causal variables C,, I...,.. K.
This lemma intuitively states that the minimal causal variable s (C;) is modeled in the latent representation Zé. for any representation that maximizes the likelihood objective. Note that this does not imply exclusive modeling yet, meaning that 2,h, can contain more information than just s v (C;). We will discuss this aspect Appendix D.5.3.
Proof. In order to prove this lemma, we first review some relations between the conditional and joint entropy. Consider two random variables A, B of arbitrary space and dimension. The conditional entropy between these two random variables is defined as (A) = HA, B) H HB) (Cover et al., 2005). Further, the maximum of the joint entropy is the sum of the individual entropy terms, H(A, B) (A) + BB) (Cover et al., 2005). Hence, we get that (AAB)) = (A,, B) - (B) s ((A) + H(B) (B) (A). In other words, the entropy of random variable A can only become lower when conditioned on any other random variable B.
Using this relation, we move now to identifying the minimal causal variables. If a minimal causal variable the empty set, i.e,, 8 (C,) = 0, for instance due to not having observed interventions on C₁ the lemma is already true by construction since no information must be modeled in 2 Thus, focus on cases where 8 svar (C+) + 0, which implies that C++11
Ie+1. Therefore, the following
inequality must strictly hold:
H(CH+1C (Ct+) HCC++CC* C411 11t1
for all = I,... K. Additionally, based on the assumption that the observational and interventional distributions share the same support, we know that the intervention posterior, i.e., p(I++1 p([++IXX-1) cannot be deterministic for any data point X and intervention target 1"+1, Thus, we cannot derive 1 from the observation X Thirdly, because every latent variable is only conditioned on exactly one intervention target in iCITRIS and there exist no deterministic function between any pair of intervention targets, one cannot identify 1!+1 in any latent variables except Zue.. Therefore, the only way in iCITRIS to fully exploit the information of the intervention target 1 is to model its dependent information in žy,- As this information corresponds to the minimal causal variable, sar (Ct), any representation function must model the distribution p( var (C:))...) ....) in p(2,, |, to achieve the maximum likelihood solution. This is independent of the modeled causal graph structure, meaning that if there exist representation functions with different graphs in A', then all of them must model 8 yar (C+) in Z Finally, using assumption 4 (Appendix D.2.4), we obtain that this distributional relation implies a functional independence of s q(ar(C.) in 2 to any other latent variable. Thus, there exists deterministic map from Zdh to s g (C2) in any of the maximum likelihood solutions.
D.5.3 DISENTANGLING THE MINIMAL CAUSAL VARIABLES
The previous subsection showed that 2Φ. models the minimal causal variable sVa((C)), This, however, is not necessarily the only information in 2,..- For instance, for two random variables A, B RR the following distributions are identical:
P(A)
p(A) p(B + Al p(A) P(B,
(40)
The second distribution add additional information about A arbitrarily to B without changing the likelihoods. This is because the distribution is conditioned on A, and the conditional entropy of random variable to itself is H(AAA) = H(A, A) (AA) H(A) H(A) 0. Hence, for
29

---

Published as a conference paper at ICLR 2023
C1
C
C1
C2
C1
C2
C1
C2
Cs
C3
C3
C3
(a) Observational regime
(b) C1 intervened
(c) C2 intervened
(d) Cs intervened
Figure 8: Example instantaneous causal graph between 3 causal variables C1 C2, C2, and the aug- mented graphs under different single-targe interventions that remove instantaneous parent dependen- cies. The augmented graphs have the edges to the intervened variables removed. For readability, the intervened variables are colored in red in the graphs.
arbitrary autoregressive distributions, we cannot identify the variables from each other purely by looking at the likelihoods.
However, iTRIS, we are given interventions under which variables are strictly independent of their instantancous parents. With this, we postulate the following lemma:
Lemma D.5. For all representation functions in the class A', Ze;; does not contain information about any other minimal causal variable s (C)) $ i, except s (C.). i.c., H (2 ( (Ct)) H (2 s (C.), gaa (C2)).
Proof. In order to prove this lemma, we consider all augmented graph structures that are induced by the provided interventions on the instantaneous causal graph. Specifically, given graph G = (V, F)
with V being its vertices and E its edges, and a set of binary intervention targets / = {L1, k we construct an augmented DAG G' = (V', E"), where V' V and E' E E (fpaa((VV) 1,..../V ...,,VV,,4 = 1}. In other words, the augmented graph G' has all its input edges to intervened variables removed. An example for graph of three variables and its three single-target interventions is shown in Figure 8.
ViH
representation function in the class A must model the optimal likelihood for all intervention- augmented graphs of its originally learned graph G since it cannot achieve lower likelihood for any of the graphs than the ground truth. For every pair of variables C₁ Cj. assumption 2 (Appendix D.2.2) ensures that there exist one out of three possible experiment sets: (1) we observe I 1,
and 1 = 0 = 1, (2) 1: = 0, = 0, 1 = 1, 1 = O, and 1 1 = 1, or (3) 1 = 0, 0, 0, = 1, and 1/ 11 1, = 1. In all cases, there exist at least one augmented graph in which C L C₁. and hence 2/. L 2/0. must hold since (2) and (3) observe joint interventions on both variables (T! = 1, = 1). In (1), constant connection between the two variables would require both edges C. + Cj and C3 + C, to be present in the graph, which implies cycle in graph violating our acyclicity assumption 5. Under the augmented graph, where 2t JL 2 . likelihood can only be achieved if the distribution of 2/. is actually independent of = thus not containing any information about s yar (C₃). The same holds for ZØ., Hence, representation function in the class A must identify the minimal causal variables in the latent space.
the optimal
D.5.4 DISENTANGLING THE REMAINING VARIABLES
In Appendix D.5.2 and Appendix D.5.3, we have shown that for any solution in the class A we can ensure that 2g,, models the minimal causal variable 8 var (C,), and none other. Still, there exist more dimensions that need to be modeled. The causal variables without interventions, the invariant parts of the causal variables, sinv (C,), as well as the noise variables E are part of the generative model that influence an observation X All these variables share the property that they are not instantancous children of any minimal causal variable, and can only be parents of them. This leads to the situation that any of these variables could be modeled in the latent representation of 2 for an arbitrary i 1,... K as long as C, is the parent of the same variables. The reason for this is that the distribution modeling of such variables is independent of interventions.
30

---

Published as a conference paper at ICLR 2023
To exclude them from the causal variable modeling, follow the same strategy as in CITRIS (Lippe et al., 2022b) by taking the representation function that maximizes the entropy of Zxp
Lemma D.6, For all representation functions in the class A that maximize the information content of 2o I according to LDDP, the latent representation zw, models exclusively the minimal causal variable 8 gaa (Ct) all causal variables Ci i == 1,...KK, K.
Proof. Using Lemma D.4 and Lemma D.5, we know that the only remaining information be- sides the minimal causal variables are the causal variables without interventions, invariant parts of the causal variables, sinv (C.), as well as the noise variables E², All these variables cannot be children of the observed, intervened variables, the assumption 1 (Appendix D.2.1) states, Thus, the remaining information M = { (C1)..... gav (Cx), ,, E can be optimally modeled by p(M (22)) ..., 25x [M, 2", I. This implies that there exist a solution where 2vo M, which can be found by searching for the solution with the maximum entropy of p (O)). In this solution, the latent representation 2 Zgb does not model any subset of M, hence modeling the minimal causal variables exclusively.
L
The overall result is that we identify the minimal causal variables in : Sex. and all remaining information is modeled in Zvie. Note that the causal variables without interventions, the noise variables and the invariant part of the causal variables can be arbitrarily entangled in v Furthermore, since there exist variables in Ze% that may not have any temporal parents (e.g., the noise variables and invariable parts of the intervened causal variables). we cannot rely on assumption 4 (Appendix D.2.4) to ensure functional independence. Hence, while the distribution of p( p(=zu/=²)) is independent of E ...., z there may exist dependencies such that for a single data point, a change in 2v; can result in change of the noise or invariable parts of the causal variables in the observational space.
D.6 THEOREM 3.4 PROOF STEP 3: IDENTIFIABILITY OF THE CAUSAL GRAPH
In this step of the proof, we discuss the identifiability of the causal graph under the previous findings. In the first subsection, we discuss what graph we can optimally find under the identification of the minimal causal variables. In the second part, we then show how the maximum likelihood objective is sufficient for identifying the instantaneous causal graph. Finally, we discuss the identifiability of the temporal causal graph.
D.6.1 CAUSAL GRAPH ON MINIMAL CAUSAL VARIABLES
The identification of the causal graph naturally depends on the learned latent representations of the causal variables. In A Appendix D.5, we have shown that one can only guarantee to find the minimal causal variables in iTRIS. Thus, we are limited to finding the causal graph on the minimal causal variables 8 y (C1), 8 yaE (C2),..., 8 var (Cx) and the additional variables modeled in Z The graph between the minimal causal variables is not necessarily equal to the ground truth graph. For instance, consider a 2-dimensional position (z, y) and the color of an object as two causal variables. the x- position causes the color, but the minimal causal variable of the position is only g v (C1) y. then the color has only g inv (Ci) as parent, not 8 (C1). In the learned graph on the latent representation, would mean that we do not have an edge between Zuh and 2gz, but instead z - Zuzz Hence, we might have a mismatch between the ground truth graph on the full causal variables, and the graph on the modeled minimal causal variables.
Still, there are patterns and guarantees that one can give for how the optimal, learned graph looks like. Due the nature of the interventions, the invariable part of a causal variable, s inv (C;), cannot have any instantancous parents. Thus, the instantancous parents of minimal causal variable var (C,) are the same ground truth causal variables as in the true graph, i.e., pa(C.) = pa(s var (C.)) The difference is how the parents are represented. Since each parent Cj E pa(C) C2) is split into variable and invariable part, any combination of the two can represent a parent of s yar (C.). Thus, the learned set of parents for a yar (C.). i.e., pa(=g.), must be a subset of fs (C,)CC, E pa(C;) U ( This implies that if there is no causal edge between two causal variables C. and C, in the ground truth causal graph, then there is also no edge between their minimal causal variables s (Ca) and (C,). The causal graph between the true variables and the minimal causal variables therefore shares a lot of similarities, and in practice, is often almost the same.
31

---

Published as a conference paper at ICLR 2023
Exp.
11
12
Exp.
I1
12
Exp.
4
12
Fg
0
Fg
0
0
E
0
E1
0
E1
0
1
C2
E1
1
32
1
E2
1
(a) Causal graph
(b) Experimental setting 1
(c) Experimental setting 2 (d) Experimental setting 3
Figure 9: Identifiability of a causal relation between two variables C1, C2 under different interven- tional settings. (a) The causal relation to consider. The discussion is identical in case of the reverse orientation by switching the variable names C₁ and C2. (b-d) The tables describe the minimal sets of experiments, i.e., unique combinations of [1, 12 in the dataset, that guarantee the intervention targets to be unique, i.e., not vt, 1 = 1z Under each of these sets of experiments, we show that the maxi- mum likelihood solution of p(C1, C2//1, 12) uniquely identifies the causal orientation.
The additional latent variables Zvo summarize all invariable parts of the intervened variables, the remaining causal variables without interventions, and the noise variables. Therefore, Aoho cannot be an instantancous child of any minimal causal variable, and we can predefine the orientation for those edges in the instantaneous graph.
Next, we can discuss the identifiability guarantees for the graph on the minimal causal variables. For simplicity, in the rest of the section, refer to identifying the causal graph the minimal causal variables as identifying the graph on C1, CK-
D.6.2 OPTIMIZING THE MAXIMUM LIKELIHOOD OBJECTIVE UNIQUELY IDENTIFIES THE INSTANTANEOUS CAUSAL GRAPH UNDER INTERVENTIONS
Several causal discovery works have shown before that causal graphs can be identified when given sufficient interventions (Brouillard et al., 2020; Eberhardt, 2007: Lippe al., 2022a; Pearl, 2009). Since the identification of the causal variables already requires interventions that render variables independent of their instantaneous parents, we can exploit these interventions for learning and identifying the graph as well. In assumption 5 (Appendix D.2.5), we have assumed that the causal graph to identify faithful. This implies that any dependency between two variables, C1, C2, which have causal relation among them (C1 C2 C2 + C1). cannot be replaced by conditioning Ci and/or C2 on other variables. In other words, in order to optimize the overall likelihood p(C1, Ck). require graph that has causal edge between two variables if they are causally related. Now, are interested in whether we can identify the orientation between every pair of causal variables that have causal relation in the ground truth graph, which leads us to the following lemma:
Lemma D.7. iTRIS, the orientation of an instantaneous causal effect between two causal variables C₁ Cj can be identified by solely optimizing the likelihood of p(C,, CJ I,).
Proof. To discuss the identifiability of the causal direction between two variables C1, C2, need to consider all possible minimal sets of experiments that fulfill the intervention setup in assumption (Appendix D.2.2). These three sets are shown in Figure 9, For all three sets, we have show that the maximum likelihood of the conditional distribution p(C1. C2 12) can only be achieved by modeling the correct orientation, here C1 - C2. For cases where the true graph is C2 C1, the same argumentation holds, just with the variables names C1 and C2 swapped. As an overview, Table 3 shows the distribution p(C,, C2 , I2) under all possible experiments and causal graphs.
Experimental setting 1 (Figure 9b) In the first experimental setting, we are given single target interventions on C₁ and Cz. In the experiment Eo which represents interventions on C₁ and passive observations on C2. the dependency between C₁ and C2 persists in the ground truth, i.e., C L C2 4 1, 12 = O. Hence, only causal graphs that condition C2 on C1 under interventions on Ci can achieve the maximum likelihood in Eg. From Table does this is C1 + C2. Thus, when single-target interventions on C1 are observed, we can uniquely identify the orientation of its outgoing edges.
see that the only causal graph that
Experimental setting 2 (Figure 9c) The second experimental setting provides the observational regime (Fe), interventions on Cj with C2 being passively observed (F1), and joint interventions
32

---

Published as a conference paper at ICLR 2023
Table The probability distribution pCC, C2)41; for all possible causal graphs among the two causal variables C1. C2 under different experimental settings. Observational distributions are denoted with p(...), and interventional with p(.... Note that under interventions, it is enforced that p(...) is not conditioned on any parents, since we work on the instantaneous graph.
Interventions
Causal graph C CL
I
12
0
0
P(C))((C)IC))
P(C))p(C2) ((C))
0
((DD)(CC)))))
((C2)
0
1
P(C))
C(C))
p(C))p(C2) p(C))p(C()
((C))((C))
C1 and C2 (F2). Since the experiment F1 gives us the same setup in experimental setting can directly conclude that the causal orientation C1= - C2 is yet again identifiable.
Experimental setting 3 (Figure 9d) In the final experimental setting, C1 is only observed to be jointly intervened upon with C2, not allowing for the same argument as in the experimental settings I and 2. However, the causal graph yet remains identifiable because of the following reasons. Firstly, the experiment Eg with its purely observational regime cannot be optimally modeled by causal graph without an edge between C₁ and C2, reducing the set of possible causal graph G2 - C2 and C2 + Ct. Under the joint interventions F.z, both causal graphs model the same distribution. Still, under the experiment E1 where only C2 has been intervened upon, the two distributions differ. The graph with the anti-causal orientation compared to the true graph, C2 C1, uses the same distribution as in the observational regime to model C₁ i.e., p(Ci (C2) In order for this to achieve the same likelihood the true orientation, it would need to be conditioned on Ig the following derivation from the true distribution p(C1, C2/71, I2) shows:
p(CI, 12) = p(C2)41,12) p(C,1C2,, 11, 12)
(41)
I1, = (P(Gg
I2 =1
(42)
C2,
if I2 = 0
This derivation shows that p(Ci I, I2) strictly depends on 12 if p(C,)C,, p(C1 CC2, 41, 12 0). which ensured by C1. C2 not being conditionally independent in the ground truth graph. As the causal graph C2 + C₁ models Cj independently of 12, it therefore cannot achieve the maximum likelihood solution in this experimental settings. Hence, the only graph achieving the maximum likelihood solution is C1 + C2, such that the orientation can again be uniquely identified.
+
All other, possible experimental settings must contain one of the three previously discussed experi- ments as subset, due to assumption 2 (Appendix D.2.2). Hence, we have shown that for all valid experimental settings, optimizing the maximum likelihood objective uniquely identifies the causal orientations between pairs of variables under interventions.
Based on these orientations, we can exclude all additional edges that could introduce a cycle in the graph, since we strictly require an acyelic graph. The only remaining non-identified parts of the graph are edges among variables that are independent, conditioned on their parents. In terms of maximum likelihood, these edges do not influence the objective since for two variables Ci, C2 with
C1 1 C2 p(C1) p(C²)) p(Ci (2) p(C2) p(C1) (1). Hence, the equivalence class in terms of maximum likelihood includes all graphs that at least contain the true edges, and are acyclic. By requiring structural minimality, i.e., taking the graph with the least amount of edges that fully describes the distribution, we can therefore identify the full causal graph between C1,..,, CK-
D.6.3 IDENTIFYING THE TEMPORAL CAUSAL RELATIONS BY PRUNING EDGES
So far, have shown that the instantaneous causal relations can be identified between the minimal causal variables. Besides the instantaneous graph, there also exist temporal relations between C* and (++1, which also aim to identify:
Lemma D.8. In iTRIS, the temporal causal graph between the minimal causal variables can be
33

---

Published as a conference paper at ICLR 2023
identified by removing the edge between any pair of variables 2/y 2+ with i, [Oo.K]. if z00, IL b 120 (xb5))
Proof. The prior in Equation (2) conditions the latents variables 2 on all variables of the previous time step, 2¹. Thus, this corresponds to modeling fully connected graph from </yu 26, ..... dpx to s+1 2411 Peo 24 ..... "xx Since any temporal edge must be oriented from to 2*+1, is clear that the true temporal graph, Gr. must be a subset of this graph. Further, since in assumption 5 (Appendix D.2.5), we have stated that the true causal model is faithful, we know that two variables, 2/e and 243 s+1 are only connected by an edge, if they are not conditionally independent of each other: =ye L s+1 1a ; ,>pa (=(tt(( This implies that all redundant edges must be between two, conditionally independent variables with: zy. 1 ++1
,+1,
2/; |pa' (2 (2f+"), ), (= with pa p= being a subset of
Thus, we can find the true temporal graph by iterating through all pairs of variables af. and 2+1
and remove the edge if both of them are conditionally independent given , pa aa 2+1).
D.7 THEOREM 3.4 PROOF STEP 4: FINAL IDENTIFIABILITY RESULT
Using the results derived in Appendix D.4, Appendix D.5 and Appendix D.6, we are finally able to derive the full identifiability results. In Appendix D.5, have shown that any solution that maximizes the likelihood pa.0,a (x.+1 | , re+1) identifies the minimal causal variables of Cı.... Cк in 2dby ..... x. Further, we able to summarize all remaining variables in 2yo by maximizing the entropy (LDDP) of p (==++112)) I="). In Appendix D.6, we have used this disentanglement condition to show that the causal graph that maximizes the likelihood must have at least the same edges the ground truth graph on the minimal causal variables. To obtain the full ground truth graph, we need to pick the one with the least edges.
These aspects together can be summarized into the following theorem:
Theorem D.9. In iTRIS, a model M' = =(0,,'' o" ,b",""), identifies a causal system S = (C, E, h) (Definition 3.3) if M under the constraint of maximizing the likelihood Ps.0.a(X** X [++]);
G*)
(I) maximizes the information content (==+++12²) in terms of the LDDD (Jaynes, 1957; 1968).
(2) minimizes the number of edges in G", and
(3) no intervention variables 17 Ij are deterministically related, i.e., Vj vt: = f(1)))
34

---

Published as a conference paper at ICLR 2023
4 variables
variables
variables
Figure 10: Example sequences of the Voronoi benchmark for the different graph sizes. Each image of 32 x 32 is partitioned into K patches. The values of the K true causal variables have been transformed by a two-layer normalizing flow, which result into the hues of the K patches - gT. The hues are finally mapped into the RGB space, resulting in the images above.
E DATASETS
The following section gives detailed overview of the dataset and used hyperparameters in all settings. Appendix E.1 contains the description of the Voronoi benchmark, for which the experimental results are shown in Section 5.2. Appendix E.2 discusses the Instantaneous Temporal Causal3Dldent dataset, and Appendix E.3 the Causal Pinball dataset.
E. VORONOI BENCHMARK
The purpose of the Voronoi benchmark is to provide flexible, synthetic dataset where can evaluate causal representation learning models on various settings, such as number of variables and graph structure (both instantaneous and temporal). For each dataset, generate one sequence with 150k time steps, in between which single-target interventions may have been performed. We sample the interventions with 1/(K- +2) for each variable, and with 2/(K + 2) purely observational regime. A visual example of the Voronoi benchmark is shown in Figure 10, and we describe its generation steps below.
E.1.1 NETWORK SETUP
In the Voronoi benchmark, we need a data generation mechanism for the conditional distributions (CC++I[pa(((+))) that support any set of parents. For this, we deploy randomly initialized neural networks which models arbitrary, non-linear relations between any parent set and causal variable. We visualize the network architecture in Figure 11. As simplified setup, we use the neural networks to parameterize a Gaussian distribution. Specifically, the neural networks take as input a subset of C C*+1 according to the given graph structure (see next subsection for the graph generation), and output a scalar representing the mean of the conditional distribution NV (C, | the standard deviation is
C. f )), )),))) where 0.3. We have also experimented with having the (log) standard deviation as an additional output of the network. However, we experienced that this leads to the true causal variables be the optimal solution when modeling K conditionally independent factors. Hence, both iCITRIS and the baselines were able to identify the causal variables well, making the task easier than anticipated. The interventional distribution is thereby set to (00, 1) for all causal variables.
On the causal variables, we apply a normalizing flow which consisted of six layers: Activation Normalization, Autoregressive Affine Coupling, Activation Normalization. Invertible 1x1 convolution, Autoregressive Affine Coupling, Activation Normalization. The Activation Normalization (Kingma al., 2018) layers are initialized once after the Batch Normalizations of the distribution neural networks have been set, and ensure that all outputs roughly have a zero mean and standard deviation of one. The Autoregressive Affine Coupling layers use randomly initialized neural networks, with the average standard deviation of the outputs being 0.2. The coupling layer is volume preserving, i.e., do not use scaling term in the affine coupling, to prevent any issues with the image quantization. The Invertible Ixl convolution (Kingma et al., 2018) is initialized with random, orthogonal matrix, entangling all causal variables across dimensions. Hence, each output the normalizing flow is influenced by all causal variables.
35

---

Published as a conference paper at ICLR 2023
Masked input C*, CC+1
Linear .32)
BatchNorm (32)
LeakyReL ((0.1)
Linear
BatchNorm(33
LeakyReLUUOO.)
Linear(32.
BatchNormi
Mean
Figure 11: Network architecture of the randomly initialized neural networks in the Voronoi benchmark, modeling the conditional distributions p(C,+) Ipa(C;=1), I 0) ([C;+>/((pa C o²) with a 0.3. The BatchNorm layers (loffe et al., 2015) are initialized by sequentially sampling 100 batches of the causal variables, using each the input to the next batch. This ensures that the marginal distribution p(C, (+++) has a mean close to zero and standard deviation of one.
C1
C2
C1
C2
C4
C3
C4
C
Ca
(a) Graph structure random
(b) Graph structure chain
(c) Graph structure full
Figure 12: Example instantaneous causal graphs with four variables for the three graph structures. The causal ordering for the causal variables is randomly sampled for each graph to prevent any structural biases.
Finally, the outputs the normalizing flow are transformed by the function f(x) [T tanh ($) This function maps all values to range of [- _T, [ which we can use as hues in the patches of the Voronoi diagram. The division by 2 of i performed reduce the number of data points in the saturation points of the tanh. The Voronoi diagrams are generated by sampling K points on the image, which have distance contrast to just mapping the colors into grid, the Voronoi diagram is an irregular structure. Hence, the mapping from images to the K color non-trivial and does not transfer across datasets. Once the Voronoi diagram was created, we have used matplotlib (Hunter, 2007) to visualize the structure as an RGB image.
least pixels between each other, and fixed within dataset. In
1.2 GRAPH GENERATIONS
For the instantaneous causal graph, we have considered three graph structures: random, chain, and full. example of each is visualized in Figure 12.
The random graph samples an edge for every possible pair of variables C., Cj, i * with a chance of 0.5. Thereby, we ensure that the graph is acyelic by sampling undirected edges, and directing them according randomly sampled ordering of the variables. This way, the average number of edges in the graph is K[K-1] For small graphs of size this results in variables eventually having incoming or outgoing edges, testing also the model's ability on conditionally independent variables.
The chain graph connects the variables sequence, where each variable the parent of the next one in the sequence. This leads to each graph having K - edges, i.e., the sparsest, yet continuously connected graph.
The full graph represents the densest directed acyclic graph possible. We first sample an ordering of variables, and then add an edge from each variable all others that follow it in the sequence. Thus, it has the most possible edges
DAG, namely 2
K(K-1)
Finally, the temporal graph is sampled similar to the random graph. However, the orientations pre-determined by the temporal ordering, and no cycles can occur. We therefore sample a directed edge between any pair of variables CJ2, Cj C++1, including
with chance of 0.25. This leads
36

---

Published as a conference paper at ICLR 2023
Figure 13:
Example sequence from the training set of the Instantaneous Temporal Cadsal3DDdent dataset (from left right, top to bottom). Each image is of size 64 x 64 pixels. One can the instantancous effects of the background influencing the object color, for instance, or the object color again influencing the rotation of the object.
average number of edges of K² Additionally, we ensure that every variable has at least one temporal parent, to prevent variance collapses in the neural network distributions.
E.2 INSTANTANEOUS TEMPORAL CAUSAL3DIDENT
The creation of the Instantaneous Temporal Causal3Dident dataset closely followed the setup of Lippe et al. (2022b); von Kügelgen et al. (2021), and we show an example sequence the dataset in Figure 13. We used the code provided by Zimmermann et al. (2021)' I to render the images via Blender (Blender Online Community, 2021), and used the following seven object shapes: Cow (Crane, 202 I), Head (Rusinkiewicz al., 2021), Dragon (Curless al., 1996), Hare (Turk al., 1994), Armadillo (Krishnamurthy et al., 1996). Horse (Praun et al., 2000), Teapot (Newell, 1975). As a short recap, the seven causal factors are: the object position as multidimensional vector [x. y,
[-2, 2]; the object rotation with two dimensions [a,, 8] E [0, )x; the hue of the object, background and spotlight in [0, 2T); the spotlight's rotation in [0, 2 ;; and the object shape (categorical with seven values). We refer to Lippe et al. (2022b, Appendix C. for the full detailed dataset description of Temporal Cadsal and describe here the steps taken to adapt the datasets towards instantaneous effects.
The original temporal causal graph of the Temporal Causal3Dident dataset contains 15 edges, of which 8 are between different variables over time. Those relations form an acyclic graph, which we can directly move to instantaneous relations. Thus, the adjacency matrix of the temporal graph an identity matrix, while the instantaneous causal graph is visualized in Figure 14. The causal mechanisms remain unchanged, except that the inputs may now be instantaneous. For instance, the spotlight rotation adapted as follows:
r Instantaneous version: rot_s = (atan2(pos_ t+1 pos_yttl), r
Previous version: rot_s" s-_ss=11 = (atan2 x", pos y²), pos_y"),
(43)
(44)
where f(a, b, = - + c. The causal parents of other variables, here pos_x and pos_y are now instantaneous instead of the previous time step. Hence, an intervention on the position will lead to an instantaneous effect on the rotation of the spotlight.
In the original dataset, all interventions have been perfect with uniform distribution. To relax these interventions to partially-peffect interventions, we instead use interventions that are centered around
httpp://ggtte thub, wtt.u...cmmbbeeeeeeeeeeeee www/brendel group/cl-is
37

---

Published as a conference paper at ICLR 2023
hue_b
hue_s
rot_st
pos 0
rot
hue
obj_s
Figure 14: The instantancous causal graph in the Instantaneous Temporal Causal3Dident dataset. The graph contains several common sub-structures, such as a chain rot_o->pos__ 1---pos_o-rot__ o-trots)s fork (hue_o,hue save_oohue__i _.hue_b->rot__ b_>rot_oo)) and confounders (hue b->bue_s,huee .). The most difficult edges to recover include rot_ toso--pos___ since the object orientation has complex, non-linear relation to the observation space which difficult to model and prone to noise. Further, the edge que_,,hue___ b.hue_s-thue holds for two object shapes (Hare and Dragon), for which the background and spotlight hue have an influence on the object color. For the other five object shapes, the object color independent of the other two parents.
only
D
DF
An example sequence of the Pinball dataset, from left right, top to bottom. The paddles, ie., the two gray rectangles the bottom center, are accelerated forwards under interventions such that they make large jump within an image. For instance, in image 5, the right paddle has been intervened upon and hits the ball (gray circle). It is accelerated immediately, showcasing the instantancous effect between the two. When no interventions on the paddles are given, they slowly move backwards. In image 8, the ball hits bumper (5 circle centers with light red filling) which lights up. This represents the scoring of a point, the instantaneous increase in points shows in image 8 (the digits in the bottom right corner). Note that technically, there is no winning or losing state here since we not focus on learning a policy, instead a causal representation of the components. Further, not shown here, there exist fourth channel representing the ball's velocity.
Figure 15:
the previous time step value. Specifically, for circular values, we use C ~ ~NCC, of) with o, For the position variables, we use c ~ (0. C₁ o?) with T denoting a truncated Gaussian [- prevent objects leaving the canvas, and a; 1.5. All remaining aspects of the dataset generation are identical to the Temporal Causal3DIdent dataset.
E.3 CAUSAL PINBALL
The Causal Pinball dataset is simplified environment of the popular game Pinball, as shown in Figure 15. In Pinball, the user controls two fixed paddles on the bottom of the playing field, and tries to the ball such that collides with various objects scoring points. There are several versions of Pinball, but for this dataset, we limit it to the essential parts representing the five, multidimensional causal variables:
The ball is defined by four dimensions: the position on the x- and y-axis, and its velocity in and Both are continuous values, with the position being limited to the available spots on the field.
The left paddle y-position (paddle_lef) describes the position of the left paddle. Its
38

---

Published as a conference paper at ICLR 2023
paddle_lef
ball
bumpers
score
paddle_rigg
The instantancous causal graph in the Causal Pinball dataset. An intervention on the paddles can have an immediate effect on the ball by changing its position and velocity. A change in the ball's position again influences the bumpers, whether their light is activated or not. Finally, when the bumpers are activated, the score increases in the same time step.
Figure 16:
maximum is close-to the top of the black border next to it (e.g.. image 7 in Figure 15), and its minimum is close to the bottom (e.g., image 10 in Figure 15).
The right paddle y-position ganddle_right is similar to paddle_ left, just for the right paddle.
The bumpers represent the activation, i.e., the light, of all 5 bumpers. It is a five-dimensional continuous variable, each dimension being between 0 (light off, e.g., image in Figure 15)
and (light fully on, e.g., image 8 in Figure 15).
The score is a categorical variable summarizing the number of points the player has scored. Its value ranges from 0 maximum of 20.
The dynamics between these causal factors resembles the standard game dynamics of Pinball, which results in the instantaneous causal graph in Figure 16. The ball can collide with the paddles, borders, and bumpers. When it collides with the borders, it is simply reflected, and reduce its velocity by 10% (i.e., multiply by 0.9). Under collisions with the paddles, we distinguish between a collision where the paddle has been static or moving backwards, versus collision where the paddle was moving. When the paddle was static, we use the same collision dynamics as the borders, except that we reduce its y-velocity by 70% to reduce oscillations around the paddle position. When the paddle was moving, we instead set the y-velocity of the ball to the y-velocity of the paddle. Finally, when the ball collides with a bumper, it activates the bumper's light and reflects from it, similar the borders. When a bumper's light is turned on, we increase the score by one, but include a 5% chance that the score is not increased to introduce some stochastic elements and faulty components in the game. Next to the collisions, the ball is influenced by gravity towards the bottom, adding constant every time step to its y- y-velocity, and friction that reduces its velocity by 2% after each time step.
In terms of interventions, chance that would correspond more closely to the game dynamics. Specifically, we intervene on the paddles in 20% of the frames, 10% the ball, and 5% each the score and bumpers. An intervention of the paddle represents moving forwards, from its previous position, to a randomly sampled position between the middle and maximum paddle position. Its velocity is set to the difference between the previous position and new position. Since these interventions are usually elements of the standard Pinball game play, we sample them rather often with 20%. An intervention the ball represents stopping it at the position it is, and give it slightly random velocity towards the bottom. In real-life, this would correspond to a player interfering with the ball by stopping with their hand. To prevent instantaneous effects from the paddles, we move the ball slightly up if it is in reach of the paddles. An intervention on the bumpers is that we randomly activate bumper with a 25% chance, while leaving others untouched and maintaining their original dynamics. Finally, intervention on the score resets it to a random value between and 4.
sample the interventions the five elements independently, but with
To render the images, we use matplotlib (Hunter, 2007) and a resolution of 64 x 64 pixels. The images are generated by having a single sequence of 150k images.
39

---

Published as a conference paper at ICLR 2023
Algorithm 1 Pseudocode of the training algorithm for the prior and graph learning in iCITRIS with NOTEARS as graph learning method. For efficiency, all for-loops are processed parallel the code.
Require: batch of observation samples and intervention targets: B {x², xlt1, [ [ m==1
1: for each batch element x², x xx+1 1 do
2: Encode observations into latent space: z' go(="), 2 go (1)
3: Differentiably sample one graphs G: Gi ~ GumbelSoftmax - o (Nt) o
4: Sample latent causal assignments from &i for each batch element
5: for each causal variable C, do
6: Determine parent mask from G: S E [0,1)AM 1,,,),,,,,
,, GeOi. 12 2tt C I1,,
7: Calculate nll, =- loggs
8: end for Backpropagation loss L Et= nll;
9:
10: end for
11: Acyclicity regularizer: Ceycle = tr (exp(o(7))) K
12: Sparsity regularizer: L 1 K² Σx Σj=1 (nis)
Update parameters @, v. with V 60r07 Acycle L
Axparse L 1 D
& Cm)
F EXPERIMENTAL DETAILS
In this section, we give further details on implementation details of iCITRIS and hyperparameters that were used for the experiments in Section 5.
F.1 ICITRIS MODEL DETAILS
Similar to CITRIS, iCITRIS can be either implemented as a VAE or as a normalizing flow trained on the representation of pretrained autoencoder. The core elements to implement in iCITRIS are:
The map ge from observations r' to latents z' and back (iCITRIS VAE: convolutional encoder-decoder of the VAE iCITRIS-NF: autoregressive normalizing flow)
The assignment function of latents causal variables (matrix of IR Mx(++) from which we sample via Gumbel softmax)
The prior distributions pa (MLPs for conditional Gaussians)
The
causal discovery method for learning the instantaneous causal graph (ENCO or NOTEARS, see below)
The first three are the same in CITRIS, with the last being novel in iCITRIS. Thus, we diseuss implementation details of this graph learning below, as well as the mutual information estimator, which is only necessary for perfect interventions and extra optimization stability. For the two graph learning methods, we additionally discuss the specific setup used to learn the prior distributions pe.
Graph Learning NOTEARS The full training algorithm of iCITRIS with the NOTEARS graph parameterization is shown in Algorithm 1. The adjacency matrix is parameterized by 7 E (((+1))(X++), where (Tig), with a being the sigmoid function, represents the probability of having the edge 20; + Z3y in the instantaneous graph. To prevent self-loops, we set 7 -cc, 00 0,..., K, and Yoo = -0, = 1,... K to guarantee an empty instantaneous parent set for Zeoo At each training iteration, we sample an adjacency matrix per batch element using the Gumbel Softmax trick (Jang et al., 2017). These matrices are used to mask out the inputs to the prior, and therefore obtain gradi- ents by optimizing the likelihood of the prior. Further, NOTEARS requires two regularizers. First, the acyclicity regularizer takes the matrix exponential of the edge probabilities, (ry). The trace of this matrix exponential has a minimum of K, which is only achieved the matrix does not contain any cycles. All operations in this regularizer are differentiable, and weigh this regularizer in the loss by Acyete- This weighting factor follows a scheduling over training, which starts with value of exp(-6) R 2.5c 3. and reaches maximum of exp(4) ss 54.6. In experiments, this maximum factor ensured the graph to be approximately acyelic. Finally, the second regularizer is a sparsity regu- larizer, that removes redundant edges and is implemented I.I regularizer on the edge probabilities.
40

---

Published as a conference paper at ICLR 2023
Encoder do
T
0
A+1
Ysz
Encoder go
Decoder
113
B2s
V1s
8++1
Figure 17: A visualization of iCITRIS as a VAE framework with using ENCO in its prior. Similar to CITRIS, iCITRIS uses an encoder-decoder structure to map images x1+1 to latents 2*1 and back. The assignment function v splits the latent vector 2 into K parts (here K = 3), one per causal
variable. Between these, we learn a causal graph with ENCO, and condition the variables additionally on the intervention targets r(+1 according to v and the previous time step z'.
Graph Learning ENCO The full training algorithm of iCITRIS with the ENCO graph parame- terization is shown in Algorithm 2. The adjacency matrix is parameterized by two sets of parame- ters, with (K (K+) representing the edge existence parameters, and 0 E [R(K +++)x((++) the orientation parameters, with 0, = -,,c. The probability of an edge EΦp + Egy in the instanta- neous graph is determined by or a(%j) )- a ). Similar to NOTEARS, we prevent self-loops by set- ting Yi = - 00, = O.... K, and fix the orientations of the edges of vvo by setting 0.0- = 00: = -00, 1,., .... K. In contrast to NOTEARS, this parameterization leads to initial edge probabilities of 0.25. We found beneficial to initialize the edge probabilities closer to 0.5, which we implement by initializing %a 4, * i. E [[L..]] (ar (4) R 0.98). At each training iteration, graphs from ENCO. For all experiments, we found L = 8 to be sufficient. For each of these graphs, evaluate the negative log likelihood of all variables. Note that in contrast to NOTEARS, this does not need to be differentiable with respect to y and 0. Once all graphs are evaluated, we can determine the average negative log likelihood of 2gy under graphs with the edge Z + Zoly. versus graphs where this edge was missing. We use this to determine the gradients of Nij and oij. if zw, has not been intervened upon. For the gradients of 0.j, we further mask out gradients for batch samples in which 24, has not been intervened upon. With these gradients, we can update the graph parame- ters, while distribution parameters are updated based on the differentiable negative log likelihood. Note that the sparsity regularizer, Asparse is integrated in the update of the parameters.
sample L
Prior networks Both graph learning algorithms use a prior network of the form l4+1), To implement this efficiently in neural network setting, consider for each latent 2m m E 1M 2-layer neural network (hidden size 32 in all experiments), that take as in- put 2f, 2++1 I and mask on zf+ and r++1 Therefore, its input size + K +M M 3.M + 2K. The mask on 1 depends on which causal variable the latent 2m has been assigned to, Sw,
1411/2 [+1 Po Fo 1 vs
i.e,, ='+1 should only depend on 7,+11, The mask on 2++ depends on the graph that was sampled, in
combination with the causal variable assignment, i.e., only leave _t+1 unmasked. Further, can use Znm
autoregressive prior the potentially multiple dimensions of 2 by leaving previous latents unmasked that have been assigned to the causal variable C₁. We use this autoregressive variant for the Instantancous Temporal Causal3DIdent and Causal Pinball dataset, since the multiple dimensions in those causal factors may not be independent.
Mutual information estimator The full training algorithm of the mutual information estimator is shown in Algorithm 3. The MI estimator is 2 layer network, that takes as input the latent parents of a causal variable, and its current value, and has a single output value. This value indicates whether the current causal variable and its parents match not, i.e., is a the value of C, time step + based on observing the parents z' and w or not. We train this network by a binary classification
1+1
41

---

Published as a conference paper at ICLR 2023
Algorithm 2 Pseudocode of the training algorithm for the prior and graph learning in iCITRIS with ENCO as graph learning method. For efficiency, all for-loops are processed in parallel the code.
Require: batch of observation samples and intervention targets: B = {x', xx 1+ j==1
1: for each batch element x², x= xx*+1 _ do
2: Encode observations into latent space: z' = (,(),, 2t+1 go (x*+1)
3: Sample L graphs G G',... G from C ~ (o
4: Sample latent causal assignments from & for each batch element for each graph G' do for each causal variable C; do Determine parent sets for graph G²: 24+1
5:
6:
7:
(zft je [I.MM, E pagi(()
8:
Calculate nll = log
1/0
1-1
11"
t+1
14+1
9: end for
10:
end for
Backpropagation loss L 1 Eim Σ/= nll
11:
Σis mly
Σ C neg neg Σt (1-G",
12: Average nll for C: - A Cj: pos_nlll'j
レ- -i-1 Ciy
13: end for
14: Theta gradients: V(O,,) = a (7g)) (0()))
E (1 11 15) (pos__nl) nlll) neg_ neg_nll"))) Em= 13)) ppos_nlll neg nll'' + )
15: Gamma gradients: V(%) = (0,,))
16: Update theta and gamma with the gradients calculated above
17: Update distribution and assignment parameters o, s with V N Σm C"
211 2,, 241 to a randomly picked problem, where the model compares the true set of values, i.e., Sw
time step T, to deduce from the values of the causal variables whether they match not. Under interventions, we know that for the true causal variables, the optimal performance of this binary classifier is 0.5, because C++1 is independent of all its parents under perfect interventions. Thus, the gradients of the latents is to move the classifier closer to 0.5, which is equal to trying to increase the misclassification rate the MI estimator. During training, we need to sample instantaneous graphs G from our graph parameterization. Since especially in the beginning, this graph is close to random, and the true causal variables still depend on their children, for instance, under interventions, it can lead to unstable behavior to train the MI estimator on all parents from the start. Thus, instead, we initially train the MI estimator with an empty instantaneous causal graph and try to make =+1 independent of 2", Le., its temporal parents. Over the progress of training, we introduce the instantaneous parents, similar to the graph learning scheduling, such that at the end of training, the MI estimator is fully trained on both temporal and instantaneous parents.
TT+ -v. =,,, 2++1 Since the model does not have the precise time step / or 7 as input, has "0
2
F.2 HYPERPARAMETERS
We have summarized an overview of all hyperparameters in Table 4. Additionally, we discuss the main hyperparameter choices for all models here.
Base VAE architecture For all VAE-based methods, we have applied the same VAE to have fair comparison between methods. In particular, we have used a VAE with a normalizing flow prior (Rezende et al.. 2015), inspired by the inverse autoregressive flows (Kingma al., 2016). The encoder outputs the parameters for M independent Gaussian distributions. A sample of these Gaussians used as input to the decoder to reconstruct the original image, but also as input to a four-layer autoregressive normalizing flow. This flow consists of sequence of Activation Normalization (Kingma et al., 2018), Invertible 1 x Convolutions (Kingma et al., 2018), and autoregressive affine coupling layers. The outputs of the flow are used as input to prior, which is conditioned on the latents of the previous time step and the intervention targets. For iCITRIS, this prior follows the structure of Equation (2) including causal discovery. For CITRIS, this prior is similar to Equation (2), except that no instantancous parents are modeled. For the iVAE. the prior is a 3-layer MLP that outputs M independent Gaussian distributions. Finally, for the iVAE-AR, the prior is 2-
42

---

Published as a conference paper at ICLR 2023
Algorithm 3 Pseudocode of the training algorithm for the mutual information estimator in iCITRIS.
For efficiency, all for-loops are processed parallel the code.
Require: batch of observation samples and intervention targets: B = { x2tt r++1 n=
1: Encode all observations into latent space: 2f = go(x²) (ox")), 2 ()(
2: Sample an instantaneous graph G' from graph parameterization
3: Sample latent to causal assignments from tb
4: for each causal variable C; do
5: Filter out all batch elements for which 1,+ =
6: for each element in the filtered batch do
7:
Determine parent sets for graph G: =(+1 = (z) |j E [I..M], v() E pa (1)
8:
Calculate logits of positive pairs: Cpps = NNMI (+1 ,- _1+1
9:
For each batch element, sample a different, random time step in the batch, Calculate logits of negative pairs: eneg NNMI (= =²,, (41)
10:
11:
Calculate loss for MI estimator: L. = epos + log [exp(<pos) + exp )
12:
Calculate loss for latents: LIMI -ueg + log |exp(epes) + exp exp(ence encg
end for
14:
end for
Update parameters of NNMI according avg loss C CNNMI
16: Backpropagate gradients of latents according to avg loss LAMI
autoregressive NN predicting N Gaussian distributions in sequence. The reconstruction loss is based on the McannSquareed Error (MSE) objective, which provided much better results than learning flexible distribution over the output images. The specific architecture of the encoder and decoder depends on the dataset, where we used simpler models where possible to reduce computational cost without losing significant performance. For the Voronoi benchamrk, we use a 5-layer CNN. For the Instantancous Temporal Causal3DIdent dataset and the Causal Pinball dataset, we used a 10-layer CNN for the encoder, and a 5-layer ResNet (He et al., 2016) as decoder.
Autoencoder + Normalizing flow architecture For iCITRIS and CITRIS, training a normalizing flow on a pretrained autoencoder for the Instantancous Temporal Causal3DIdent and Causal Pinball dataset. The autoencoder uses the same encoder and decoder architecture as the VAE, except that we increase the decoder size since it can be trained much faster than the VAE (does not require any temporal dimension), and, in contrast to the VAE, lead to improvements in the reconstruction for the two datasets. The autoencoder is trained on reconstructing the input images, where we add Gaussian noise with a small standard deviation (0.05) to the latents simulate distribution. Additionally, we apply a small L2 regularizer on the latents to prevent that the autoencoder counteracts the noise in the latents by artificially scaling up the standard deviation of the latents. For CausaI3DDdent,, we use a weight of le-5 on this regularizer, and 1e-6 for the Causal Pinball since its reconstructions obtain much lower losses. The normalizing flow, applied on it, follows the same architecture in the VAE.
the variation of
Optimizer For all models, we use the Adam optimizer (Kingma et al., 2015) with a learning rate of 1e-3. Additionally, we warmup the learning rate for the first 100 steps. Afterwards, we follow cosine annealing learning rate scheduling, that, over the course of the training, decreases the learning rate to 5e-5.
Frameworks All models have been implemented and trained using PyTorch v1.10 (Paszke et al., 2019) and Torch Lightning v 1.6.0 (Falcon et al., 2019).
F.3 EVALUATION METRICS
For the details on the correlation matrix evaluation, we refer to Lippe et al. (2022b, Appendix C.3. The causal graph evaluation is performed for each model in the same way. For each model, we use the checkpoint of the best training loss, and encode all observations to the latent space. Next, we need to separate the latent space into the causal variables. For iCITRIS and CITRIS. learned assignment function & to assign latent variables to causal variables. Since the iVAE models do not learn such latent-to-causal assignment, instead assign each latent variable the causal
the
43

---

Published as a conference paper at ICLR 2023
Table 4: Summary of the hyperparameters for all models evaluated on the Voronoi benchmark, Instantaneous Temporal Causal3DIdent dataset, and the Causal Pinball dataset. all methods, we performed a hyperparameter search over the individual, most crucial hyperparameters (e.g. KLD factor in iVAE). The smaller networks in the latter two datasets for the iVAE architectures were chosen because they require training the full encoder, decoder, and NF at the same time, and larger networks did not show any noticeable improvements. The graph learning warmup in iCITRIS is equal for all datasets, although deviations to e.g. 5k, 15k or 20k often work equally well. Further, the weighting parameters of target classifier and mutual information estimator for iCITRIS such that, for instance, equally good results were achieved with smaller (5) or higher (20) weights on the Voronoi benchmark. For the graph sparsity regularizer, we used the same value for all graph structures of the same size.
Voronol benchmark
Hyperparameter
CIITRI ENCO
iCITRIS- ICITRIS-NTTARR
NOTEARS
CITRIS
iVAE
IVAE-AR
Learning rate
le-3
Learning rate warmup
100 steps
Optimizer
Adam (Kingma et al. 2015) 512
size
Number of epochs
400
KLD Factor (9)
Num latents
2x number of causal variables
Model variant
VAE with NF prior
Encoder
layer CNN + lincar layers
NF-hased prior
layer, mutoregressive affine coupling (Kingma et al., 2016)
Prior dependencics
Independent Gaussions
Autorogressive
Decoder
layer (decom-XCNN + lincar layers
Hiddeen dimensionality
32
Activation function
Swish (Ramachundran et al., 2017)
Target classifier weight
10
11D.
n.a.
MI weight
n.a.
n.a.
Graph learning warmup
10k
B.a.
R.S.
I.S..
Graph spussity oeg.
0.02 (K=46)/0000004 (K=9)
0.002 (K=4,6) 00.0004 (K=9)
0.a.
0.8.
Instantaneuus Temporal Cansal3Dident dataset
Hl perpurrrrete
CITRIS ENCO
iCITRIS NOTEARS
CITRIS
iVAE
IVAE-AR
Learning rate
1e-3
Learning rate warmup
100 stops
Optimizer
Adam (Kingma al., 2015)
Batch size
512
Number epochs
500
250
KT Factor (1)
Num latents
32
Model variant
AE NF
with NE peior-
Encoder
10-layer CNN (AE)
10-layer CNN (VAE)
Num flow layers
layers
layers
Prior dependencies
Autoregressive per Eo.
Independent Autoregressive
Decoder
10-layer ResNet-
S-layer ResNet
Hidden dimensionality
64 (VARAAE) 32 (NF)
Activation function
Swish (Ramachandran ct al., 2017)
Target classifier weight
MI weight
n.a.
m...
n.S.
Graph learning warmup
10k
Graph spansity reg.
0.0004
n.a.
n.a.
Causal Pinhall dataset
llyperpurameter
CITROSSENCO
CUTRIS-NOTEMA CITRIS
iVAE
IVAE-AR
Learning rate
le.3
Learning rate warmup
100 steps
Optimizer
Adam (Kingma et al., 2015)
Batch size
512
Number epochs
250
KLD Factor (?)
Num latents
24
Model variant
AE NF
VAE with NF peior- 10-layer CNN (VAE) layers Independent Autoregressive
Finedder
10-l CNN (AF)
Num flow layens
layers
Price dependencies
Autoregressive per Ep.
Decoder
10-layer ResNet
S-layer ResNet-
Hidden dimensionality
64 (VAEAE)/ 32 (NF)
Activation fanction
Swish (Ramschandran et al., 2017)
Target classifier weight
n.a.
n.a.
MI weight
0
n.a.
n.a.
n.a.
Graph learning warmap
10k
n.a.
Graph spansity reg.
0.02
0.001
n.a.
n.a.
n.a.
factor that it has the highest correlation to. This requires using the ground truth values of the causal variables, and hence gives the iVAE advantage over iCITRIS and CITRIS. With this separation,
44

---

Published as a conference paper at ICLR 2023
apply ENCO (Lippe et al., 2022a) learn the temporal and instantaneous graph. Since iCITRIS already learns an instantaneous graph, we reuse the learned orientations of the model, and only relearn the edge existence parameters, 7, for potential pruning. In general, found that the graphs predicted by iCITRIS have a few redundant edges between ancestors and descendants, which occur due to correlations in the early training iterations, and can easily be removed in this post-processing step.
As an additional metric to jointly evaluate the disentanglement of the causal variables and the learned causal graph, we use the learned causal graph and distributions by ENCO sample new data point under novel interventional settings. For each data point in the test dataset, we use the trained model sample the latents the next time step, and map them back the true causal variable space. This mapping is done by a small neural network, trained on the latents of the training dataset. To evaluate how well these samples match the interventional distributions of the true causal model, we train small discriminator network which tries to distinguish between the true data points in the test set, and the newly generated ones from our model. Only a model that has disentangled the causal variables, and learned the correct causal graph, can perform well this metric. We show the results for this metric on the Voronoi benchmark and the Instantancous Temporal Causal3DIDent dataset in Appendix G.
45

---

Published as a conference paper at ICLR 2023
Table 5: Experimental results of the large-scale study on the Voronoi benchmark with standard deviations over 5 seeds. We report the R correlation (R² diag/ / R² sep), the SHD between the predicted and ground truth graph (instantaneous temporal), and the accuracy of a discriminator distinguishing between true intervention samples and generated ones from the individual models (optimal 50%).
Graph Cluin
Model
Evariables
Randen
SHID
Disc.
R²
Dis.
0.00
0.007
06.71%
0,99 FO.00
05.615%
0.97/ 00000
0.00/
56,25%
6-1.52
(10.251
6-1.96
ICITRIS NOTEARS
0.01
0.007
07.43%
F0.08
1,00/180
63.28%
0.94/6.02
7.20
36,4290
6-44
(ABLEHCABLL)
2.30
CITRIS
1.0072.01
0.83/0112
120
WAA
0.76/0.20
0.6070.24
120 TRITEE
0.02
0867032
1.003000
77.005C
ICTTRIS-ENCO
59.1%%
(20 IENVITITE
KCTERIS-NOTAARS
0.803.811
61 11.485
FO00
65.17%
73.24%
120 MOVTEE
120 TALT AND
CITRIS
0.8/00133
5.0/11,20
0.80/
65.54%
11:20/ 102
07 03-000
120 MAITEE
155 2500441222000 27
WAA
0.70/0222
10,00
00-00%
0.70//03
09.45%
12400 1120
70.16%
120 CEETEE
150 71
0.7//021
0.81/023
5,007
9064%
07577034
12-00/ 1000
83.755
101 381185 4M
(27.4
KCTTRIS-ENCO
0:207
03.295%
0.007
02.11%
12007900
65.705
121
KCTERIS-NTTARRS
1.40/2.00
0,00/
6252%
07470009
14.079.00
70.715
120 R00000011
159
123.000
(50
CITRIS
0.71/6.17
14.00/18400
70.799
0.81/0.10
3.00/13.00
07.44%
0.68/0.11
32.40/22.0
73.02%
12n TEEFEE
125.
1211 700
IVAE
0.6G/
13.60/2400
71.78%
4.80/
71.80%
0.61/0.15
32.40 /9.20
74.22%
IVAE-AR
0.69/60.2
17.60/21 00
90.25%
70.38
16.00
70.12%
0.70/0.21
34.00/
85,8355
G ADDITIONAL EXPERIMENTAL RESULTS AND ABLATION STUDIES
In this section, we list the detailed results of the experiments in Section 5, including the standard deviations over multiple seeds. We further provide results on the metric for predicting intervention outcomes, described in Appendix F.3. Moreover, we present ablation studies on the Voronoi benchmark to further investigate the limitations of iCITRIS. Finally, we include a visualization of the predicted graph by all models on the Instantaneous Temporal Causal3DIdent dataset and the Causal Pinball environment.
G.1 VORONOI BENCHMARK
The full experimental results for the Voronoi benchmark can be found in Table 5. Compared the results in Figure 2, we also show the results of the discriminator that is trained on newly generated samples from the models. It is apparent that crucial factor for simulating the true intervention distributions is to have low entanglement across other factors (It2 sep). Both iVAE and especially iVAB-AR have strong entanglement between factors, and show a significant gap between the true distribution and their modeled ones. For instance, on the random graphs of size 9, 90% of the samples can be correctly classified from iVAE-AR, indicating that the distributions do not overlap much. In comparison, iCITRIS achieves close-to optimal scores on the small graphs with 55% accuracy only. Note that 50% already random performance, i.e., the optimum that could be achieved. Still, with larger graphs, we see that the performance also goes down for iCITRIS, although it still outperforms all baselines.
Furthermore, to show the specific failure types of the different models on graph prediction, we additionally list the recall and precision of the graph prediction (instantaneous Table 6, temporal Table 7). A high recall (max 1.00) reflects that the models are able recover all edges, while high precision (max 1.00) shows that the model do not overpredict false positive edges. One key characteristic on all models is that they tend to have higher recall than precision for the temporal
46

---

Published as a conference paper at ICLR 2023
Table 6: Experimental results of the large-scale study on the Voronoi dataset for predicting the instantaneous graph, including the recall and precision to highlight false negative and positive predictions.
Gruph stracture Chan
Model
#vortables
Random recall
Full
precrson
SHD
recall
prrcision
SHD
rocall
KTIRIS ENCO
1.00
1.00
0.00
1,00
1.00
0.00
1,00
KTTRISSOTEARS
0.00
1.00
1.00
1:40
0.73
0.80
0.00
1.00
1.00
CITRIS
1,00
0.60
0.55
2.00
0.77
0.62
2,50
0.53
0.50
03.25
110.55
120
120
120 64
150.50 951
WAA
1.00
0.57
0.75
2.80
0.47
0.38
3.60
0.40
0.95
155-40
1211
165-30)
941
165-20
131
IVAE AR
1.80
0.33
0.40
3.20
0.40
0.58
4.50
0.25
0.42
61140
110-4)
0.0 501
113.90
0.0.0
110430
110.00
KTTRIS ENCO
0.20
0.98
1.00
0.00
1,00
1.00
0.50
0.05
1.00
110.0
0.0
118.00
0.0.00
110.00
11000)
KYTRIS-NOTFARS
0.50
0.92
1.00
0.80
0.92
0.93
6.75
0.68
121
18030
1211
181.50
100 11
150-000
153 m1
131-20
CITRS
0.48
0.70
2.80
0.61
0.80
11.20
0.25
0.95
0.336
110.30
113.00
110:25
012:51
116.00
IVAE
3.40
0.56
0.66
12.00
0.20
03330
110.30)
0.0.00
113110
C.0.0
110:20
111.8
116/20
121
180.91
120.40
155/90
1211
121/30
ETTRIS-NOTAAN
0.96
0.40
1,00
0.96
14:50
CITRIS
14.00
0.088
3,00
0.52
0.81
IVAE
13.50
0.27
0.68
4.80
0.55
0.66
32:40
0.10
0.06
IVAE-AR
17.60
0.13
0.30
7.80
0.53
0.65
34.00
0.06
0.55
180.99
083.30
(22)
120-90
111
Table 7: Experimental results of the large-scale study on the Voronoi dataset for predicting the temporal graph, including the recall and precision to highlight false negative and positive predictions.
Graph strncture Cluin
Pvariables
Randum rocall
prociiion
SHD
musil
presision
SHD
rocall
prozision
CITRISENNCO
0.20
1.00
0.97
0.00
1.00
121
185/90
(25118)
(28-20
182-90
124 M9
151-90
1291 111
ITTTII NOTEARS
1.60
0.50
034.05
110.00
0.0.00
113.77
0.0.5
118:20
012331
110.00
1.0
CITRIS
IVAR
0.55
100
IVAE-AR
2:40
1,00
0.73
KTTRIS ENCO
0.92
1,00
0.98
KITRIS-NOTEARSS
0.79
1.00
1.00
0.95
7.75
1.00
CITRIS
11.20
0.94
0.50
9.80
0.52
0.62
12:40
0.98
0.45
146.22
IVAE
14.00
0.70
0.42
13.20
0.51
0.44
17:20
0.87
0.30
m
125.00
025.0
18t 1.0
120.00
IVAE-AR
9.20
0.85
0.56
8.20
0.96
0.56
9.00
0.90
0.54
653.4)
146.233
6-0 131
142.40
Co0 000
145.00
160.00
146.00
111
XCTTRIS-ENCO
1.20
1.00
0.54
0.00
1,00
1.00
9.50
1.00
0.71
164.35
146.00
(-0
140.00
145.00
10:544
145.00
100.31 111
KITRIS NOTBARS
2.40
1.00
0.52
0.20
1,00
0.90
9.00
0.95
0.72
125.37
125.90
120 111
120.40
125.00
120 00
120.90
120 001
CITRIS
18.40
0.84
0.33
33,00
1,00
0.62
22. 40
0.92
0.48
186.330
(201
183.30
165/800
102 431
195480
IVAE
24.00
0.48
27. 27.00
0.90
0.46
29.20
0.86
185.330
(201
188.27)
185-47)
155 731
155.30
IVAE
0.55
0.60
157 26
155.07)
119.35
(20
158.175
136
110.00
120.0h
graph. In other words, many mistakes due to predicting too many edges, which easily occurs when causal variables are entangled. However, on the instantaneous graphs, we clearly see that the baselines, CITRIS and iVAF, predict a sparse graph by having low recall. This also underlines that
47

---

Published as a conference paper at ICLR 2023
Table 8: Results of three ablation studies on the Voronoi benchmark, performed on the random graph with 6 variables. Left (Intervention noise): We introduce noise on the intervention targets by introducing 10% false positive cases in the intervention targets, i.e., 1 although C, sampled from the observation distribution. iCITRIS-ENCO performs almost as well as before. Middle (No instantancous): We apply all methods on graphs with an empty instantaneous graph. iCITRIS- ENCO obtains almost perfect disentanglement along with CITRIS and iVAE, showing that iCITRIS can be used as a replacement of them under perfect interventions. Right (No temporal): The most difficult setup is when the variables have purely instantancous relations and samples between time steps are independent. In this case, no method can disentangle the variables well, showing that different optimization strategies than iCITRIS are needed in this setting.
Ablation study
Model
Intervention noise
No instantaneous
No temporal
R
SHD
R
SHD
R
SHD
ICITRIS-ENCO
0.96 / 0.00
0.20 / 4.00
0.99 / 0.00
0.00 0.00
0.55 / 0.21
6.80 / 0.00
620 (22.00)
(20.009/120000)
(20.000/c2t) 00)
(20.00):
(12.399/129.00)
CITRIS
0.78 / 0.14
5.00 / 10.60
0.99/ / 0.00
0.00 / 0.00
0.54 / 0..1
6.80 / 0.00
(200 (58.00)
(20.000/cc00
(400.06) (26000)
(#2.399/(=50.00)
iVAE
0.69 0..2
5.40/ / 15.00
0.99 / 0.00
0.00 0.00
0.49 / 0.20
7.00 0.00
(20. (1)001))
(133.00 / (:2399
[22 (200 00)
(20 (20.000/c20..om 00)
(20 (400.05) (200002)
(b (22.22970220000
iVAE-AR
0.84 / 0.22
7.40 / 8.60
0.79/ / 0.23
2.80 / 9.00
0.54 / 0.28
7.80 / 0.00
(40.30)/($0.00)
(#=2.07)7((3.44))
1d:0. ka 06)
(#:0 (500455//444.44
(4:0 (20000))
(#:3.27/7050.00)
the false positive edges in the temporal graph cannot be simply removed by increasing the sparsity regularizer in the causal discovery method, since otherwise, even more edges would be lost in the instantaneous graph. iVAE-AR, on the other hand, has low precision and recall on the instantaneous graphs; showcasing that it predicts a very different graph with anticausal edges. Meanwhile, only (CITRIS-ENCO obtains a higher recall and precision across the different graph structures and sizes.
Next, we look at ablation studics that investigate the applications and limitations of iCITRIS.
G.1.1 ABLATION 1: NOISY INTERVENTION TARGETS
In the first ablation study, we focus the dependency of iCITRIS on accurate intervention targets. In practice, performing perfect interventions is a difficult task, and is prone to noise. While we can easily observe whether we pushed button or did external actions influence dynamical system, we do not know for sure whether the intervention succeeded or not. This corresponds to a case where the intervention targets, He, are noisy and tend have false positives, i.e., I = 1 although the intervention did not succeed. How sensitive is iCITRIS to such noise?
To investigate this question, we repeat the experiments of the Voronoi benchmark on the random graphs of size 6, but simulate that in 10% of the cases when H. we actually do not intervene on C and instead sample the value from its observational distribution. The results are summarized in Table 8 (left two columns), and clearly show that iCITRIS can yet work well in this setting. The variables are almost as well as before disentangled as before. The additional temporal variables are partially also because of noisy interventions in the post-processing causal discovery setting.
G.1.2 ABLATION EMPTY INSTANTANFOUS GRAPH
The main aspect of iCITRIS in contrast to the baselines that supports instantaneous effects. However, in practice, we might not know whether instantaneous effects are in the data not. Thus, this ablation study investigates, whether iCITRIS can yet be used as a replacement of the baselines like CITRIS, when perfect interventions are provided. For this, we repeat the experiments of the Voronoi benchmark on causal models with an empty instantaneous of size 6. As the results in Table show, iCITRIS. CITRIS. and iVAE all are able identify the causal variables and the graph. This show that iCITRIS can indeed be used as a replacement of CITRIS and iVAE, even in the setting that the variables independent, conditioned on the previous time step.
48

---

Published as a conference paper at ICLR 2023
Table 9: Experimental results on the Instantaneous Temporal Causal3D dataset, with standard deviations across three seeds.
Model
Spearman
Triplets
SHD (Instant)
SHD (Temp)
EETRIS-ENCC
0.96 0.07 (10.00)/(20.00)
0.96/0.15
0.11
1.67
5.67
(10.00)
(10.01)
(1.0.58)
(111115)
iCITRIS-NOTEARS
0.95 / 0.10
0.95 0.14
0.15
4.33
6.33
(#0.01//20000)
(=0000))((0001))
(20.02)
(21.15)
(:1115)
CITRIS
0.90 / 0.23
0.89 0.26
0.20
5.67
12.67
(10.001//200. (20.02)
(20.01) 02))
(20.01)
(4:0 58)
(20.58) 58)
iVAE
0.79 0.24
0.76 / 0.24
0.27
6.00
15.00
(:0. (20.02))(20004)
(#000)) (#000)0 02)
(#0.00)
(#1000)
(#1.00)
IVAE-AR
0.74 0.29
0.72 / 0.36
0.31
10.67
12.33
(10.02)/c20.03)
(20000)/(b0.00)
(200000)
(20.55
(24.51)
G.1.3 ABLATION EMPTY TEMPORAL GRAPH
As final ablation study, we consider the most difficult setup, namely having no temporal relations all. In this case, all relations between causal variables purely instantaneous, and we cannot use any information of the previous time step, i.e., 2². as an initial guidance for disentangling the variables. Once more, we repeat the experiments of the Voronoi benchmark on the random graphs of size 6, but with an empty instantaneous graph, and summarize the results in Table 8 (right columns). Due to the difficulty of the task, none of the methods was able to identify the causal variables. Since the probabilities of the edges in iCITRIS are initially around 0.5, the model focuses independent factors of variations instead the causal variables. The balance that is crucial for the temporal setup is that the knowledge of the interventions and previous time step is more important than the instantaneous effects for some variables, which, in this case, does not hold. Hence, to overcome this problem, different optimization strategies than the ones discussed in Section 4.3 are needed. Interestingly, even the autoregressive iVAE fails at going beyond finding K independent factors, underlining the difficulty of the task.
finding K
G.2 INSTANTANEOUS TEMPORAL 3A
We report the full experimental results for the Instantaneous Temporal Causal3DIdent,, including standard deviations across three seeds for all models, are shown in Table 9. Next to the correlation and graph prediction metrics, we also list the results of the triplet evaluation, following Lippe et al. (2022b). The triplet distance measures how well we can perform combinations of causal factors in latent space without causing correlations among different factors.
G.2.1 ABLATION STUDY 4: ORIGINAL TEMPORAL CAUSAL WITHOUT INSTANTANEOUS EFFECT
To verify that iCTTRIS can be used replacement to CITRIS, even in environments where no instantaneous effects are present, we apply iCITRIS to the original Temporal Causal3DIdent dataset. This dataset has the same causal variables and relations, but instead of instantaneous effects, all effects are temporal time steps. The results shown in Table 10. iCITRIS achieves almost identical results to CITRIS, verifying that iCITRIS generalizes CITRIS. In terms of the causal graph prediction, iCITRIS occasionally predicted instantaneous edge between the object shape and the object rotation; an edge that CITRIS incorrectly predicted over time. This is to be expected due to the visual complexity of the dataset.
G.2.2 ABLATION STUDY 5: PERFECT INTERVENTIONS IN INSTANTANEOUS TEMPORAL CAUSAL3DIDENT
As ablation to the shown dataset, we conduct experiment on the Causal3DIdent dataset, where all interventions are perfect. The experimental results for this dataset, including standard deviations across three seeds for all models, are shown in Table 11. We additionally show the discriminator
49

---

Published as a conference paper at ICLR 2023
Table 10: Experimental results on the original Temporal Causal3DIdent dataset. Results for CITRIS and iVAE are taken from Lippe et al. (2022b). iCITRIS performs on par with CITRIS, despite not assuming an empty instantaneous causal graph.
Model
R
Spearman
Triplets
iCITRIS -ENCO
0.97 0.04
0.97 0.09
0.08
CITRIS
0.98
0.04
0.97 0.08
0.07
iVAE
0.80/ 0.299
0.77 / 0.28
0.27
Table 11: Experimental results on the Instantaneous Temporal Causal3D dataset, with standard deviations across three seeds. The results for iCITRIS-ENCE are shown with and without using the Mutual Information Estimator.
Model
Spearman
Triplets
SHD (Instant)
SHDD (Temp)
Disc, Acc.
iCITRIS-ENCO
0.96 / 0.05
0.96 0.10
0.09
1.33
5,00
55.41%
(.50000//22..00
(20 000 00)
(20000
(21.15)
(21-73)
(2:0 87%)
iCITRIS-ENCO No MI
0.96 0.15
0.95 0.21
0.13
3.00
8.33
60.85%
(=0.00)/tt0.00)
(#0:0:)/(20-00)
(:1000
(22.00)
(22.331%)
iCITRIS-NOTEARS
0.95 0.09
0.14
4.00
5.00
61.05%
#40.000//t0.00)
(5.0001)(ct0.0))
(#0.00)
(+1.00)
(#1778)
(1:4448%)
CITRIS
0.92 / 0.19
0.90 0.22 0.22
0.19
4.67
10.00
67.51% 16%
(+00011)(+0000))
(#0.02)/(+0.00)
(#0.01)
(+0.58)
(+2.000
(+55990%)
IVAE
0.82/ / 0.20
0.80/ 0.22
0.27
6.67
15.33
86.87%
((0002)
(5.0.02 (550002)((00)) (2.0.01)
(40.01)
(#2.52)
(#:1.63
(d:2.7196
IVAEAAR
0.79 / 0.29
0.78/ 0.33
0.27
11.00
12.67
87.07%
(L0.01)/16..00)
L100 1400002)/L000000
(40.00)
(1:1000)
(411633)
(10090%)
accuracy of distinguishing between true and fake interventional samples. The results indicate that while this makes the task in general bit easier, iCITRIS- -CITRIS-ENCO still performs the best. The small differences in entanglement between iCITRIS-ENT ENCO and ECTRIS-NOTEARE lead to considerable difference for generating new combinations of causal factors, highlighting the importance of strong disentanglement between causal factors. Similarly, the discriminator accuracy shows that iCITRIS- ENCO can accurately model the distribution of the true causal model, while clear differences to the VAE-based baseline, iVAE and iVAE-AR, visible.
To get an intuition on what graphs the different models identify, we have visualized on example of each model in Figure 18. In general, we see that iCITRIS-ENCO misses only one edge which is sparse anyway, since hue_b b affects hue_o only for two shapes. Similarly to the results of Lippe al. (2022b), we find that the object shape is false positive parent of the rotation of the object. For CITRIS, we see that it starts to predict incorrect orientations due to correlations among factors. Finally, iVAE and iVAE-AR predict graphs that have little in common with the ground truth one.
To show the importance of the mutual information estimator in iCITRIS, we experiment with iCITRIS but without the MI estimator. The results Table 11 show that the MI estimator is indeed a crucial performance to reach iCITRIS's ' strong performance. Without the MI estimator, we experience higher correlations between different latent representations and causal variables. In the end, this also leads to worsened graph estimation for both instantaneous and temporal effects.
G.3 CAUSAL PINBALL
Finally, the full experimental results for the Causal Pinball environment can be found in Table 12. Besides the correlation and graph metrics, we again report the triplet evaluation, which shows once more that iCITRIS and CITRIS both work well here.
G.3.1 ABLATION STUDY PERFECT INTERVENTIONS IN CAUSAL PINBALL
As an ablation study, we simulate a Causal Pinball environment where all partially-perffee interven- tions replaced by perfect interventions. With this, make use of the Mutual Information es-
50

---

Published as a conference paper at ICLR 2023
(a) Original ground truth
(b) SECTRRISEENC
buu
huue
broe
reoLo
inLL
juhj
(c)
(d) CITRIS
bree.
buc
Hue
PLL
(moL
miL
roLLi
hon
abj
(e) iVAE
(I) iVAE-AR
Figure 18: Learned instantaneous graphs the Instantaneous Temporal Causal3DIdent dataset for all five models for a single seed. Red arrows indicate false positive edges, and dashed red arrows false negatives.
(a) The ground truth of the dataset. (b) iCITRIS-ENTT ENCO achieves for one score a perfect recovery of the graph, and for the other two graphs, miss one edge to huc since huc b only affects it for certain object shapes, and have an additional from the object shape to the rotation due to the complexity of the problem. (c) OCITRIS-NOTEARS had more false positive and negative edges than SUITRIS-ENCO However, all orientations were correct. (d) CITRIS had in general a sparser graph than the graph, but in contrast iCITRIS, also obtained wrong orientations several times (e.g. between pos_o and ro(_o). (e) The iVAE obtains very different graphs from the ground truth, with many incorrectly edges. (f) Due the autoregressive prior in iVAE-AR, we observed significant amount of false positive edges, with occasional incorrect orientation as well.
Table 12: Experimental results the Causal Pinball dataset three seeds.
Model
R²
Spcarman
Triplets
SHD (Instant)
SHD (Temp)
iCITRIS-ENCO
0.19
0.99 / 0.25
0.03
0.67
3.00
(20.000/(20.04
(#0.00)/c20.02)
(20.01)
(t:.15)
(#0.74)
iCITRIS-NOTEARS
0.98 0.18
0.99 0.38
0.18
3.33
4.67
(#0.00) /c:0.0))
(30000)
(#00003
(t:162)
(t00558)
CITRIS
0.90/
0.95 0.50
0.11
3.00
7.67
(22000)
(d0.01)/c20.00)
(20.01)
(1:1 .00
(42.31)
IVAE
0.44/ 0.05
0.47 / 0.05
0.61
4.33
4.67
120.09//120.00
(20.10)/c20-00
(20004)
(20.58)
(1: 58)
IVAE-AR
0.47 0.15
0.52 0.40
0.63
8.00
3.67
(#0.10//(+0004)
(#0.11)/(+0.10))
(60.04)
(#2.00)
(+1165)
timator iCITRIS. The results on this dataset are shown in Table 13. Besides identifying the causal variables well, iCITRIS-ENCO identifies the instantaneous causal graph with minor emors. Interest- ingly, CITRIS obtains a good correlation score as well here. This is likely due to the instantancous ef- fects being very sparse, and perfect interventions giving very strong preference towards independent variables in this case. Yet, there is still a gap between iCITRIS CCIRRI-EENCO ENCO and CITRIS in the instanta- neous SHD, showing the benefit of learning the instantaneous graph jointly with the causal variables.
51

---

Published as a conference paper at ICLR 2023
Table 13: Perfect interventions in the Causal Pinball dataset. Experimental results are shown over three seeds.
Model
Spearman
Triplets
SHD (Instant)
SHD (Temp)
iCITRIS-ENCO
0.98 / 0.04
0.99 / 0.17
0.02
0.67
3.67
(#0.00)((+000))
(#0.00)/(+0.00)
(+0.00)
(+0.58)
(+1.15)
iCTTRIS-NOTEARS
0.98 0.06
0.99 0.19
0.02
2.33
3.67
2.0000/1220000
(50.00) 140.000/c200000
(20.00)
(20.58)
(10 (8)
CITRIS
0.98 0.04
0.99 0.18
0.02
2.67
4.00
(2000))(22000))
(#0.00//ct0002
(20.00)
(2:155)
(21.00)
IVAE
0.55 0.04
0.58 0.14
0.55
2.33
4.33
(10.08)
(#0.09//cb0.00
(20.06)
(20.58)
(E:115)
iVAE-AR
0.53 0:15
0030
0.56
4.33
6.33
(40.08) (1:0.09)
(60.00)
(#1.53)
(#1.03)
puddie_let left
paddle Jeft
hall
humpers
score
ball
humpers
paddie right
puddle right
(a) Original ground truth
(b) iCITRIS-ENCO
puddle left
puddle left
ball
bumpers
scure
ball
humpers
score
puddile right
poddlee right
(c) ieITRIS-NOTEARE NOTEARS
(d) CITRIS
puddle lef'
puddle Jefi
ball
bumpers
scure
ball
bumpers
ScUTE
paddik_ right
paddk_rrihh
right
(c) iVAE
(f) iVAE-AR
Figure 19: Learned instantancous graphs in the Causal Pinball dataset for all five models for a single seed. Red arrows indicate false positive edges, and dashed red arrows false negatives. (a) The ground truth of the dataset. (b) iCITRIS-ENCO recovered the graph for one seed perfectly, and for the other two seeds, incorrectly oriented edge between the ball and paddles. (c) iCTTRIS-NOTEARS
commonly has some incorrect orientations between the paddles and the ball. (d) CITRIS, similar to other experiments, tends have a sparser instantaneous graph. (e) iVAE has sparser graph, similar to CITRIS, but with additional false positive edges. (f) iVAE-AR predicts a causal graph that has no edge common with the true graph.
Further, visualize the predicted causal graphs of the different methods. In general, found that the most difficult relations are between the paddles and the hall, in particular their orientation. This is due the deterministic relations between the two factors, such that if the ball has been hit by the paddle, we can already predict it just from the ball position. Further, in many states, the ball and paddle do not affect each other, such that state where the paddle would have hit the ball, but the ball was intervened upon in the same time step, is extremely rare. Overall, all models suffered from this problem, but iCITRIS showed handle it.
52


Reviews:
[{"review_id": "nBCmmJxeegF", "reviewer": "Reviewer_4Ntv", "summary": "", "questions": "", "limitations": "", "rating": 8, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "", "summary_of_the_paper": "The authors deeply dived into CAUSAL REPRESENTATION LEARNING on temporal sequences of observations and argue that temporal sequences of observations may still contain instantaneous causal relations in practice. \nIn order to model the causal structure of the temporal sequences of observations with  INSTANTANEOUS AND TEMPORAL EFFECTS, the authors propose a new method iCITRIS based on CITRIS.", "summary_of_the_review": "In the paper, the authors investigate a new problem setting, which is a novelty.\nMeanwhile, a detailed discussion is stated and an effective model for causal variables identification is proposed.", "strength_and_weaknesses": "Strength of the paper:\n1. The paper investigates a new problem setting in which the causal variables may contain instantaneous causal relations. \n2. The paper offers a detailed explanation of the motivation and method for modeling the instantaneous effects in causal variables.\n3. Sufficient experiments are conducted to evaluate the effectiveness of the proposed model in modeling the instantaneous effects of causal variables.\n4. A detailed SUPPLEMENTARY MATERIAL is attached after the main paper, in which some proofs for assumptions and theories are also contained.\n\n\nWeakness of the paper:\n\nThere is little description of related models, such as CITRIS, however, that is not a big issue.", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"correctness": "4: All of the claims and statements are well-supported and correct.", "technical_novelty_and_significance": "3: The contributions are significant and somewhat new. Aspects of the contributions exist in prior work.", "empirical_novelty_and_significance": "3: The contributions are significant and somewhat new. Aspects of the contributions exist in prior work."}, "extra_text": {}}, {"review_id": "RV7EQiwUMYx", "reviewer": "Reviewer_1H3n", "summary": "", "questions": "", "limitations": "", "rating": 8, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "", "summary_of_the_paper": "This paper describes iCITRIS, a generalization of CITRIS [Lippe et al 2022] to allow for “instantaneous effects”: i.e. they allow dependencies between the latent variables within a given time step, as opposed to enforcing conditional independence of the latents at time t given the latents at time t-1. They start by showing that it is impossible to identify latents using only the CITRIS assumptions (known intervention targets & time series observations), because we can’t distinguish between independent latents $P(z_1, z_2) = P(z_1)P(z_2)$ and entangled non-independent latents that have the same joint. However, if you have access to “partially perfect” interventions that break the instantaneous dependencies, then you can recover the same identifiability results as those provided in CITRIS. They support their theory with a series of nice empirical demonstrations that show strong disentanglement on data generating processes with slightly weaker assumptions than their theory.", "summary_of_the_review": "This is a nice paper that deals with an important problem and has strong experimental results. My main issues are with presentation - there are a lot of elements and the presentation could help the reader keep track of them all better.\n \n", "strength_and_weaknesses": "Strengths\n - Instantaneous effects are an important problem to address if we are going to make causal representation learning practical, so I like the clear articulation of the setting and the problems that arise. The light / light switch example is a really nice simple illustrative example, but you could probably strengthen this motivation by finding more settings where this arrises: e.g. it is common in natural science applications where you can only measure a system infrequently (very common in biology); you want to argue that this is very common in practice (I think this is likely), and not just a feature of things that occur at the speed of light.\n - There’s a really nice discussion of the assumptions in the appendix. Space is always an issue, but as much as possible of this should be moved into the text, because it does a better job than the main text of articulating what we’re ruling out with the assumptions. \n\nWeaknesses:\n - The assumption that we know the intervention target is still very strong. This was true of CITRIS and is true here too. I don’t think it’s grounds for rejection, but I do think it severely limits what we can apply this method to.\n - Assumption 5’s symmetry breaking condition seems tricky to verify. The appendix articulates a nice example of a failure case, and gives sufficient conditions for gaussian (interventions need to modify means) - do you have sufficient conditions for any other distributions? You say, “this assumption will likely hold in most practical scenarios” - is that because most practical scenario involve Gaussians? Or some other reason?\n - The assumption that there is a bijection between X and both C and E is strong given the critique of methods that rely on counterfactual information (Locatello et al 2020, Brehmer et al 2022 and Ahjua et al 2022). It implies that the challenges of counterfactuals are reduced to “just” solving disentanglement (of course this isn’t an easy task), because we succeed in disentangling, there is no uncertainty about the noise variables in the abduction step, so counterfactuals are known exactly. I don’t mind this assumption in this context — it’s just saying that we see all the latents rendered visually — I’m having had time deciding whether it really is weaker than the perturbation assumptions. \n - A lot of the detail about the actual implementation of the approach is hidden in a very long appendix - I know it’s just maximum likelihood wrt to the appropriate interventional distributions, but we need a little more detail in the main text to know how that translates to a loss function (e.g. are the distriubiutions over latents conditionally gaussian? Something else? ) and how the various components of the method are parameterized.\n - I think a running example would really help. There are a lot of moving pieces, so if you could have an example to keep referencing, it’d keep track of all the variables and relationships. E.g. C = [light state, switch state, …], X = pixel level image of the light scene, G = … and so on. If you just look at how many variables show up in Definition 4.3, you can see that it’s hard for a reader to track all the pieces. \n - Minimal causal variables were not well explained. I know this is a concept from Lippe et al 2022, but I would have liked to see an explanation here too, in order to make this paper self-contained.\n", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"correctness": "4: All of the claims and statements are well-supported and correct.", "technical_novelty_and_significance": "3: The contributions are significant and somewhat new. Aspects of the contributions exist in prior work.", "empirical_novelty_and_significance": "3: The contributions are significant and somewhat new. Aspects of the contributions exist in prior work."}, "extra_text": {}}, {"review_id": "KgfTuSAnFS", "reviewer": "Reviewer_4Gh8", "summary": "", "questions": "", "limitations": "", "rating": 5, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "", "summary_of_the_paper": "The paper aims at learning the causal representation from time-series data with the access of interventions. The formulation is a state-space model, which the causal process is unobserved and the observation is a funtion of the unobserved causal process. It generalizes CITRIS (Lippe et al., 2022b) to include instantaneous effects in the underlying causal graph. To identify the unique representation, the paper requires the access to the intervensions on each latent variable. The causal graph and the causal representation (or variables) can be identified with the access of the interventions by maximizing likelihood, maximizing the information content, and minimizing the edges of a causal graph.\n", "summary_of_the_review": "My main concern is the assumption of access to the interventions and the significance of the contributions can be limited. Moreover, it would be more convincing with real-world data experiments. ", "strength_and_weaknesses": "The strength:\n\n(+) The work spends a lot of effort on the proof of identifiability.\n\n(+) The problem is important: the generalization of CITRIS (Lippe et al., 2022b) in the presence of instantaneous effects.\n\n\n(-) The identifiability relies on the access to the interventions on each causal variable, which can not be easy to get in practice. This in turn limits the contribution of the work by the setting of the solution.\n\n(-) The paper claims to identify the causal representation; however, the experiments didn't show what is the learned causal representation. Perhaps, besides showing the performance of causal discovery in the setting, it can be helpful to explain the causal representations. If there is any in the appendix, it would be good to include part of the analysis in the main content. Moreover, it would be more convincing to include a ", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"correctness": "3: Some of the paper’s claims have minor issues. A few statements are not well-supported, or require small changes to be made correct.", "technical_novelty_and_significance": "3: The contributions are significant and somewhat new. Aspects of the contributions exist in prior work.", "empirical_novelty_and_significance": "2: The contributions are only marginally significant or novel."}, "extra_text": {}}, {"review_id": "5TfpjaQIqr", "reviewer": "Reviewer_R1KQ", "summary": "", "questions": "", "limitations": "", "rating": 6, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "", "summary_of_the_paper": "This paper proposes a causal representation learning method, iCITRIS, to solve the instantaneous effects of time series data via latent intervention. Specifically, iCITRIS learn a temporal process using VAE, where the latent variables can be identified by the construction. Then the structure learning methods such as NoTears and Enco are employed to learn the relations of latent variables.\nBeyond that, to solve the instantaneous effects, it assumes that the intervention targets are observed. Overall, as an extension of CITRIS, this method shows the ability to allow instantaneous effect in both theoretical and experimental perspectives.\n", "summary_of_the_review": "Overall, I almost like this paper and think it focuses on an important challenge. However, the rationality of the assumptions that the intervention of latent variables can be accessed seems an issue.\nBalancing the positive and negative points, I think this paper should be further polished. More experiments using real-world data are needed.\n\n\nMost concerns are addressed in the responses. I would like to raise my score accordingly. ", "strength_and_weaknesses": "Strength:\n1) The task to identify the latent variables are very important to understand the temporal dynamic process.\n2) The problem of the instantaneous effect is challenging and ubiquitous, which has the potential for improving time-series representation.\n3) Both theoretical and experimental evidence shows that the proposed method is effective to solve the instantaneous effect problem.\n\nWeaknesses:\n1) This method assumes that the intervention targets are observed.  However, these interventions act on the latent variables.  How can we intervene the latent variables? Is it possible for the real-world application to observe the intervention targets?\n2) All experiments are conducted on the simulation environment, such as Temporal Causal3DIdent and Pinball. These datasets are far away from real-world videos where the latent variables are unseen and uncontrollable.  Could this method is used for real-world video benchmarks, such as traffic event (Sutd-trafficqa[1]) or human actions (Oops[2]) . These datasets are more important to evaluate causal representation learning.\n3) The contribution of this paper should be further highlighted. It is suggested to separate the original CITRIS method and this extension to avoid overlap and confusing contribution, e.g. adding a baseline method section to introduce CITRIS.\n\n[1]Xu L, Huang H, Liu J. Sutd-trafficqa: A question answering benchmark and an efficient network for video reasoning over traffic events[C]//Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition. 2021: 9878-9888.\n\n[2]Epstein D, Chen B, Vondrick C. Oops! predicting unintentional action in video[C]//Proceedings of the IEEE/CVF conference on computer vision and pattern recognition. 2020: 919-929.\n ", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"correctness": "3: Some of the paper’s claims have minor issues. A few statements are not well-supported, or require small changes to be made correct.", "technical_novelty_and_significance": "4: The contributions are significant, and do not exist in prior works.", "empirical_novelty_and_significance": "4: The contributions are significant, and do not exist in prior works."}, "extra_text": {}}]

Author rebuttal:

Decision: Accept: poster