Active Bayesian Causal Inference
Authors: Christian Toth
Lars Lorch
Christian Knoll
Andreas Krause
Franz Pernkopf
Robert Peharz
Julius Von Kügelgen
Venue: NeurIPS 2022 Accept (2022)
Keywords: Bayesian methods
causal inference
causal discovery
causal reasoning
active learning
experimental design
probabilistic machine learning
Gaussian processes

Abstract: Causal discovery and causal reasoning are classically treated as separate and consecutive tasks: one first infers the causal graph, and then uses it to estimate causal effects of interventions. However, such a two-stage approach is uneconomical, especially in terms of actively collected interventional data, since the causal query of interest may not require a fully-specified causal model. From a Bayesian perspective, it is also unnatural, since a causal query (e.g., the causal graph or some causal effect) can be viewed as a latent quantity subject to posterior inference—quantities that are not of direct interest ought to be marginalized out in this process, thus contributing to our overall uncertainty. In this work, we propose Active Bayesian Causal Inference (ABCI), a fully-Bayesian active learning framework for integrated causal discovery and reasoning, i.e., for jointly inferring a posterior over causal models and queries of interest. In our approach to ABCI, we focus on the class of causally-sufficient nonlinear additive Gaussian noise models, which we model using Gaussian processes. To capture the space of causal graphs, we use a continuous latent graph representation, allowing our approach to scale to practically relevant problem sizes. We sequentially design experiments that are maximally informative about our target causal query, collect the corresponding interventional data, update our beliefs, and repeat. Through simulations, we demonstrate that our approach is more data-efficient than existing methods that only focus on learning the full causal graph. This allows us to accurately learn downstream causal queries from fewer samples, while providing well-calibrated uncertainty estimates of the quantities of interest.

Active Bayesian Causal Inference
Christian Toth
Lars Lorch ETH Zürich
Christian Knoll TU Graz
Andreas Krause ETH Zürich
Franz Pernkopf TU Graz
TU Graz
Robert Peharz TU Graz
Julius von Kügelgen"
MPI for Intelligent Systems, Tübingen University of Cambridge
Abstract
Causal discovery and causal reasoning are classically treated as separate and con- secutive tasks: one first infers the causal graph, and then uses to estimate causal effects of interventions. However, such a two-stage approach is uneconomical, especially in terms of actively collected interventional data, since the causal query of interest may not require fully-specified causal model. From Bayesian per- spective, it is also unnatural, since a causal query (e.g., the causal graph or some causal effect) can be viewed as latent quantity subject to posterior inference other unobserved quantities that are not of direct interest (e.g., the full causal model) ought to be marginalized out in this process and contribute to our epistemic uncertainty. In this work, we propose Active Bayesian Causal Inference (ABCI), a fully-Bayesian active learning framework for integrated causal discovery and rea- soning, which jointly infers a posterior over causal models and queries of interest. In our approach to ABCI, focus on the class of euusall-suuffccee additive noise models, which we model using Gaussian processes. We sequentially design experiments that maximally informative about target causal query, collect the corresponding interventional data, and update our beliefs to choose the next experiment. Through simulations, demonstrate that our approach is more data-efficient than several baselines that only focus on learning the full causal graph. This allows us accurately learn downstream causal queries from fewer samples while providing well-calibrated uncertainty estimates for the quantities of interest.
nonlinear
1 Introduction
Causal reasoning, that is, answering causal queries such as the effect of a particular intervention, fundamental scientific quest [3. 36, 39, 49]. A rigorous treatment of this quest requires reference causal model, typically consisting at least of (i) causal diagram, or directed acyclic graph (DAG), capturing the qualitative causal structure between system' variables [55] and (ii) a joint distribution that is Markovian w.r.t. this causal graph [75]. Other frameworks additionally model (ii) the functional dependence of each variable on its causal parents in the graph [56, 83]. If the graph is not known from domain expertise, causal discovery aims to infer it from data [48, 75]. However, given only collevel observational data and no assumptions on the senaraeenn process, causal discovery is limited to recovering the Markov equivalence class (MEC) of DAGs implying the conditional independences present in the data [75]. Additional assumptions like linearity can render the graph identifiable [37, 61, 71, 86] but are often hard to falsify, thus leading risk of
Shared author.
Correspondence to: (christian. toth, robert peharz (ecaarz/@tturees @tugraz at, jvketue mpg de causal inferennfeeeee
Code available at: http://wwww.giuuu.oo/////iiooooe
36th Conference on Neural Information Processing Systems (NcurIPS 2022).

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misspecification. These shortcomings motivate learning from experimental (interventional) data, which enables recovering the true causal structure [16, 17, 31]. Since obtaining interventional data is costly in practice, we study the active learning setting, in which we sequentially design and perform interventions that are most informative for the target causal query [1, 26, 31, 32, 50, 79].
Classically, causal discovery and reasoning are treated as separate, consecutive tasks that are studied by different communities. Prior work on experimental design has thus focused either purely on causal reasoning- is, how to best design experimental studies if the causal graph is known?-or purely on causal discovery, whenever the graph is unknown [35, 61]. In the present work, we consider the more general setting in which we are interested in performing causal reasoning but do not have access to a reference causal model a priori. In this case. causal discovery can be seen as means to an end rather than as the main objective. Focusing on actively learning the full causal model to enable subsequent causal reasoning can thus be disadvantageous for two reasons. First, wasting samples on learning the full causal graph is suboptimal if we are only interested in specific aspects of the causal model. Second, causal discovery from small amounts of data entails significant epistemic uncertainty-for example, incurred by low statistical test power multiple highly-scoring DAGs-which is not taken into account when selecting a single reference causal model [2. 21].
In this work, we propose Active Bayesian Causal Inference (ABCI) a fully- Bully-Bayssian framework for integrated causal discovery and reasoning with experimental design. The basic approach is to put a Bayesian prior over the causal model class of choice, and to cast the learning problem as Bayesian inference over the model posterior. Given the unobserved causal model, we formalize causal reasoning by introducing the larget causal query, function of the causal model that specifies the set of causal quantities we are interested in. The model posterior together with the query function induce a query posterior, which represents the result of our Bayesian learning procedure. It can be used, e.g., in downstream decision tasks or to derive a MAP solution or suitable expectation. To learn the query posterior, we follow the Bayesian optimal experimental design approach [10, 42] and sequentially choose admissible interventions on the true causal model that are most informative about our target query w.r.t. our current beliefs. Given the observed data, we then update our beliefs by computing the posterior over causal models and queries and use them to design the next experiment.
Since inference in the general ABCI framework is computationally highly challenging, we instantiate nonlinear additive Gaussian noise models [37],
our approach for the class of which we model using Gaussian processes (GPs) [22, 82). To perform efficient posterior inference in the combinatorial space of causal graphs, we use a recently proposed framework for differentiable Bayesian structure learning (DiBS) [45] that employs continuous latent probabilistic graph representation. To efficiently maximise the information gain in the experiment design loop, we rely on Bayesian optimisation [46, 47, 73]. Overall, we highlight the following contributions:
We propose ABCI as a flexible Bayesian active learning framework for efficiently inferring arbitrary sets of causal queries, subsuming causal discovery and reasoning as special cases ($ 3).
We provide fully Bayesian treatment for the flexible class of nonlinear additive Gaussian noise models by leveraging GPs, continuous graph parametrisations, and Bayesian optimisation ($ 4).
We demonstrate that our approach scales to relevant problem sizes and compares favourably to baselines in terms of efficiently learning the graph, full SCM, and interventional distributions ($ 5).
2 Related Work
Causal discovery and reasoning have been widely studied in machine learning and statis- tics [27, 35, 61, 81]. Given an already collected set of observations, there is a large body of literature on learning causal structure, both in the form of point estimate [9, 30, 43, 59, 60, 71, 75] and a Bayesian posterior [2. 4, 12. 14. 21, 33, 45]. Given known causal graph, previous work studies how to estimate treatment effects or counterfactuals [56, 67, 69]. When interventional data is yet to be collected, existing work primarily focuses on the specific task of structure learning without its downstream use. The concept of (Bayesian) active causal discovery was first considered in discrete [50, 79] or linear [11, 53] models with closed-form marginal likelihoods and later extended to nonlinear causal mechanisms [78, 80), multi-target interventions [77], and general models by using hypothesis testing [23] or heuristics [68]. Graph theoretic works give insights on the interventions
required for partial or full identifiability [15-17, 31, 38, 40, 70, 84].

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estimate as Baj (p(Y MM)
Bayesian Exper- imental Design
Target Causal Query Y = g(M)
perform do(a.)
inform
SCM M over X (X1,,.n Xa} a² pliiat) (X M*)
Interventional Data x
Posterior over SCMs p(M x: x!
observe outcome
infer
Figure 1: Overview of the Active Bayesian Causal Inference (ABCT) framework. At each time step we Bayesian experimental design based perform. We then collect finite data sample from interventional distribution induced by the environment, which we assume to be described by an unknown structural causal model (SCM) M over set of observable variables X. Given the interventional data xt collected from the true SCM M and prior distribution over the model class of considcration, we infer the posterior expressed function of the causal model. For example, we may be interested in the graph (causal discovery), the presence certain edges (partial causal discovery), the full SCM (causal model learning). a collection of inteerventional distributions or treatment effects (causal reasoning), or any combination thereof.
current belief's choose maximally informative intervention a
target causal query Y = ((M) that can be
Beyond learning the complete causal graph, few prior works have studied active causal inference. Concurrent work of Tigas et al. [78] considers experimental design for learning a full SCM parameterised by neural networks. There are significant differences to our approach. In particular, our framework ($ 3) is not limited to the information gain over the full model and provides fully Bayesian treatment of the functions and their epistemic uncertainty ($ 4). Agrawal al. [1] consider actively learning function of the causal graph under budget constraints, though not of the causal mechanisms and only for linear Gaussian models. Conversely, Rubenstein et [66] perform experimental design for learning the causal mechanisms after the causal graph has been inferred. Thus, while prior work considers causal discovery and reasoning as separate tasks, ABCI forms an integrated Bayesian approach for learning causal queries through interventions, reducing to previously studied settings in special cases. We further discuss related work in Appx. A.
Active Bayesian Causal Inference (ABCI) Framework
In this section, we first introduce the ABCI framework in generality and formalize its main concepts and distributional components, which are illustrated in Fig. 1. In $ 4, we then describe our particular instantiation of ABCI for the class of causally sufficient nonlinear additive Gaussian noise models.
denote random variables and their realizations, respectively. Sets and vectors are written in bold face, X and x. We use p(-) to denote different distributions, densities, which are distinguished by their arguments.
Notation. We upper-case X and lower-case
rigorous way, we first need to postulate mathematically well-defined causal model. Historically hard questions about causality can then be reduced to epistemic questions, that is, what and how much is known about the causal model. A prominent type of causal model is the structural causal model (SCM) [56]. From Bayesian perspective, an SCM can be viewed as hierarchical data- Bata-generating process involving latent random variables.
Causal Model. To treat causality
Definition 1 (SCM). An SCM M over observed endogenous variables X = (XI...., Xa} and unobserved exogenous variables U = (UI...., Ua} consists of structural equations, or mechanisms, X; = (P,,, U.),
for EE (1.....d),
(3.1)
which assign the value of each X; deterministic function Si of its direct causes, or causal parents, Pa, C X (X.) and U;; and a joint distribution p(U) over the exogenous variables.
Associated with each SCM a directed causal graph G with vertices X and edges X, + X, and only if X, E Pa,, which assume to be acyclic. Any acyclic SCM then induces unique observational distribution p(X IM) over the endogenous variables X, which is obtained as the pushforward measure of pUU) through the causal mechanisms in Eq. (3.
Interventions. crucial aspect causal models such as SCMs that they also model the effect of interventions- external manipulations to more of the causal mechanisms in Eq. which.
3

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Uezz)) with I S [d] in general, are denoted using Pearl's do-operator [56] as do([X, - f(Pa, f and suitably chosen fi(.). An intervention leads to new SCM, the so-called interventional SCM, in which the relevant structural equations in Eq. (3.1) have been replaced by the new, manipulated ones. The interventional SCM thus induces a new distribution over the observed variables, the so-called interventional distribution, which is denoted by pdo(a) (X (M) with a denoting the (set of) interven- tion(s) {x; = fe (Pa,, Kee. Causal effects, that is, expressions like E j X; = 3)], can then be derived from the corresponding interventional distribution via standard probabilistic inference.
Being Bayesian with Respect to Causal Models. The main epistemic challenge for causal reasoning stems from the fact that the true causal model M is not or not completely known. The canonical response to such epistemic challenges is a Bayesian approach: place a prior p(M) over causal models, collect data D from the true model M and compute the posterior via Bayes rule:
p(M D) P(D] M))(M)
I(D MMM P M) ((M)
(3.2)
P(D)
A full Bayesian treatment over M is computationally delicate, say the least. We require way to pa- rameterise the class of models MM while being able to perform posterior inference over this model class. In this paper, we present a fully Bayesian approach for flexibly modelling nonlinear relationships ($ 4).
Bayesian Causal Inference. In the causal inference literature, the tasks of causal discovery and causal reasoning are typically considered separate problems. The former aims learn (parts of) the causal model M typically the causal graph G", while the latter assumes that the relevant parts of M' are already known and aims to identify and estimate some query of interest, typically using only observational data. This separation suggests a two-stage approach of first performing causal discovery and then fixing the model for subsequent causal reasoning. From the perspective of uncer- tainty quantification and active learning, however, this distinction is unnatural because intermediate, unobserved quantities like the causal model do not contribute to the epistemic uncertainty in the final quantities of interest. Instead, we define a causal query function q. which specifies a target causal query Y (MM) as a function of the causal model MM. This view thus subsumes and generalises causal discovery and reasoning into a unified framework. For example, possible causal queries are:
Causal Discovery:
Y = (M) G. that is, learning the full causal graph G;
Partial Causal Discovery:
Y gred(M) (M) (G)), that is, learning some feature of the graph, such as the presence of a particular (set of) edge(s);
Causal Model Learning:
Y = (M) = M, that is, learning the full SCM M;
Y (MM(M) = fx (xx0((=))==5)))) Teeg, that is, learning a set of interventional variables Xy induced by M under do (x())) = wyy).
Causal Reasoning:
Given a causal query, Bayesian inference naturally extends to our learning goal, the query posterior:
DD) - f p(Y p(M = IEMI [M)].
(3.3)
Evidently, computing Eq. (3.3) constitutes a hard computational problem in general, as we need marginalise out the causal model. In $ 4, we introduce a practical implementation for a restricted causal model class, informed by this challenge.
Identifiability of causal models and queries. A crucial concept is that of identifiability of model class, which refers to the ability to uniquely recover the true model in the limit of infinitely many observations from it [25]. In the context of our setting, if the class of causal models M is identifiable, the model posterior p(M) 'D) in Eq. (3.2) and hence, assuming q(-) is deterministie,
"The return value of is a set of realisations of the respective random variables. In principle, the set I can be uncountable, subsuming interventional distributions for continuous set of intervention values, possibly on different variables. However, instead of having an uncountable set J for continuous of intervention values, it may be more practical to have a finite set J for intervention targets and to assume a distribution over intervention values 4b, ~ Py (+b) as we do in $ 4.2 and $.5.
'It is worth pointing out that the term "identifiability" sometimes used differently in the causal inference literature: within causal discovery, it typically refers to structure identifiability, that is, recovering only the causal graph: in the context of causal reasoning, on the other hand, it typically refers to whether an interventional (or counterfactual) query can be expressed in terms of known quantities, usually involving only the observational distribution. Here, we will use the term in its (original) statistical sense refer to identifiability of models.

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also the query posterior p (D) in Eq. (3.3) will collapse and converge to point mass on their respective true values M and (MM)), given infinite data and provided the true model has non-zero mass under our prior, p(M) > O. Given only observational data, causal models are notoriously unidentifiable in general: without further assumptions on (() and the structural form of Eq. (3.1), neither the graph nor the mechanisms can be recovered. In this case, p(M (D) may only converge to an equivalence class of models that cannot be further distinguished. Note, however, that even in this case, p(Y (D) may still sometimes collapse, for example, if the Markov equivalence class (MEC) of graphs is identifiable (under causal sufficiency) and our query concerns the presence of particular edge which is shared by all graphs in the MEC.
Active Learning with Sequential Interventions. Rather than collect large observational dataset, we seek to leverage experimental data, which can help resolve some of the aforementioned identifiability issues and facilitate learning our target causal query more quickly, even if the model identifiable. Since obtaining experimental data is costly in practice, we study the active learning setting in which we sequentially design experiments in the form of interventions az." At each time step 6, the outcome of this experiment o is a batch x² of Ne i.i.d observations from the true interventional distribution:
x² far f.
a.t lid p (dn(ar)(( (M)
(3.4)
1==1
Crucially, we design the experiment a, to be maximally informative about our target causal query Y. In our Bayesian setting, this is naturally formulated as maximising the myopic information gain from the next intervention, that is, the mutual information between Y and the outcome X' [10, 42]:
maxa, I(Y; X" //=:(=11
(3.5)
where X follows the predictive interventional distribution of the Bayesian causal model ensemble time t - 1 under intervention de, which is given by
X pdolar) (x 121:-11 plo(ac) (X MM P(M 1 d.M d.M.
(3.6)
By maximising Eq. (3.5), collect experimental data and infer our target causal query Y in highly efficient, goal-directed manner.
Tractable ABCI for Nonlinear Additive Noise Models
Having described the general ABCI framework and its conceptual components, we now detail how to instantiate ABCI for a flexible model class that still allows for tractable, approximate inference. This requires us to specify (i) the class of causal models we consider in Eq. (3.1). (ii) the types of interven- tions a, we consider at each step and the corresponding interventional likelihood in Eq. (3.4), (iii) our prior distribution p(M) over models, (iv) how to perform tractable inference of the model posterior in Eq. (3.2), and finally (v) how to maximise the information gain in Eq. (3.5) for experimental design.
Model Class and Parametrisation. In the following, we consider nonlinear additive Gaussian noise models [37] of the form
X; = (Pa.) + U,, with U o7 for E (I,...,d),
(4.1)
where the fi's are smooth, nonlinear functions and the U,'s are assumed to be mutually independent. The latter corresponds to the assumption of causal sufficiency, or no hidden confounding. Any model M in this model class can be parametrised a triple M = (G. f. o²), where G is causal DAG, f = (f.,.... fa) is a vector of functions defined over the parent sets implied by G, and o² (oi (-...... .., , contains the Gaussian noise variances. Provided that the fi are nonlinear and not constant in any of their arguments, the model identifiable almost surely [37, 62].
Interventional Likelihood. We support the realistic setting where only subset W C X of all variables are actionable, that is, can be intervened upon. 5 We consider hard interventions of the form do(a.)= = do( X - = æT) that fix subset Xx C W to a constant ΦT. Due to causal sufficiency, the interventional likelihood under such hard interventions a, factorises over the causal graph and is given by the g- g-formula [64] or truncated factorisation [75]:
dolas) (X G, (G = 11x, = æz} II
PXx (fPa)))())) jgz
(4.2)
4 "Note that restricting to a = Ø amounts to learning from observational data as a special case. In principle, the set of actionable variables might even change over time, in which case they denoted We.
5

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The last term in Eq. (4.2) is given by (XX (Pa)), 0,)), due to the Gaussian noise assumption. Let wi be the entire dataset, collected up time t. The likelihood of æ is then given by
t pla I - II
p dola.) (x "elI,ff,
II
I p do(a=) (2 (oT""G,,
(4.3)
Structured Model Prior. To specify our prior, we distinguish between root nodes X,, for which Pa, 2 and thus fi const, and non-root nodes Xj. For given causal graph G, we denote the index set of root nodes by R(G)- {i E [d] Pa Ø and that of non-root nodes by NR(G) = [d] RGG). We then place the following structured prior over SCMs M (G. o²))
((M) - p(G)) G) - (GG) *ER(G)
II o( (
II JENR(G)
pls (U)()) G). (4.4)
Here, p(G) is prior over graphs and p(f, o² (G) is a prior over the functions and noise variances. We factorise our prior conditional on G as in Eq. (4.4) not only to allow for separate treatment of root vs. non-root nodes, but also share priors across similar graphs. Whenever Pa Pa we set p(fi, of G1) = p(fa. of G2). As consequence. the posteriors are also shared, which substantially reduces the computational cost practice (see Appx. E.2 for details). Our prior also encodes the beliefs that {fi, ) 11 ofv G for + E [d] and that f3 .1. o3 G for j E NR(G) which motivated by the principle of independent causal mechanisms [61] and the causal sufficiency assumption. Our specific choices for the different factors on the RHS of Eq. (4.4)
are guided by ensuring tractable inference and described in more detail below.
Model Posterior. Given collected data 2;2: we can update our belief's and quantify our uncertainty in M by inferring the posterior p(M =:::)) over SCMs M = -GG, -G, o²), which can be written as
p(M 12 = p | 1t
p(S. of 1 .G) (ER(G)
II
II JENR(C)
p(3,03 x ,G).
(4.5)
For root nodes E R(G), posterior inference given the graph is straightforward. We have fi const, so fi can be viewed as the mean of U.. We thus place conjugate normal-inverse-gann
N-r- As, a. ) priors on p(f. o² [G,, which allows us to analytically compute the root node
posteriors p(f.,, of x G') in Eq. (4.5) given the hyperparameters (M, X, a", 3*) [51].
The posteriors over graphs and non-root nodes j NR(G) are given by
p(G x litt)) p(x P(G
p(f,, | G) p(x 1st (f |G) (4.6)
p(x
p(
Computing these posteriors is more involved and discussed in the following.
4.1 Addressing Challenges for Posterior Inference with GPs and DiBS
The posterior distributions in Eq. (4.6) are intractable to compute in general due the marginal likelihood and evidence terms p(x1: G) and p(x=::), respectively. In the following, we will address these challenges by means of appropriate prior choices and approximations.
Challenge Marginalising out the Functions. The marginal likelihood 2: 1: |G) reads
p( 1:t GG) = p(a G, (G.f P(f) [G) Gf
(4.7)
and requires evaluating integrals over the function domain. We use Gaussian processes (GPs) [82] as an elegant way to solve this problem, as GPs flexibly model nonlinear functions while offering conve- nient analytical properties. Specifically, we place a GP(O, 0, (-()))) prior on p(f GG), where (.--) is a covariance function over the parents of X, with kernel parameters Ke,. As is common, we refer to (Ki,, of) as the GP-hyperparammtters. In addition, we place Gamma (az, 189) and Gamma (o5,37)
priors on p G) and p(R;) G) and collect their parameters in (a ,.)).
'To avoid further complicating the notation, we write all posteriors and likelihoods in terms of the full data x 1t However, only observations of X; and X, Paf matter for E R(G) and E NR(G).

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The graphical model underlying all variables and hyper- parameters is shown in Fig. For our model class, GPs provide closed- form expressions for the GP- GP-marginal likelihood p(ar l:t G, 7 Kz), as well as for the GP posteriors p(fy x lit G, oz. Ry) and the predictive pos- teriors over observations p( X |a G, G. k) [82], see Appx. B for details.
(n, A, a² 3")
G
Z
(a
ft
ot
f
R2
qj
R(G)
NR(()
Marginalising out the GP- Hyperparameters. While GPs allow for exact posterior inference stance of Kig). evaluating expressions such as p(fs a G) requires marginalising out these from the GP-posterior. In general, this is intractable to do exactly, as there is no analytical expression for p(o}; Kÿ w G). To tackle this, we approximate such terms using a maximum a posteriori (MAP) point estimate (0) R.)) obtained by performing gradient ascent on the unnormalised log posterior
Challenge 2:
do(a-) )
TM T
conditional on a fixed in-
N.
Figure 2: Graphical model of GP-DiBS-ABCL.
logp(o); K2 =: G) = logp(æ GG, K,)) + V Vlog Ki G)
(4.8)
according to a predefined update schedule, see Alg. 1. More specifically,
p(fs x lit G) /(fs x G. 1Y) Ky x lit G) doG dr, 2e plfs x wel""",G,, G, R2)
Challenge 3: Marginalising out the Causal Graph, The evidence p(artit) is given by
p( (t)
Σ p(al (G)
(4.9)
and involves a summation over all possible DAGs G. This becomes intractable for d 2 5 variables as the number of DAGs grows super-epponentiin in the number of variables [65]. To address this challenge, we employ the recently proposed DiBS framework [45]. By introducing a continuous prior p( Z)) that models G' via p(G Z) and simultancously enforces acyclicity of G, Lorch et al. [45] show that we can efficiently infer the discrete posterior p(G x a:l:t) via p( Z
x 1:))) as
EG z 1p(x 1st |G) ((G)
EG ti [o(G)) = Ez aiti
(4.10)
EG 2 pl p 1:1 []]
where ø is some function of the graph. Since p( Z x x1:)) is continuous density with tractable gradient estimators, we can leverage efficient variational inference methods such Stein Variational Gradient Descent (SVGD) for approximate inference [44]. Additional details on DiBS are given in Appx. D.
4.2
Approximate Bayesian Experimental Design with Bayesian Optimisation
Following $ 3, our goal is to perform experiments o, that are maximally informative about our target query = (MM) by maximising the information gain from Eq. (3.5) given our hitherto collected data D = x lit-1 In Appx. C, we show that this is equivalent to maximising the following utility function:
U(a) (XX' D) + EMID [X
[log EM D [p(x', Y [M)]]]].
(4.11)
where
H(X' EMID ExxIMM Elo I [p(X' [M]]]]]
denotes the differential entropy of the experiment outcome which depends on a and distributed as in Eq. (3.6). This surrogate objective can be estimated using nested Monte Carlo estimator as long as we can sample from and compute p(Y _M), or alternatively, p(Y X G, D). Refer
to Appx. C for further details. For example, for (cM(M) = x- =d)(( with &ú p(1)) a distribution over intervention values, we obtain:
UeR (a) EG B [x.I - log E I [p( X DD G")]
(4.12)
+ Ep [EX] Xx,G, Eo aa [p(x' | [D dal (ox X G, G)D)
7

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Algorithm 1: GP-DiBS-ABCI for nonlinear additive Gaussian noise models
Input: # of experiments T, batch sizes (Ne)-=1 # of latent particles M, # of MC samples K, particle resampling schedule (rx (rr) hyperparameter update schedule {s:]<
Output: Posterior over target causal query p(Y x (TT)
=D 2(
s>sample initial particles; Eq. (D.12)
for do
d; < arg ((T,r= U(a U(a 1
Ddesign experiment; Eq. (4.11)
x' + (a(hm) ~ (o(()) (X MM
>perfrrm experiment; Eq. (3.4)
z' < =²-1
rf then
z' resampleeparticleee (z*)
Deee Appx.E
end
repeat
G - (G ~ = M
sample graphs; Eq. (D.11)
m=1
k. - aeteiheeee_..eerrprrrtttttttrere G)
Deee Eq. (4.8)
x' - svgd_step(2"'''' G, G
uupdate latent particles
until cove_conee
x= now approximate p(Z x =
end
Importantly, for specific instances of the query function q(-) discussed in 3. can derive simpler utility functions than Eq. (4.11). For example, for gen (M) = G and geMu (JV) = M, we arrive at
Ucb(a) = EGID [Ex+
G,D [logp(X* p(X' DD G) - log EE [p[ x DD, G')l]]+
(4.13)
Uema.C (a) = EMI Ex
M [logp(X* X [M) - log E. ID [p( [']]]]]]
(4.14)
where the entropy Ex IM [logp(X ( X MM]] can again be efficiently computed given our modelling choices. For brevity, defer derivations and estimation details Appxs. C and D.
Finding the optimal experiment aj = (I*, requires jointly optimising the utility function cor- responding our query with respect to (i) the set of intervention largets I and (ii) the corresponding intervention values Tr. This lends itself naturally to a nested, bi-level optimisation scheme [80]:
I arg maxy U U(IIxt))
where VI:
2 E arg max UZZ æ1),
(4.15)
In the above, we first estimate the optimal intervention values for all candidate intervention targets and then select the intervention target that yields the highest utility. The intervention target I may contain multiple variables, which would yield a combinatorial problem. For simplicity, we consider only single-node interventions, [Z] 1. To find æij employ Bayesian optimisation [46, 47, 73] to efficiently estimate the most informative intervention value zÿ. see Appx. D.
Experiments
Setup. We evaluate ABCI by inferring the query posterior on synthetic ground-truth SCMs using several different experiment selection strategies. Specifically, we design experiments w.r.t. Uep (causal discovery), UCML (causal model learning), and UCa (causal reasoning); see 4.2. We compare against baselines which (i) only sample from the observational distribution (OBS) or (ii) pick an intervention j uniformly at random from [d] (Ø) and set X; = 0 (RAND FIXED, weak random baseline used prior work) or draw Xj ((-777) (RAND) X; & Z. All methods follow our Bayesian GP-DiBS-ABCI approach from $ 4. We sample ground truth SCMs over random scale-free graphs [6] of size d = 20, with mechanisms and noise variances drawn from our model prior in Eq. (4.4). In Appx. G. we report additional results for both scale-free and Erdōs Renyi random graphs over 10 resp. d = 20 variables. For specific prior choices and simulation details, see Appx. D.
Metrics. As ABCI infers posterior over the target query Y, a natural evaluation metric is the Kullback-Leibler divergence (KLD) between the true query distribution and the inferred query posterior, |MM) interventional distributions (qeR). As a proxy for the KLD of the SCM posterior (qoMr.), we report
x (=)))). We report Query KLD, a KLD estimate for target
'The SCM KLD is either zero, if the SCM posterior collapses onto the true SCM, or infinite, otherwise.

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Expected SHD
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Figure 3: Causal Discovery and SCM Learning. Comparison experimental design strategies for causal discovery (Uen) and causal model learning (Uean) with random and observational baselines on simulated ground truth models with 20 nodes. We initialise all methods with 50 observational samples. and then perform experiments with batch size of Nr 5. Lines and shaded areas show means and 95% confidence intervals (CIs) across 15 runs (5 randomly sampled ground-truth SCMs with 3 restarts per SCM). CIs for OBS and RAND FIXED baselines
shown aid readability; Fig. in Appx. for the full figure. (a) ESHD. Both our objectives significantly outperform the observational and random baselines. (b) Average I-KLD, U significantly outperforms the baselines, whereas Uea.. performs only marginally better than RAND. (c) AUPRC. Both our strategies perform consistently better than the uninformed selection strategics.
Query KLD
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Figure 4: Learning Interventional Distributions. (left) Comparison of different methods w.r.t. learning set of interventional variables X 4xt Xa=e) with z0 ~ 22(2, on simulated ground truth models with fixed causal graph (right). We initialise all methods with 5 observational samples, and then perform experiments with batch size of N, Lines and shaded
show means and 95% confidence intervals (CIs) across 30 runs (10 randomly sampled ground truth SCMs with 3 restarts each). Cls for OBS and RAND FIXED basclines are shown aid readability; see Figs. and 10 in Appx. G for the full figure. All nodes actionable. Uen significantly outperforms all other methods as expected. Ueaa. performs second best which, in conjunction with the results from Fig. 3, suggests that Uron. yields solid base model for performing downstream causal inference tasks. (b) Xa not actionable. In this setting. where we cannot directly intervene on the treatment variable interest, Uex clearly outperforms all other methods for 2 10 experiments.
the average KLD across all single node interventional distributions (x))(()=)((()) with sb (1),77 (Average I-KLD). We also report the expected structural Hamming distance [13], ESHD = Eg|=.. SSD((G, G²)], commonly used causal discovery metric, and the area under the precision recall curve (AUPRC), See Appx. F for further details.
Causal Discovery and SCM I. Learning (Fig. 3). In our first experiment, we find that all ABCI-based methods are able to meaningfully learn from small amounts of data, which validates our Bayesian approach. Moreover, performing targeted interventions using experimental design indeed improves performance compared to uninformed experimentation (OBS, RAND FIXED, RAND). Notably, the stronger random baseline (RAND), which also randomises over intervention values, performs well the considered setting. As expected by the theoretical grounding of the information gain utilities, UcD identifies the true graph the fastest (as measured by ESHD), whereas UCML exhibits good scores across all metrics. Further details are given in the caption of Fig. 3.

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Learning Interventional Distributions (Fig, 4). In our second experiment, we investigate ABCT's causal reasoning capabilities by randomly sampling ground SCMs as described above over the fixed graph shown in Fig. 4 (right), which is not known to the methods. Our target query is the set of interventional random variables, or "distributional treatment effects", X xx_l((x==5)) for treatments 3 ~ 2/2, 5]. The results show that our informed experiment selection strategies significantly outperform the baselines at causal reasoning as measured by the Query KLD. In accordance with the results from Fig. 3 and considering that, once we know the true SCM, we can compute any causal quantity of interest, UeMI. seems to provide a reasonable experimental strategy in case the causal query of interest is not known a priori. However, our results indicate that if we do know our query of interest, then UeR provides a more efficient experiment design strategy for its estimation, even when the treatment variable of interest is not directly intervenable. In this case, the task is indeed more difficult, as highlighted by the larger Query KLD values across all considered methods.
Discussion
Assumptions, Limitations, and Extensions. In $ 4, we have made several assumptions to facilitate tractable inference and showease the ABCI framework in a relatively simple data-generating process. In particular, our assumptions exclude heteroscedastic noise, unobserved confounding, and cyclic relationships. On the experimental design side, we only considered hard interventions, but for some applications soft interventions [18] are more plausible. On the query side, we only considered interventional distributions. However, SCMs also naturally lend themselves to counterfactual reasoning, so one could also consider counterfactual queries such as the effect of the treatment on the treated [34, 72]. In principle, the ABCI framework as presented in 3 extends directly to such generalisations. In practice, however, these can be non-trivial to implement, especially with regard to model parametrisation and tractable inference. Since actively performed interventions allow for causal learning even under causal sufficiency violations, we consider this promising avenue for future work and believe the ABCI framework to be particularly well-suited for exploring it.
Reflections on the ABCI Framework. The main conceptual advantages of the ABCI framework are that it is flexible and principled. By considering general target causal queries, we can precisely specify what aspects of the causal model we are interested in. This conceptual framework offers a fresh perspective on the classical divide between causal discovery and reasoning: sometimes, the main objective may be to foster scientific understanding by uncovering the qualitative causal structure underlying real-world systems; other times, causal discovery may only be a means to an end to support causal reasoning. Of particular interest in the context of actively selecting interventions is the setting in which we cannot directly intervene on variables whose causal effect on others we are interested in (see Fig. 4), which connects to concepts such as transportability and external validity [7, 57]. ABCI is also flexible in that it easily allows for incorporating available domain knowledge: if we know some aspects of the model priori (as assumed in conventional causal reasoning) [53] or have access to large observational sample (from which we can infer the MEC of DAGs) [1], we can encode this in our prior and only optimise over smaller model class. The principled Bayesian nature of ABCI comes at a significant computational cost: most integrals are intractable and approximating them with Monte-Carlo sampling is computationally expensive and can introduce bias when resources limited, though cf. [85] for recent efforts to address such intractability. We discuss the computational complexity of our implementation in more detail in Appx. E.3. On the other hand, in many real- world applications, such as in the context of biological networks, active interventions are possible but only experi-
significant cost [11, 53]. In such cases in particular, a careful and computationally-het mental design approach as presented in the present work is warranted and could be easily amortised.
Acknowledgments and Disclosure of Funding
We thank Paul K Rubenstein, Adrian Weller, and Bernhard Schölkopf for contributions to an early version of this work [80], and the anonymous reviewers for helpful feedback. This work was supported by: the German Federal Ministry of Education and Research (BMBF): Tübingen AI Center, 011SI 18039B; the Machine Learning Cluster of Excellence, EXC number 2064/1 Project number 390727645; the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation program grant agreement no. 815943; the Swiss National Science Foundation under NCCR Automation, grant agreement SINF40 180545; and the Graz University of Technology LEAD project "Dependable Internet of Things in Adverse Environments".
FKZ:
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Reviews:
[{"review_id": "hyIqEyGdBvB", "reviewer": "Reviewer_JJRE", "summary": "This paper presents a framework to simultaneously infer a causal graph as well as the specific causal model, including specific relationships between the variables from observational data. \nTo do so, a bayesian approach for joint inference is outlined in the contribution.\nThe approach also enables learning from additional, new experimental data - in sense of a standard active learning methodology.\nA experimental setup justifies the framework as it can show the success of the algorithm.", "questions": "* Personally, I consider the terminology using exogenous and endogenous variables for latent and observed variables (L. rather confusing (Def. 1)  somewhat confusing as my understanding is more aligned with variables determined by the model and variables not determined by the model (see https://en.wikipedia.org/wiki/Exogenous_and_endogenous_variables )\n\n ", "limitations": "Limations, especially regarding feasability and tractabilty, are well discussed. ", "rating": 7, "confidence": 3, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "\n\n## Strengths\n\n* The problem is well structured, with precise mathematical notation to distinguish certain problem classes\n\n* The choice of the problem - i.e. to extend inferring from only observational to observational and experimental data - is somewhat novel in a sense, that most research in the field is focussed on inferring from observational data alone\n\n* Recent advances, such as the DiBS approach, are considered and brought to use in context of the framework.\n\n* An accompanying codebase provides a computational implementation of the approach, including jupyter notebooks which facilitate understanding the \"how-to\" of the codebase.\n    the problem is well formulated and the manuscript is clear and well written\n\n    the formal style with highlighted Theorems as well as the choice of notation is consistent with the relevant literature of the field\n\n    The real world datasets are well chosen and fit the problem, for the experiments several methods for comparison are included\n\n## Weaknesses\n\n* As the approach outlined is somewhat experimental - in that a \"real\" active learning approach would necissate manipulation / intervention into the setting - I would have expected a stronger / more elaborate discussion setting, \n\n\n*   I would have expected to have more discussion of state-of-the-art / comparable methods. The methods for comparison introduced only later could be already discussed theoeretically in an earlier section\n\n* The idea of causal reasoning as formulated in the paper (i.e. via do-formalism interventions) would benefit from some more explanation, ideally using examples.", "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": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"ethics_review_area": ""}, "extra_text": {}}, {"review_id": "bDxoCE7wFE", "reviewer": "Reviewer_T1JL", "summary": "This work proposes Active Bayesian Causal Inference (ABCI) that is a fully-Bayesian active learning framework for integrated causal discovery and reasoning, which jointly infers a posterior over causal models and queries of interest. They implemented ABCI for the class of nonlinear additive Gaussian noise models using Gaussian processes. To parametrize the combinatorial space of causal graphs and having tractable posterior, they applied a continuous latent probabilistic graph representation. ", "questions": "Please see the above comments.", "limitations": "Please see the above comments.", "rating": 6, "confidence": 4, "soundness": 3, "presentation": 4, "contribution": 3, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "They studied a practical and important question that is answering to a causal query without knowing the causal graph. The paper is well-written. \n\nThe proposed approach, unlike most exiting methods, does not learn the causal structure and then answer to the causal query of interest. This makes it both novel and more data efficient. \n\nAlthough the proposed ABCI is a general framework but as it is mentioned it is not a trackable algorithm for general causal models and hence it is implemented for nonlinear Gaussian models. How can this algorithm be used for model classes that contain discrete variables?\n\nApplying information gain framework for selecting the next best intervention is standard in Bayesian approaches but it is suboptimal as it is greedy. Is there any guarantee of how far the solution is from the optimal solution (e.g., if the objective is sub-modular then it is a constant approximation)? Moreover, as it is suggested by the experiments, there is not so much gain compared to the random selection method considering the additional computational cost of design experiment line in ABCI algorithm.\n\nThe overall method is computationally intense which makes its scalability questionable. A discussion on the computational complexity of ABCI would be interesting. The experiments are also for small graphs. \n\nAs it is mentioned, this work is built on causal sufficiency which is a limiting assumption and in most real-world applications it does not hold. Having said this fact, suppose that the target causal query is structure learning. In this case, having observational data, we can learn the essential graph of the corresponding MEC. On the other hand, the authors of “Learning Causal Graphs with Small Interventions” show that designing minimum number of interventions with limited size to learn the causal structure is NP-hard. When the actable set of variables are limited in size, then it is interesting to see how ABCI algorithm performs compared to the aforementioned work.\n\nUnder the causal sufficiency and using the rules of do-calculus, all interventional queries become identifiable from observational data given the causal graph. It is important to see whether ABCI algorithm in the learning interventional distribution task learns the whole graph or only a part of it before finding the interventional distribution of interest. \n\nMinor:\nIntroduction: \"Additional structural assumptions (e.g., linearity)\". Linearity is not a structural assumption. It is an assumption on the underlying generative model. \n", "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": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"ethics_review_area": ""}, "extra_text": {}}, {"review_id": "8E7Zazn9Ol", "reviewer": "Reviewer_8rNg", "summary": "The paper introduces a new framework, ABCI, for end-to-end Bayesian causal inference. Starting from no data, ABCI iteratively designs experiments (interventions) to maximize the information gain for the target causal query. ABCI maintains uncertainty over the causal model over time by maintaining a set of latent particles, corresponding to causal graphs, which are used to compute the information gain (through a Gaussian-process based marginal likelihood). ABCI is distinguished from previous works by 1) targeting (and optimizing interventions) for a specific causal query $q(\\mathcal{M})$, which can be any function of the SCM; and 2) modeling linear non-Gaussian SCMs. Empirically, ABCI methods are shown to provide improved performance compared to random-intervention baselines. \n", "questions": "Key questions:\n- What are the essential novel insights/results that we should take away from this paper, that were not previously known? \n- What is the justification (e.g. computational) for the choices made? For instance, can other models be used in place of Gaussian processes? Why was Bayesian optimization chosen? Can alternative Bayesian structure learning methods be used in place of DiBS?\n- In Appx D.7, what likelihood $p(x|z)$ is used for SVGD, and why? Is it the GP marginal-likelihood?\n- In Appx D.7, it is mentioned that only 5 latent particles are maintained while using DiBS; intuitively, this seems insufficient to maintain enough graphical uncertainty. Have the authors tested the effect of changing this hyperparameter?\n    \nOther:\n- Is it possible to use gradient-based optimization for the utility (Eqn 4.11)? E.g. score function estimator?\n- Have the authors considered using a Laplace approximation for the GP hyperparameter posterior? \n- In line 268, it is shown that simplified utilities are available for cases where the query is either the full graph, or full SCM. Are there any practical benefits (e.g. in terms of estimation) in these cases?\n", "limitations": "The authors have appropriately addressed the limitations through discussion. ", "rating": 6, "confidence": 4, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "This is a solid work which tackles the important problem of designing optimal experiments for answering causal queries, which are not identifiable from observational data. Bayesian inference is a principled approach through which to maintain uncertainty, and the ABCI framework is a sensible and well-engineered end-to-end solution for inferring a target causal query. In that sense, ABCI fills an practical gap that previous works did not fully address.\n\nOn the positive side, ABCI brings together a number of previous ideas, such as maintaining uncertainty through graph samples, maximizing information gain, and using Gaussian processes to model non-linear relationships with tractable marginal likelihoods, to form a useful end-to-end framework. The paper investigates targeting specific queries by developing utility functions (information gain) for arbitrary functions of the SCM. The benefits are demonstrated experimentally, where $U_{CR}$ is shown to converge faster than $U_{CML}$ on the specific causal query.\n\nMy key concern is that there does not appear to be significant novel components introduced in this work; nor does there appear to be significant novelty in the approach to integrating the components: for instance, computation appears to be the key limiting factor, and there does not appear to be theoretical advances in this regard. Alternatively, comprehensive empirical analysis could provide justification and insight regarding the efficiencies and bottlenecks in the ABCI pipeline, e.g. ablation studies considering different approaches to each step. \n\nMore technically, I have some concern over the nature of the uncertainty representation maintained by ABCI. In particular, ABCI maintains a set of latent particles $\\textbf{Z}$, corresponding to graphs, and it appears that the GP marginal-likelihood and posterior is computed fresh in each timestep $t$. This is concerning for two reasons. Firstly, the number of samples maintained by DiBS (Appx D.7) is chosen to be very small (5), which seems unlikely to provide a sufficiently diverse uncertainty. Secondly, the idea of maintaining a set of graphs for evaluating information gain already appeared in Agrawal et al. [1], and the method proposed appears to just use this without taking advantage of modeling the full SCM and targeting a specific query (e.g. maintaining GP posterior also), which is the key novel component of this work.  \n\nFrom a presentation standpoint, the problem is well motivated and the prose is generally of very high-quality; I found the paper to be very pleasant to read. The description of Bayesian causal inference is thorough, perhaps the clearest that I have seen in the literature. However, in my view the paper goes too far in this direction; too much of the main paper is describing (the use of) existing work, without spending enough time highlighting the novel contributions, and perhaps even more importantly, justifying the choices being made (see Questions below). As a result, just reading the main paper, it’s difficult to tell what are the key novel insights.  \n\nOverall, I think the work is solid and a good practical contribution, but I am not entirely convinced that the novelty of the framework itself, or the analysis and insights thereof, is sufficiently strong.\n", "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": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"ethics_review_area": ""}, "extra_text": {}}, {"review_id": "-mc9FEf0D4", "reviewer": "Reviewer_LPpv", "summary": "This paper introduces a Bayesian approach to causality that combines causal discovery (of the causal graph and mechanisms) and inference (about a question of interest). They use GPs with additive noise for the mechanisms to be able to have analytics posteriors for them. They use the previously introduced DiBS framework to optimize over a soft version of the graph (using a continuous latent variable Z that conditions independent edge choices) using a variational method. They propose a generic method for handling different kinds of causal inference queries (e.g. to discover the graph vs to discover a particular edge or estimate a causal effect on some variable given some hypothetical interventino). An active learning Bayesian experimental design algorithm is proposed which exploits the estimated posteriors to estimate and optimize information gains wrt possible interventions.", "questions": "1. first sentence after eq 3.3 may be confusing for some readers, since Y may be a function of some condition or intervention; what the authors probably mean is that q(M) is a posterior that concentrates mass on a single distribution (whereas 3.3 is a mixture and P(Y|M) is not a Dirac).\n2. line 203: why do we need to distinguish between root nodes and non-root nodes? (see also next 2 questions)\n3. Eq 4.4: why can we factor p(f,sigma|G) for NR but not for R?\n4. line 212: why is this true for NR but not for R?\n5. sec 4.1: do we need to solve (i.e. O(n^2) time) for the GPs of all the nodes after each sample of graph G? is that the computational bottleneck?\n6. how does the overall algorithm complexity scale with the size of the graph and the number of examples? what are the computational bottlenecks in the overall procedure?\n", "limitations": "- see the above questions about computational complexity, which should be discussed in the limitations section too\n", "rating": 9, "confidence": 5, "soundness": 4, "presentation": 4, "contribution": 4, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "Pros:\n- The approach is very appealing because it is highly principled, based on a Bayesian framework that is important to \n- It is original in addressing together the causal discovery and the causal inference questions, something very important in because in general (especially for finite data and the generally unidentifable settings) there will be many more than one graph G that is consistent with the given data.\n\nCons:\n- the additive noise and GP assumptions are convenient but it would be good to think about getting rid of them (although this is clearly for future work, this paper is already quite dense)\n- the computational complexity of the proposed approach should be clarified (see below): the fact that experiments are done with d=8 variables and starting with n=5 examples and minibatches of size 3 suggests that computational scaling may be an issue that should be better covered in the limitations section.\n\n", "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": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"ethics_review_area": ""}, "extra_text": {}}]

Author rebuttal:

Decision: Accept