Original: Causal Discovery from Conditionally Stationary Time Series
Revised: Causal Discovery from Conditionally Stationary Time Series

Original: Abstract
Revised: Abstract

Original: Causal discovery, i.e., inferring underlying causal relationships from observational data, has been shown to be highly challenging for AI systems.
Revised: Causal discovery, i.e., inferring underlying causal relationships from observational data, has been shown to be highly challenging for AI systems.

Original: In time-series modeling context, existing causal discovery methods mainly consider constrained scenarios with fully observed variables and/or data from stationary time-series.
Revised: In time-series modeling context, existing causal discovery methods mainly consider constrained scenarios with fully observed variables and/or data from stationary time-series.

Original: We develop a causal discovery approach to a wide class of nonstationary time-series that are conditionally stationary , where the non-stationary behaviour is modeled as stationarity conditioned on a set of (possibly hidden) state variables whose dynamics may be dependent on the observed sequence.
Revised: We develop a causal discovery approach to a wide class of nonstationary time-series that are conditionally stationary , where the non-stationary behaviour is modeled as stationarity conditioned on a set of (possibly hidden) state variables whose dynamics may be dependent on the observed sequence.

Original: Named
Revised: Named

Original: State-Dependent Causal Inference (SDCI)
Revised: State-Dependent Causal Inference (SDCI)

Original: , our approach is able to recover the underlying causal dependencies, provably with fully-observed states and empirically with hidden states.
Revised: , our approach is able to recover the underlying causal dependencies, provably with fully-observed states and empirically with hidden states.

Original: The latter is confirmed by experiments on both synthetic linear system and spring-connected particle interaction data, where SDCI achieves superior performance over baseline causal discovery methods.
Revised: The latter is confirmed by experiments on both synthetic linear system and spring-connected particle interaction data, where SDCI achieves superior performance over baseline causal discovery methods.

Original:  INTRODUCTION
Revised:  INTRODUCTION

Original: Deep learning has achieved profound success in vision and language modelling tasks [Brown et al., 2020, Nichol et al., 2021].
Revised: Deep learning has achieved profound success in vision and language modelling tasks [Brown et al., 2020, Nichol et al., 2021].

Original: Still, it remains a grand challenge and a prominent research direction to enable deep neural networks to perform causal discovery and reasoning [Yi et al., 2020, Girdhar and Ramanan, 2020, Sauer and Geiger, 2021], which is an inherent mechanism in human cognition [Spelke and
Revised: Still, it remains a grand challenge and a prominent research direction to enable deep neural networks to perform causal discovery and reasoning [Yi et al., 2020, Girdhar and Ramanan, 2020, Sauer and Geiger, 2021], which is an inherent mechanism in human cognition [Spelke and

Original: Kinzler, 2007].
Revised: Kinzler, 2007].

Original: Specifically for analysing time series data, causal discovery involves identifying the underlying temporal causal structure of the observed sequences.
Revised: Specifically for analysing time series data, causal discovery involves identifying the underlying temporal causal structure of the observed sequences.

Original: Many existing causal discovery approaches for time series assume stationarity [Granger, 1969, Peters et al., 2017, Löwe et al., 2020, Li et al., 2020, Tank et al., 2021], which is restrictive as sequence data from real-world scenarios are often nonstationary with potential hidden confounders.
Revised: Many existing causal discovery approaches for time series assume stationarity [Granger, 1969, Peters et al., 2017, Löwe et al., 2020, Li et al., 2020, Tank et al., 2021], which is restrictive as sequence data from real-world scenarios are often nonstationary with potential hidden confounders.

Original: Recent works introduce a number of different assumptions to tackle causal discovery for non-stationary time series [Zhang et al., 2017,
Revised: Recent works introduce a number of different assumptions to tackle causal discovery for non-stationary time series [Zhang et al., 2017,

Original: Ghassami et al., 2018, Huang et al., 2019] but in general, causal discovery on non-stationary time series under mild and realistic assumptions is an open problem.
Revised: Ghassami et al., 2018, Huang et al., 2019], but in general, causal discovery on non-stationary time series under mild and realistic assumptions is an open problem.

Original: This work aims at addressing this open challenge by proposing a causal discovery algorithm for condionally stationary time series , for which the dynamics of the observed system change depend on a set of “state” variables.
Revised: This work aims at addressing this open challenge by proposing a causal discovery algorithm for condionally stationary time series , for which the dynamics of the observed system change depend on a set of “state” variables.

Original: This assumption holds for many real-world scenarios, e.g., with people who behave differently and take different decisions depending on underlying factors such as mood, previous experience, and the actions of other agents.
Revised: This assumption holds for many real-world scenarios, e.g., with people who behave differently and take different decisions depending on underlying factors such as mood, previous experience, and the actions of other agents.

Original: The causal discovery task for such conditionally stationary time series poses different challenges depending on the observability of the states, which is classified into 4 different scenarios:
Revised: The causal discovery task for such conditionally stationary time series poses different challenges depending on the observability of the states, which is classified into 4 different scenarios:

Original: Scenario class 1 concerns the simplest case, where the states are observed and their dynamics are independent on other observed time series data (Figure 1a).
Revised: Scenario class 1 concerns the simplest case, where the states are observed and their dynamics are independent on other observed time series data (Figure 1a).

Original: In Scenario class 2 , the states are unobserved and directly dependent on observed variables.
Revised: In Scenario class 2 , the states are unobserved and directly dependent on observed variables.

Original: One example is to consider an agent moving in a room where different behaviors are observed depending on their location.
Revised: One example is to consider an agent moving in a room where different behaviors are observed depending on their location.

Original: Figure 1b illustrates a similar setting.
Revised: Figure 1b illustrates a similar setting.

Original: Scenario class 3 is more challenging: the state depends on earlier events, and thus cannot be directly inferred from observations.
Revised: Scenario class 3 is more challenging: the state depends on earlier events, and thus cannot be directly inferred from observations.

Original: Figure 1c illustrates this regime with particles that change state on wall collision.
Revised: Figure 1c illustrates this regime with particles that change state on wall collision.

Original: Also in a football game the action of one player is triggered by an earlier action by another player.
Revised: Also in a football game the action of one player is triggered by an earlier action by another player.

Original: Finally, a large share of real-world scenarios (e.g., Figure 1d) are governed by underlying states that are not fully identifiable from the observations over time.
Revised: Finally, a large share of real-world scenarios (e.g., Figure 1d) are governed by underlying states that are not fully identifiable from the observations over time.

Original: Here the states can be unknown confounders to the observed time series, thus the causal discovery task is ill-defined.
Revised: Here the states can be unknown confounders to the observed time series, thus the causal discovery task is ill-defined.

Original: 1 1 2 2 0 0 0ss T 1 1 1 1 2 2 0 0 2 2 0 0 1 1 22 0 2 1 1 0 0 2 0 1 0 0 2 2 11 1 2 1 12 2 0 0 12 

Revised: 1 1 2 2 0 0 0ss T 1 1 1 1 2 2 0 0 2 2 0 0 1 1 22 0 2 1 1 0 0 2 0 1 0 0 2 2 11 1 2 1 12 2 0 0 12

Original: [Figure]
Revised: [Figure]

Original: [Figure]
Revised: [Figure]

Original: Our approach, named State-Dependent Causal Inference (SDCI), is based on discovering the summary graphs conditioned on states given observed sequences.
Revised: Our approach, named State-Dependent Causal Inference (SDCI), is based on discovering the summary graphs conditioned on states given observed sequences.

Original: It fits a graph neural network based variational auto-encoder [Löwe et al., 2020] to the non-stationary time series data, which enables efficient amortization for causal discovery across multiple observation sequences.
Revised: It fits a graph neural network based variational auto-encoder [Löwe et al., 2020] to the non-stationary time series data, which enables efficient amortization for causal discovery across multiple observation sequences.

Original: We prove identifiability results for cases with fully-observed states; empirically SDCI also applies to cases with hidden states, which is confirmed by experiments on both synthetic linear datasets and spring data (See Figures 1b & 1c), covering scenario classes 1-3.
Revised: We prove identifiability results for cases with fully-observed states; empirically SDCI also applies to cases with hidden states, which is confirmed by experiments on both synthetic linear datasets and spring data (See Figures 1b & 1c), covering scenario classes 1-3.

Original: Compared to baselines, SDCI achieves better accuracy in identifying the underlying causal graph structure as well as forecasting future trajectories from historical observations.
Revised: Compared to baselines, SDCI achieves better accuracy in identifying the underlying causal graph structure as well as forecasting future trajectories from historical observations.

Original:  RELATED WORK
Revised:  RELATED WORK

Original: Causal discovery aims to identify causal relationshipsfrom observational data [Glymour et al., 2019].
Revised: Causal discovery aims to identify causal relationships from observational data [Glymour et al., 2019].

Original: Constraint-based methods rely on conditional independence tests to recover the underlying DAG structure of the data.
Revised: Constraint-based methods rely on conditional independence tests to recover the underlying DAG structure of the data.

Original: Representative approaches include the PC algorithm [Spirtes et al., 2000] and Fast Causal Inference (FCI)
Revised: Representative approaches include the PC algorithm [Spirtes et al., 2000] and Fast Causal Inference (FCI)

Original: [Spirtes, 2001], and their extension to time series [Entner and Hoyer, 2010, Runge, 2018].
Revised: [Spirtes, 2001], and their extension to time series [Entner and Hoyer, 2010, Runge, 2018].

Original: Score-based methods, such as Greedy Equivalence
Revised: Score-based methods, such as Greedy Equivalence

Original: Search (GES)
Revised: Search (GES)

Original: [Chickering, 2002], define and optimize score functions of causal graphs to identify the underlying causal structure.
Revised: [Chickering, 2002], define and optimize score functions of causal graphs to identify the underlying causal structure.

Original: Regarding time series, these methods are reformulated as learning Dynamic Bayesian Networks (DBNs) from data [Murphy et al., 2002].
Revised: Regarding time series, these methods are reformulated as learning Dynamic Bayesian Networks (DBNs) from data [Murphy et al., 2002].

Original: A recent approach in this line is DYNOTEARS [Pamfil et al., 2020].
Revised: A recent approach in this line is DYNOTEARS [Pamfil et al., 2020].

Original: Functional causal model-based methods represent the effect as a function of its direct causes and their independent immeasurable noise [Shimizu et al., 2006, Zhang and Hyvärinen, 2009, Peters et al., 2014, Glymour et al., 2019].
Revised: Functional causal model-based methods represent the effect as a function of its direct causes and their independent immeasurable noise [Shimizu et al., 2006, Zhang and Hyvärinen, 2009, Peters et al., 2014, Glymour et al., 2019].

Original: For time series, these approaches fit a dynamic model, which is often constrained in terms of its functional form and connection sparsity in favor of identifiability guarantees [Peters et al., 2013].
Revised: For time series, these approaches fit a dynamic model, which is often constrained in terms of its functional form and connection sparsity in favor of identifiability guarantees [Peters et al., 2013].

Original: Our work is concerned in modelling non-stationary time series using state variables as entities responsible for changing the dynamics along the sequence.
Revised: Our work is concerned in modelling non-stationary time series using state variables as entities responsible for changing the dynamics along the sequence.

Original: Saggioro et al.
Revised: Saggioro et al.

Original: [2020] uses similar ideas for reconstructing regime-dependent dynamics.
Revised: [2020] uses similar ideas for reconstructing regime-dependent dynamics.

Original: Most relevant to ours is Amortized Causal Discovery (ACD)
Revised: Most relevant to ours is Amortized Causal Discovery (ACD)

Original: [Löwe et al., 2020], which assumes stationary time series and amortizes summary graph extraction process from samples with different graphs but shared dynamics.
Revised: [Löwe et al., 2020], which assumes stationary time series and amortizes summary graph extraction process from samples with different graphs but shared dynamics.

Original: Similar ideas are also proposed in Li et al.
Revised: Similar ideas are also proposed in Li et al.

Original: [2020] for video applications.
Revised: [2020] for video applications.

Original: We extend ACD by allowing the underlying causal structure to vary depending on some state variables.
Revised: We extend ACD by allowing the underlying causal structure to vary depending on some state variables.

Original: For other related works, Huang et al.
Revised: For other related works, Huang et al.

Original: [2015] extended Gaussian Process regression for the identification of time-varying functional causal models, Zhang et al.
Revised: [2015] extended Gaussian Process regression for the identification of time-varying functional causal models, Zhang et al.

Original: [2017] used kernel embeddings to detect distribution shifts in heterogeneous data.
Revised: [2017] used kernel embeddings to detect distribution shifts in heterogeneous data.

Original: Ghassami et al.
Revised: Ghassami et al.

Original: [2018] and Huang et al.
Revised: [2018] and Huang et al.

Original: [2019] estimated the time-varying causal effects.
Revised: [2019] estimated the time-varying causal effects.

Original: However, the latter two methods are limited by their linear causal model assumptions and fixed causal structures.
Revised: However, the latter two methods are limited by their linear causal model assumptions and fixed causal structures.

Original:  STATE-DEPENDENT CAUSAL INFERENCE (SDCI)
Revised:  STATE-DEPENDENT CAUSAL INFERENCE (SDCI)

Original: We introduce SDCI to extract causal graphs from time series data where their dynamics are altered by means of a set of categorical variables, referred to as their states.
Revised: We introduce SDCI to extract causal graphs from time series data where their dynamics are altered by means of a set of categorical variables, referred to as their states.

Original: 3.1 PROBLEM FORMULATION
Revised: 3.1 PROBLEM FORMULATION

Original: We consider a dataset D , where each sample X ∼ D consists of N non-stationary time series X
Revised: We consider a dataset D , where each sample X ∼ D consists of N non-stationary time series X

Original: = { x 1 , . . .
Revised: = { x 1 , . . .

Original: , x N } of length
Revised: , x N } of length

Original: T .
Revised: T .

Original: We denote element i at time-step t as x ti ∈ R d .
Revised: We denote element i at time-step t as x ti ∈ R d .

Original: Stationary time series.
Revised: Stationary time series.

Original: We assume that the data generation process obeys a structural causal model (SCM)
Revised: We assume that the data generation process obeys a structural causal model (SCM)

Original: [Pearl, 2009], where all the observed variables x ti ∈ V 1: T are its vertices, for each time series i at each time-step t , and that there are no instantaneous effects and no hidden confounders.
Revised: [Pearl, 2009], where all the observed variables x ti ∈ V 1: T are its vertices, for each time series i at each time-step t , and that there are no instantaneous effects and no hidden confounders.

Original: Moreover, same as the definitions of Granger causality [Granger, 1969], we assume that edges of a causal
Revised: Moreover, same as the definitions of Granger causality [Granger, 1969], we assume that edges of a causal

Original: [Figure]
Revised: [Figure]

Original: graph cannot go back in time.
Revised: graph cannot go back in time.

Original: For the sake of clarity, the underlying causal structure satisfies the first-order Markov property in this work; however, one could directly extend it to the higher order.
Revised: For the sake of clarity, the underlying causal structure satisfies the first-order Markov property in this work; however, one could directly extend it to the higher order.

Original: When considering causality in time series, the corresponding causal graph of a SCM is called full time graph [Peters et al., 2017], G 1: T .
Revised: When considering causality in time series, the corresponding causal graph of a SCM is called full time graph [Peters et al., 2017], G 1: T .

Original: Note that our treatment of stationarity refers to the structure of the causal mechanisms; i.e. edges are invariant in time.
Revised: Note that our treatment of stationarity refers to the structure of the causal mechanisms; i.e. edges are invariant in time.

Original: Assumptions similar to the previous ones have been introduced in the past for the analysis of causality considering stationary time series data [Löwe et al., 2020, Li et al., 2020], where the task is to extract the summary graph , G = {V , E} , where V = { x 1 , . . .
Revised: Assumptions similar to the previous ones have been introduced in the past for the analysis of causality considering stationary time series data [Löwe et al., 2020, Li et al., 2020], where the task is to extract the summary graph , G = {V , E} , where V = { x 1 , . . .

Original: , x N } and an edge from x i to x j is included in E if there exists 1 ≤ t < t ′ ≤ T such that there is an edge from x ti to x t ′ j in the full time graph.
Revised: , x N } and an edge from x i to x j is included in E if there exists 1 ≤ t < t ′ ≤ T such that there is an edge from x ti to x t ′ j in the full time graph.

Original: Both full time graph and summary graph are identifiable from observational data as shown by Peters et al.
Revised: Both full time graph and summary graph are identifiable from observational data as shown by Peters et al.

Original: [2013] if the generative process satisfies Time Series Models with Independent Noise (TiMINo).
Revised: [2013] if the generative process satisfies Time Series Models with Independent Noise (TiMINo).

Original: We further assume first-order Markov property and an additive noise model (ANM) without instantaneous effects for the generative process:
Revised: We further assume first-order Markov property and an additive noise model (ANM) without instantaneous effects for the generative process:

Original: [Equation]
Revised: [Equation]

Original: where PA 1 j ⊆ V denotes the parents of x j and ϵ tj denotes independent noise.
Revised: where PA 1 j ⊆ V denotes the parents of x j and ϵ tj denotes independent noise.

Original: The direct causes of x tj are PA ( x tj ) =
Revised: The direct causes of x tj are PA ( x tj ) =

Original: ( PA 1 j ) t − 1 ⊂ V 1: T .
Revised: ( PA 1 j ) t − 1 ⊂ V 1: T .

Original: As here the time series is stationary, we can regard this as first querying the summary graph G to extract the parents, and then using the correct temporal index ( t − 1 ) as a superscript of each parent ( PA 1 j ).
Revised: As here the time series is stationary, we can regard this as first querying the summary graph G to extract the parents, and then using the correct temporal index ( t − 1 ) as a superscript of each parent ( PA 1 j ).

Original: Conditionally stationary time series.
Revised: Conditionally stationary time series.

Original: For non-stationary time series considered in this work, we assume that at each time-step t we have access to state variables s t = { s t 1 , . . .
Revised: For non-stationary time series considered in this work, we assume that at each time-step t we have access to state variables s t = { s t 1 , . . .

Original: , s tN } , where s ti ∈ { 1 , ..., K } controls the causal effects of x ti to future variables x t +11: N .
Revised: , s tN } , where s ti ∈ { 1 , ..., K } controls the causal effects of x ti to future variables x t +11: N .

Original: In other words, when the state s ti changes, so do the causal effects for x ti .
Revised: In other words, when the state s ti changes, so do the causal effects for x ti .

Original: We further assume that the time series is stationary if the states are held constant along the sequence, i.e., s 1 = · · · = s T .
Revised: We further assume that the time series is stationary if the states are held constant along the sequence, i.e., s 1 = · · · = s T .

Original: For the general case where s t changes through time, even though the time series is non-stationary, it is conditionally stationary given the state variables S
Revised: For the general case where s t changes through time, even though the time series is non-stationary, it is conditionally stationary given the state variables S

Original: = { s 1 , . . .
Revised: = { s 1 , . . .

Original: , s T } .
Revised: , s T } .

Original: Figure 2a illustrates the full time graph of a conditionally stationary time series example following our assumptions.
Revised: Figure 2a illustrates the full time graph of a conditionally stationary time series example following our assumptions.

Original: In this case a variable affects different sets of future variables depending its state, and the time series is non-stationary as s 1 ̸
Revised: In this case a variable affects different sets of future variables depending its state, and the time series is non-stationary as s 1 ̸

Original: = s 2 ̸
Revised: = s 2 ̸

Original: = s 3 .
Revised: = s 3 .

Original: Note that the states are not included explicitly in the SCM of the generative process (otherwise the SCM requires additional variables with their own vertices for S ).
Revised: Note that the states are not included explicitly in the SCM of the generative process (otherwise the SCM requires additional variables with their own vertices for S ).

Original: This is a valid simplification since we assume access to the states at all times.
Revised: This is a valid simplification since we assume access to the states at all times.

Original: Our theoretical results presented below do not apply when considering hidden states, since one must then explicitly model its causal structure in the SCM.
Revised: Our theoretical results presented below do not apply when considering hidden states, since one must then explicitly model its causal structure in the SCM.

Original: Conditional summary graph.
Revised: Conditional summary graph.

Original: Our goal for causal discovery is to recover the full time graph , where for stationary time series this can be achieved by identifying the summary graph under the first-order Markov assumption.
Revised: Our goal for causal discovery is to recover the full time graph , where for stationary time series this can be achieved by identifying the summary graph under the first-order Markov assumption.

Original: However for conditionally stationary time series, the summary graph can be non-informative (probably close to a fully connected graph), due to different causal effects induced by variables in different states.
Revised: However for conditionally stationary time series, the summary graph can be non-informative (probably close to a fully connected graph), due to different causal effects induced by variables in different states.

Original: As an example, Figure 2c shows the summary graph extracted from the full time graph of Figure 2a.
Revised: As an example, Figure 2c shows the summary graph extracted from the full time graph of Figure 2a.

Original: Being dense, the summary graph as defined for stationary time series is less useful in non-stationary settings.
Revised: Being dense, the summary graph as defined for stationary time series is less useful in non-stationary settings.

Original: To address this issue, for conditionally stationary time series we define the conditional summary graph .
Revised: To address this issue, for conditionally stationary time series we define the conditional summary graph .

Original: Definition 1 (Conditional summary graph, first-order
Revised: Definition 1 (Conditional summary graph, first-order

Original: Markov setting) .
Revised: Markov setting) .

Original: Given a full time graph G 1: T , its conditional summary graph is a set of K summary graphs,
Revised: Given a full time graph G 1: T , its conditional summary graph is a set of K summary graphs,

Original: G 1: K = {G k : 1 ≤ k ≤ K } , where K is the number of possible state values.
Revised: G 1: K = {G k : 1 ≤ k ≤ K } , where K is the number of possible state values.

Original: Each summary graph G k =
Revised: Each summary graph G k =

Original: {V , E k } has the same vertices V = { x 1 , . . .
Revised: {V , E k } has the same vertices V = { x 1 , . . .

Original: , x N } , and an edge from x i to x j is added to E k if there exists a time-step 1 ≤ t ≤ T such that s ti = k
Revised: , x N } , and an edge from x i to x j is added to E k if there exists a time-step 1 ≤ t ≤ T such that s ti = k

Original: and in G 1: T , x ti is connected to x t +1 j .
Revised: and in G 1: T , x ti is connected to x t +1 j .

Original: Note that both summary and conditional summary graphs do not include auto-regressive connections x ti → x t +1 i .
Revised: Note that both summary and conditional summary graphs do not include auto-regressive connections x ti → x t +1 i .

Original: Figure 2b shows the conditional summary graph extracted from the full time graph of Figure 2a.
Revised: Figure 2b shows the conditional summary graph extracted from the full time graph of Figure 2a.

Original: For k = 1 we have s 13 = 1 and there is a “red edge” connecting x 

Revised: For k = 1 we have s 13 = 1 and there is a “red edge” connecting x

Original: and x 22 , therefore for Gthere is a “red edge” in the edge set E 1 .
Revised: and x 22 , therefore for Gthere is a “red edge” in the edge set E 1 .

Original: Similar reasoning applies for G 2 .
Revised: Similar reasoning applies for G 2 .

Original: Compared to the summary graph 2c, the conditional summary graph contains a compact, yet valid representation of the causal structure, and thus being more informative.
Revised: Compared to the summary graph 2c, the conditional summary graph contains a compact, yet valid representation of the causal structure, and thus being more informative.

Original: Although the conditional summary graph can present connections to some related work [Saggioro et al., 2020], ours is more flexible.
Revised: Although the conditional summary graph can present connections to some related work [Saggioro et al., 2020], ours is more flexible.

Original: In other words, we do not restrict the entire time series to be in one of the K states, but allow each variable to be in one of the K different ones.
Revised: In other words, we do not restrict the entire time series to be in one of the K states, but allow each variable to be in one of the K different ones.

Original: State-dependent TiMINo.
Revised: State-dependent TiMINo.

Original: We extend TiMINo to conditionally stationary time series with observed states and show the identifiability of both full time graph and conditional summary graph.
Revised: We extend TiMINo to conditionally stationary time series with observed states and show the identifiability of both full time graph and conditional summary graph.

Original: Consider the update on x tj as in Eq.
Revised: Consider the update on x tj as in Eq.

Original: First for non-stationary time series the direct causes of x tj ,
Revised: First for non-stationary time series the direct causes of x tj ,

Original: PA ( x tj ) , are no longer constant in time.
Revised: PA ( x tj ) , are no longer constant in time.

Original: However, for conditionally stationary time series satisfying first-order Markov property and our assumptions specified above, PA ( x tj ) is determined by the states at the previous time-step t − 1 .
Revised: However, for conditionally stationary time series satisfying first-order Markov property and our assumptions specified above, PA ( x tj ) is determined by the states at the previous time-step t − 1 .

Original: Thus we can write PA ( x tj ) =
Revised: Thus we can write PA ( x tj ) =

Original: ( PA 1 j | s t − 1 ) t − 1 and the statedependent TiMINo as (with first-order Markov assumption, an ANM model and no instantaneous effect):
Revised: ( PA 1 j | s t − 1 ) t − 1 and the statedependent TiMINo as (with first-order Markov assumption, an ANM model and no instantaneous effect):

Original: [Equation]
Revised: [Equation]

Original: where s t − 1 i ∈ { 1 , . .
Revised: where s t − 1 i ∈ { 1 , . .

Original: , K } and C i ( k ) ⊆
Revised: , K } and C i ( k ) ⊆

Original: V denotes the children of variable x
Revised: V denotes the children of variable x

Original: i when its associated state equals to k .
Revised: i when its associated state equals to k .

Original: To illustrate, in Figure 2a we have that PA ( x 22 ) = { x 11 , x 12 , x 13 } because x 2 ∈ C 1 ( s 11 ) and x 2 ∈ C 3 ( s 13 ) ; however PA ( x 32 ) =
Revised: To illustrate, in Figure 2a we have that PA ( x 22 ) = { x 11 , x 12 , x 13 } because x 2 ∈ C 1 ( s 11 ) and x 2 ∈ C 3 ( s 13 ) ; however PA ( x 32 ) =

Original: { x 2 1 , x 2 2 } because now x 2 ∈ C 1 ( s 2 1 ) but x 2 / ∈ C 3 ( s 2 3 ) .
Revised: { x 2 1 , x 2 2 } because now x 2 ∈ C 1 ( s 2 1 ) but x 2 / ∈ C 3 ( s 2 3 ) .

Original: Identifiability of state-dependent TiMINo.
Revised: Identifiability of state-dependent TiMINo.

Original: As we assume access to the state variables at all times, the original properties of Markov assumption and causal minimality are maintained, which are satisfied by TiMINo.
Revised: As we assume access to the state variables at all times, the original properties of Markov assumption and causal minimality are maintained, which are satisfied by TiMINo.

Original: Consequently, the conditional summary graph becomes identifiable by extending the identifiability proof of Peters et al.
Revised: Consequently, the conditional summary graph becomes identifiable by extending the identifiability proof of Peters et al.

Original: [2013] to our setting and further assuming that all the states are visited at least once.
Revised: [2013] to our setting and further assuming that all the states are visited at least once.

Original: Notice that failing to observe data corresponding to a particular state of one variable makes it practically impossible to capture the underlying effect of that variable conditioned on the unvisited state.
Revised: Notice that failing to observe data corresponding to a particular state of one variable makes it practically impossible to capture the underlying effect of that variable conditioned on the unvisited state.

Original: More details for the preservation of causal minimality and Markov condition in the state-dependent TiMINo, the identifiability of both the full time graph and conditional summary graph, and extensions to incorporate other time-lagged or instantaneous effects can be found in Appendix A and B.
Revised: More details for the preservation of causal minimality and Markov condition in the state-dependent TiMINo, the identifiability of both the full time graph and conditional summary graph, and extensions to incorporate other time-lagged or instantaneous effects can be found in Appendix A and B.

Original: State-dependent causal inference.
Revised: State-dependent causal inference.

Original: We mainly focus on non-stationary causal graphs which may have different edgetypes at different times.
Revised: We mainly focus on non-stationary causal graphs which may have different edgetypes at different times.

Original: Based on our assumptions, the interaction (i.e., edge-type) x
Revised: Based on our assumptions, the interaction (i.e., edge-type) x

Original: i −→ x j can change according to the state of the variable x i .
Revised: i −→ x j can change according to the state of the variable x i .

Original: Following Kipf et al.
Revised: Following Kipf et al.

Original: [2018], Li et al.
Revised: [2018], Li et al.

Original: [2020], Löwe et al.
Revised: [2020], Löwe et al.

Original: [2020], we consider this edge-type for x i −→ x j at time t as a categorical variable z tij ∈ { 0 , . .
Revised: [2020], we consider this edge-type for x i −→ x j at time t as a categorical variable z tij ∈ { 0 , . .

Original: . , n ϵ − 1 } which can represent n ϵ interaction types between pairs of variables.
Revised: . , n ϵ − 1 } which can represent n ϵ interaction types between pairs of variables.

Original: Specifically we use edgetype 0 to denote “no causal effect” between two variables.
Revised: Specifically we use edgetype 0 to denote “no causal effect” between two variables.

Original: Notice that the edge-type differs from the state: the former controls the functional form of the causal effect and the latter allows the variables to affect others differently along time.
Revised: Notice that the edge-type differs from the state: the former controls the functional form of the causal effect and the latter allows the variables to affect others differently along time.

Original: For example, one can have x i → x j whenever z tij ̸ = 0 , but the functional form of the causal relationship can differ.
Revised: For example, one can have x i → x j whenever z tij ̸ = 0 , but the functional form of the causal relationship can differ.

Original: The main focus of our method consists on extracting a conditional summary graph G 1: K (assuming there exist K states).
Revised: The main focus of our method consists on extracting a conditional summary graph G 1: K (assuming there exist K states).

Original: Previous approaches aiming for this task assume stationary time series data for which G 1 = · · · = G K .
Revised: Previous approaches aiming for this task assume stationary time series data for which G 1 = · · · = G K .

Original: For conditionally stationary time series, we extract the k -
Revised: For conditionally stationary time series, we extract the k -

Original: [Figure]
Revised: [Figure]

Original: th summary graph including edge-types, ˜ G k
Revised: th summary graph including edge-types, ˜ G k

Original: = {V , ˜ E k } , where ˜ E k and ˜ E k ′ can differ for k ̸ = k ′ .
Revised: = {V , ˜ E k } , where ˜ E k and ˜ E k ′ can differ for k ̸ = k ′ .

Original: We further define ˜ E k
Revised: We further define ˜ E k

Original: = { w ijk ∈ { 0 , . . . , n ϵ − 1 } : x i , x j ∈ V} as the collection of edge types for variable pairs in V .
Revised: = { w ijk ∈ { 0 , . . . , n ϵ − 1 } : x i , x j ∈ V} as the collection of edge types for variable pairs in V .

Original: Note here that an edge is visualised in the conditional summary graph visualisation only when w ijk ̸ = 

Revised: Note here that an edge is visualised in the conditional summary graph visualisation only when w ijk ̸ =

Original: (see Figure 2b).
Revised: (see Figure 2b).

Original: Then the edge-type interaction x i → x j can be queried at each time-step t as follows:
Revised: Then the edge-type interaction x i → x j can be queried at each time-step t as follows:

Original: [Equation]
Revised: [Equation]

Original: Therefore, causal discovery for conditionally stationary time series requires extracting the (unknown) conditional summary graph given observations of X (and perhaps also S ).
Revised: Therefore, causal discovery for conditionally stationary time series requires extracting the (unknown) conditional summary graph given observations of X (and perhaps also S ).

Original: It not only requires to design a parametrizable function to infer the causal structure, but also to evaluate how this inference fits to the input observations.
Revised: It not only requires to design a parametrizable function to infer the causal structure, but also to evaluate how this inference fits to the input observations.

Original: 3.2 IMPLEMENTATION
Revised: 3.2 IMPLEMENTATION

Original: We introduce a probabilistic approach which models the distribution of the edge-types { z tij } given observed data.
Revised: We introduce a probabilistic approach which models the distribution of the edge-types { z tij } given observed data.

Original: Following our assumptions for conditionally stationary time series, this task can be solved by learning the distribution of W = { w ijk : 1 ≤ i, j ≤ N, 1 ≤ k ≤ K } given data, as after inferring W one can then query the edge-types z tij as in Eq. 4.
Revised: Following our assumptions for conditionally stationary time series, this task can be solved by learning the distribution of W = { w ijk : 1 ≤ i, j ≤ N, 1 ≤ k ≤ K } given data, as after inferring W one can then query the edge-types z tij as in Eq. 4.

Original: Inspired by previous approaches [Li et al., 2020,
Revised: Inspired by previous approaches [Li et al., 2020,

Original: Löwe et al., 2020, Kipf et al., 2018], our implementation is based on a variational auto-encoder (VAE)
Revised: Löwe et al., 2020, Kipf et al., 2018], our implementation is based on a variational auto-encoder (VAE)

Original: [Kingma and
Revised: [Kingma and

Original: Welling, 2014] and graph neural networks.
Revised: Welling, 2014] and graph neural networks.

Original: We first discuss our approach for the case with fully observed states, then extend the method to the hidden state regime.
Revised: We first discuss our approach for the case with fully observed states, then extend the method to the hidden state regime.

Original: A diagram of the proposed approach is visualised in Figure 3.
Revised: A diagram of the proposed approach is visualised in Figure 3.

Original: Encoder.
Revised: Encoder.

Original: Similar to Löwe et al.
Revised: Similar to Löwe et al.

Original: [2020], we use a factorized q distribution
Revised: [2020], we use a factorized q distribution

Original: [Equation]
Revised: [Equation]

Original: The encoder receives both X and S as the input, and extracts an embedding that represents the causal interaction conditioning on the state for every possible edge x i
Revised: The encoder receives both X and S as the input, and extracts an embedding that represents the causal interaction conditioning on the state for every possible edge x i

Original: [Equation]
Revised: [Equation]

Original: where f ϕ ( X , S ) is a neural network that returns outputs { ϕ ij : 1 ≤ i, j ≤ N } given the input as X concatenated with a one-hot representation of the state variable S .
Revised: where f ϕ ( X , S ) is a neural network that returns outputs { ϕ ij : 1 ≤ i, j ≤ N } given the input as X concatenated with a one-hot representation of the state variable S .

Original: The approximate posterior q ϕ ( w ijk | X , S ) is calculated as
Revised: The approximate posterior q ϕ ( w ijk | X , S ) is calculated as

Original: [Equation]
Revised: [Equation]

Original: where Θ( · ) denotes a softmax activation with temperature τ .
Revised: where Θ( · ) denotes a softmax activation with temperature τ .

Original: Note here the softmax activation is taken over the k -th row vector ϕ ijk ∈ R n ϵ in ϕ ij , which is the edge embedding for x i −→ x j at state k .
Revised: Note here the softmax activation is taken over the k -th row vector ϕ ijk ∈ R n ϵ in ϕ ij , which is the edge embedding for x i −→ x j at state k .

Original: As this returns a categorical distribution for q ϕ ( w ijk | X , S ) , we apply the Gumble-softmax trick [Maddison et al., 2017, Jang et al., 2016] to enable direct differentiation for back-propagation during training.
Revised: As this returns a categorical distribution for q ϕ ( w ijk | X , S ) , we apply the Gumble-softmax trick [Maddison et al., 2017, Jang et al., 2016] to enable direct differentiation for back-propagation during training.

Original: The construction of f ϕ ( X , S ) is based on graph neural networks, MLPs and/or CNNs.
Revised: The construction of f ϕ ( X , S ) is based on graph neural networks, MLPs and/or CNNs.

Original: We experiment with different architecture designs, and in the main text we focus on a particular one named SDCI-Static which uses full-sequence embedding in the message passing procedure.
Revised: We experiment with different architecture designs, and in the main text we focus on a particular one named SDCI-Static which uses full-sequence embedding in the message passing procedure.

Original: See Appendix
Revised: See Appendix

Original: D.1 for more details and additional results.
Revised: D.1 for more details and additional results.

Original: Decoder for X. The decoder for the observations X given edge-types W (e.g., sampled as w ijk ∼ q ϕ ( w ijk | X , S ) ) and the states S are defined following the first-order Markov assumption (with x 0 , s 0 as dummy variables):
Revised: Decoder for X. The decoder for the observations X given edge-types W (e.g., sampled as w ijk ∼ q ϕ ( w ijk | X , S ) ) and the states S are defined following the first-order Markov assumption (with x 0 , s 0 as dummy variables):

Original: [Equation]
Revised: [Equation]

Original: The mean ˜ x t +1 j is defined as follows.
Revised: The mean ˜ x t +1 j is defined as follows.

Original: Given sampled edgetypes W = { w ijk } , the decoder first queries the edge-type for element j at time t + 1 as z tij = w ijk ′ for s ti = k ′
Revised: Given sampled edgetypes W = { w ijk } , the decoder first queries the edge-type for element j at time t + 1 as z tij = w ijk ′ for s ti = k ′

Original: (also see Eq. 4).
Revised: (also see Eq. 4).

Original: Then the information along the predicted edge-type interactions is retrieved as follows
Revised: Then the information along the predicted edge-type interactions is retrieved as follows

Original: [Equation]
Revised: [Equation]

Original: where { f e } n ϵ − 1 e =1 is a set of parametrizable functions, one defined for each edge type excluding the no-edge interaction.
Revised: where { f e } n ϵ − 1 e =1 is a set of parametrizable functions, one defined for each edge type excluding the no-edge interaction.

Original: The interactions are finally integrated to model the dynamics of each variable, where the mean for x t +1 j is defined as
Revised: The interactions are finally integrated to model the dynamics of each variable, where the mean for x t +1 j is defined as

Original: [Equation]
Revised: [Equation]

Original: where f p is a neural network that aggregates the information from the previous time-step for prediction.
Revised: where f p is a neural network that aggregates the information from the previous time-step for prediction.

Original: Training objective.
Revised: Training objective.

Original: The encoder q ϕ ( W | X , S ) and decoder p ψ ( X | W , S ) are trained using a VAE objective:
Revised: The encoder q ϕ ( W | X , S ) and decoder p ψ ( X | W , S ) are trained using a VAE objective:

Original: [Equation]
Revised: [Equation]

Original: We used a factorised prior p ( W ) =
Revised: We used a factorised prior p ( W ) =

Original: (cid:81) K k
Revised: (cid:81) K k

Original: =1 (cid:81) ij p k ( w ijk ) which acts as a regularizer over the inferred edge-type distribution q ϕ .
Revised: =1 (cid:81) ij p k ( w ijk ) which acts as a regularizer over the inferred edge-type distribution q ϕ .

Original: In our experiments we set this prior to be
Revised: In our experiments we set this prior to be

Original: Uniform ( { 0 , . . . , n ϵ − 1 } ) , although in certain applications it may be useful to use different p k to encourage different sparsity levels for G k .
Revised: Uniform ( { 0 , . . . , n ϵ − 1 } ) , although in certain applications it may be useful to use different p k to encourage different sparsity levels for G k .

Original: Our method can also be extended to capture the state dynamics and also consider hidden states, despite the latter losing identifiability guarantees due to violations of our assumption.
Revised: Our method can also be extended to capture the state dynamics and also consider hidden states, despite the latter losing identifiability guarantees due to violations of our assumption.

Original: We provide the corresponding details in Appendix C.
Revised: We provide the corresponding details in Appendix C.

Original:  EXPERIMENTS
Revised:  EXPERIMENTS

Original: We evaluate SDCI on two non-stationary time series synthetic datasets.
Revised: We evaluate SDCI on two non-stationary time series synthetic datasets.

Original: Our results are compared to Amortized
Revised: Our results are compared to Amortized

Original: Causal Discovery (ACD)
Revised: Causal Discovery (ACD)

Original: [Löwe et al., 2020], TdFCM Huang et al.
Revised: [Löwe et al., 2020], TdCM Huang et al.

Original: [2015], CD-NOD [Zhang et al., 2017], and SAEM [Huang et al., 2019].
Revised: [2015], CD-NOD [Zhang et al., 2017], and SAEM [Huang et al., 2019].

Original: 4.1 EXPERIMENTS ON LINEAR DATA
Revised: 4.1 EXPERIMENTS ON LINEAR DATA

Original: We start using linear message passing operations between a number of different time series.
Revised: We start using linear message passing operations between a number of different time series.

Original: Conditioned on the states, the variables x ti , x t +1 j ∈ R are connected by an edge of n ϵ different types.
Revised: Conditioned on the states, the variables x ti , x t +1 j ∈ R are connected by an edge of n ϵ different types.

Original: Each edge-type is captured by the linear coefficients { β k ∈ R } n ϵ − 1 k =0 with the convention that β 0 =(as edge-type 0 represents no connection).
Revised: Each edge-type is captured by the linear coefficients { β k ∈ R } n ϵ − 1 k =0 with the convention that β 0 =(as edge-type 0 represents no connection).

Original: Notice that the effect of x ti on x t +1 j changes during time according to the state value s ti and the underlying conditional summary graph
Revised: Notice that the effect of x ti on x t +1 j changes during time according to the state value s ti and the underlying conditional summary graph

Original: G 1: K .
Revised: G 1: K .

Original: Details of the data generation, training specifications, and additional experiments can be found in Appendices F.1,
Revised: Details of the data generation, training specifications, and additional experiments can be found in Appendices F.1,

Original: D.2, and G.1 respectively.
Revised: D.2, and G.1 respectively.

Original: Scenario class 2.
Revised: Scenario class 2.

Original: We consider simulations following scenario class 2 (see Figure 1b), where the states are hidden from the input data (only X is given).
Revised: We consider simulations following scenario class 2 (see Figure 1b), where the states are hidden from the input data (only X is given).

Original: This enables comparisons with TdFCM, CD-NOS and SAEM which perform causal discovery based on X only.
Revised: This enables comparisons with TdCM, CD-NOS and SAEM which perform causal discovery based on X only.

Original: As these two baseline methods consider constant causal connections (i.e., a single summary graph) model the non-stationary behaviour differently, for a fair comparison we evaluate the identification accuracy of the summary graph in a way such that it only considers the existence of an edge rather than capturing the edge type.
Revised: As these two baseline methods consider constant causal connections (i.e., a single summary graph) model the non-stationary behaviour differently, for a fair comparison we evaluate the identification accuracy of the summary graph in a way such that it only considers the existence of an edge rather than capturing the edge type.

Original: We simulate N = 3 variables with 2 edge-types (2- EDGE ), 3 edge-types with constant summary graphs (3EDGE CONST ) and 3 edge-types with no constraints on the conditional summary graph (3- EDGE FREE ).
Revised: We simulate N = 3 variables with 2 edge-types (2- EDGE ), 3 edge-types with constant summary graphs (3EDGE CONST ) and 3 edge-types with no constraints on the conditional summary graph (3- EDGE FREE ).

Original: We report the (conditional) summary graph identification accuracies in Table 1.
Revised: We report the (conditional) summary graph identification accuracies in Table 1.

Original: It is clear that SDCI performs the best in extracting both the summary graph and conditional summary graph of the underlying data distribution.
Revised: It is clear that SDCI performs the best in extracting both the summary graph and conditional summary graph of the underlying data distribution.

Original: Since
Revised: Since

Original: ACD relies on stationary time series, it achieves comparable results when considering a constant causal graph.
Revised: ACD relies on stationary time series, it achieves comparable results when considering a constant causal graph.

Original: On the other hand, TdFCM, CD-NOD and SAEM perform significantly worse, as they are designed for non-stationary time series with distribution shifts and smooth time-varying coefficients, which are not suited for time series data with discrete changes in its dynamics.
Revised: On the other hand, TdCM, CD-NOD and SAEM perform significantly worse, as they are designed for non-stationary time series with distribution shifts and smooth time-varying coefficients, which are not suited for time series data with discrete changes in its dynamics.

Original: Furthermore, as compared with TdFCM, CD-NOD and SAEM which conduct causal
Revised: Furthermore, as compared with TdCM, CD-NOD and SAEM which conduct causal

Original: [Table]
Revised: [Table]

Original: discovery on each multivariate time series separately, both
Revised: discovery on each multivariate time series separately, both

Original: ACD and SDCI benefit from amortized causal graph inference which utilises shared information across different multivariate time series.
Revised: ACD and SDCI benefit from amortized causal graph inference which utilises shared information across different multivariate time series.

Original: For the 2 edge-types setting, we visualise an example time series forecasting result of SDCI and ACD given T =observed time-steps in Figure 4, and the associated causal graph identification results in Figure 5.
Revised: For the 2 edge-types setting, we visualise an example time series forecasting result of SDCI and ACD given T =observed time-steps in Figure 4, and the associated causal graph identification results in Figure 5.

Original: In this particular case, SDCI correctly extracts the conditional summary graph, agreeing with the high CSG accuracy reported in
Revised: In this particular case, SDCI correctly extracts the conditional summary graph, agreeing with the high CSG accuracy reported in

Original: Table 1.
Revised: Table 1.

Original: In contrast, ACD extracts a single causal graph (which is incorrect), and the forecasts move to 0 rapidly.
Revised: In contrast, ACD extracts a single causal graph (which is incorrect), and the forecasts move to 0 rapidly.

Original: Close inspection shows that ACD estimates α < 1 , which in this case without external effects or cycles in the summary graph, it provokes all elements dropping to 0 eventually.
Revised: Close inspection shows that ACD estimates α < 1 , which in this case without external effects or cycles in the summary graph, it provokes all elements dropping to 0 eventually.

Original: [Figure]
Revised: [Figure]

Original: [Table]
Revised: [Table]

Original: 4.2 EXPERIMENTS ON SPRING DATA
Revised: 4.2 EXPERIMENTS ON SPRING DATA

Original: We evaluate our method on the synthetic spring data adapted from Kipf et al.
Revised: We evaluate our method on the synthetic spring data adapted from Kipf et al.

Original: [2018], Löwe et al.
Revised: [2018], Löwe et al.

Original: [2020], which consists of particles connected by springs with directed impactmeaning that e.g. particle i could affect particle j with a force through a connecting spring, but leaving particle i unaffected by this spring force.
Revised: [2020], which consists of particles connected by springs with directed impactmeaning that e.g. particle i could affect particle j with a force through a connecting spring, but leaving particle i unaffected by this spring force.

Original: We consider  2 edge-types (presence/absence of directed spring) and scenario classas presented in Section 1.
Revised: We consider N = 5 particles, 2 edge-types (presence/absence of directed spring) and scenario class 1 as presented in Section 1.

Original: Details of data generation, model hyper-parameters, and results with other scenario classes can be found in Appendices F.2, D.2 and G.2 respectively.
Revised: Details of data generation, model hyper-parameters, and results with other scenario classes can be found in Appendices F.2, D.2 and G.2 respectively.

Original: Scenario class 1.
Revised: Scenario class 1.

Original: In this experiment the states are known and their dynamics are independent from the observations.
Revised: In this experiment the states are known and their dynamics are independent from the observations.

Original: For the ground truth dynamics, the state transitions incrementally into the next one every 10 time-steps.
Revised: For the ground truth dynamics, the state transitions incrementally into the next one every 10 time-steps.

Original: Table 2 shows the corresponding results, where we experiment with datasets generated with different number of states
Revised: Table 2 shows the corresponding results, where we experiment with datasets generated with different number of states

Original: K = 1 , 2 , 3 , 5 , 8 .
Revised: K = 1 , 2 , 3 , 5 , 8 .

Original: Note that when K = 1 the generated time series is stationary, and SDCI’s performance matches the results of ACD reported in Löwe et al.
Revised: Note that when K = 1 the generated time series is stationary, and SDCI’s performance matches the results of ACD reported in Löwe et al.

Original: Although performance drops as K increases, SDCI-Static is able to maintain reasonable CSG accuracy , e.g., 74 . 87% for K = 8 .
Revised: Although performance drops as K increases, SDCI-Static is able to maintain reasonable CSG accuracy , e.g., 74 . 87% for K = 8 .

Original:  CONCLUSIONS
Revised:  CONCLUSIONS

Original: We have extended the causal discovery task to a class of non-stationary time series named conditionally stationary time series.
Revised: We have extended the causal discovery task to a class of non-stationary time series named conditionally stationary time series.

Original: Key to our development is the state-dependent
Revised: Key to our development is the state-dependent

Original: TiMINo as an extension of [Peters et al., 2013] to conditionally stationary time series, and its identifiability under observed states.
Revised: TiMINo as an extension of [Peters et al., 2013] to conditionally stationary time series, and its identifiability under observed states.

Original: We have developed SDCI for amortized causal discovery utilizing the conditional summary graph, and experiments on both synthetic linear and spring data under different scenario classes show better performance in extracting the underlying causal graph and forecasting.
Revised: We have developed SDCI for amortized causal discovery utilizing the conditional summary graph, and experiments on both synthetic linear and spring data under different scenario classes show better performance in extracting the underlying causal graph and forecasting.

Original: A number of research directions are to be explored in the future.
Revised: A number of research directions are to be explored in the future.

Original: For theoretical studies, identifiability for hidden states can be derived with additional assumptions.
Revised: For theoretical studies, identifiability for hidden states can be derived with additional assumptions.

Original: For practical applications, SDCI can be applied to time series with states as auxiliary variables for (possibly) indicating non-stationarity regimes.
Revised: For practical applications, SDCI can be applied to time series with states as auxiliary variables for (possibly) indicating non-stationarity regimes.

Original: It could also be extended to videos where e.g. the interacting objects are partially and noisily observed as semantically segmented regions.
Revised: It could also be extended to videos where e.g. the interacting objects are partially and noisily observed as semantically segmented regions.

Original: This would enable efficient and effective use of neural networks for causal reasoning in challenging real-world scenarios.
Revised: This would enable efficient and effective use of neural networks for causal reasoning in challenging real-world scenarios.

Original: A PROOFS
Revised: A PROOFS

Original: This section presents the detailed proofs for the theoretical results of our paper.
Revised: This section presents the detailed proofs for the theoretical results of our paper.

Original: Again we note here that the results only apply when the states S are fully observed.
Revised: Again we note here that the results only apply when the states S are fully observed.

Original: Our theory extend on the theoretical results presented in Peters et al.
Revised: Our theory extend on the theoretical results presented in Peters et al.

Original: [2013] and Peters et al.
Revised: [2013] and Peters et al.

Original: [2017], which are presented first before our proofs.
Revised: [2017], which are presented first before our proofs.

Original: Proposition 1 (Prop. 6.36 in Peters et al.
Revised: Proposition 1 (Prop. 6.36 in Peters et al.

Original: Causal minimality is satisfied if and only if ∀ x j , ∀ y ∈ PA ( x j ) we have that x j ̸⊥⊥ y | PA ( x j ) \{ y }
Revised: Causal minimality is satisfied if and only if ∀ x j , ∀ y ∈ PA ( x j ) we have that x j ̸⊥⊥ y | PA ( x j ) \{ y }

Original: Proposition 2 (Prop. 7.4 in Peters et al.
Revised: Proposition 2 (Prop. 7.4 in Peters et al.

Original: Consider a distribution given by the following additive noise model
Revised: Consider a distribution given by the following additive noise model

Original: [Equation]
Revised: [Equation]

Original: if the functions f j are not constant in any of their arguments, then the joint distribution satisfies causal minimality with respect to the corresponding graph.
Revised: if the functions f j are not constant in any of their arguments, then the joint distribution satisfies causal minimality with respect to the corresponding graph.

Original: Recall from the main text that state-dependent TiMINo is defined as (with first-order Markov assumption, additive noise model, and no instantaneous effect):
Revised: Recall from the main text that state-dependent TiMINo is defined as (with first-order Markov assumption, additive noise model, and no instantaneous effect):

Original: [Equation]
Revised: [Equation]

Original: Below we derive the identifiability results for statedependent TiMINo given fully observed states S .
Revised: Below we derive the identifiability results for statedependent TiMINo given fully observed states S .

Original: A.1 STATE-DEPENDENT TIMINO PRESERVES MARKOV CONDITION
Revised: A.1 STATE-DEPENDENT TIMINO PRESERVES MARKOV CONDITION

Original: We show that the state-dependent TiMINo is Markov.
Revised: We show that the state-dependent TiMINo is Markov.

Original: Lemma 1.
Revised: Lemma 1.

Original: Given the states S , if X is generated by a statedependent TiMINo, then each variable is conditionally independent of each non-descendants given its parents.
Revised: Given the states S , if X is generated by a statedependent TiMINo, then each variable is conditionally independent of each non-descendants given its parents.

Original: Proof.
Revised: Proof.

Original: As the states S are given, we can retrieve the direct causes of x tj :
Revised: As the states S are given, we can retrieve the direct causes of x tj :

Original: [Equation]
Revised: [Equation]

Original: and use state-dependent TiMINo to compute x tj given its parents:
Revised: and use state-dependent TiMINo to compute x tj given its parents:

Original: [Equation]
Revised: [Equation]

Original: Given its parents P, variable x tj is therefore independent of its non-descendants.
Revised: Given its parents P, variable x tj is therefore independent of its non-descendants.

Original: A.2 STATE-DEPENDENT TIMINO PRESERVES CAUSAL MINIMALITY
Revised: A.2 STATE-DEPENDENT TIMINO PRESERVES CAUSAL MINIMALITY

Original: We show that the state-dependent TiMINo model satisfies causal minimality given observed states.
Revised: We show that the state-dependent TiMINo model satisfies causal minimality given observed states.

Original: Lemma 2.
Revised: Lemma 2.

Original: Assume all the functions f j in a state-dependent
Revised: Assume all the functions f j in a state-dependent

Original: TiMINo are not constant in any of their arguments.
Revised: TiMINo are not constant in any of their arguments.

Original: Then causal minimality is preserved given observed states.
Revised: Then causal minimality is preserved given observed states.

Original: Proof.
Revised: Proof.

Original: Again as the states S are observed, then for any x tj the direct causes of it can be retrieved by:
Revised: Again as the states S are observed, then for any x tj the direct causes of it can be retrieved by:

Original: [Equation]
Revised: [Equation]

Original: Assume causal minimality is not satisfied, then following
Revised: Assume causal minimality is not satisfied, then following

Original: Proposition 1, there exists x tj and x t − 1 i ∈ P such that
Revised: Proposition 1, there exists x tj and x t − 1 i ∈ P such that

Original: [Equation]
Revised: [Equation]

Original: Denote P \{ x t − 1 i } as P A .
Revised: Denote P \{ x t − 1 i } as P A .

Original: Then, if E [ ϵ tj ] = 0 , there exists a function c ( · ) such that
Revised: Then, if E [ ϵ tj ] = 0 , there exists a function c ( · ) such that

Original: [Equation]
Revised: [Equation]

Original: This implies that f s t
Revised: This implies that f s t

Original: [Equation]
Revised: [Equation]

Original: is constant with respect to x t − 1 i .
Revised: is constant with respect to x t − 1 i .

Original: It contradicts with the assumption that f s t − 1 j is not constant in any of its arguments.
Revised: It contradicts with the assumption that f s t − 1 j is not constant in any of its arguments.

Original: Therefore, causal minimality in statedependent TiMINo is preserved given observed states.
Revised: Therefore, causal minimality in statedependent TiMINo is preserved given observed states.

Original: A.3 IDENTIFIABILITY IN STATE-DEPENDENT TIMINO
Revised: A.3 IDENTIFIABILITY IN STATE-DEPENDENT TIMINO

Original: Using the Markov and causal minimality results, we derived the following identifiability result for state-dependent
Revised: Using the Markov and causal minimality results, we derived the following identifiability result for state-dependent

Original: TiMINo, and the proof is inspired by Peters et al.
Revised: TiMINo, and the proof is inspired by Peters et al.

Original: Theorem 1.
Revised: Theorem 1.

Original: Consider the state-dependent TiMINo, where only first-order Markov interactions are present, and there is no instantaneous effects.
Revised: Consider the state-dependent TiMINo, where only first-order Markov interactions are present, and there is no instantaneous effects.

Original: Then the full time graph G 1: T is identifiable from the distribution of data given states.
Revised: Then the full time graph G 1: T is identifiable from the distribution of data given states.

Original: Proof.
Revised: Proof.

Original: Conditioned on S , assume that x t can be computed from two state-dependent TiMINo with two different full time graphs, G 1: Tand G 1: T 2 .
Revised: Conditioned on S , assume that x t can be computed from two state-dependent TiMINo with two different full time graphs, G 1: Tand G 1: T 2 .

Original: Suppose there is an edge x t −
Revised: Suppose there is an edge x t −

Original: i −→ x tj
Revised: i −→ x tj

Original: which is in
Revised: which is in

Original: G 1: T 1 , but
Revised: G : T 1 , but

Original: not
Revised: not

Original: in G 1: T 2 . 1.
Revised: in G 1: T 2 . 1.

Original: From G 1: Tand the Markov condition (Lemma 1), we have x tj ⊥⊥ x t − 1 i
Revised: From G 1: Tand the Markov condition (Lemma 1), we have x tj ⊥⊥ x t − 1 i

Original: [Equation]
Revised: [Equation]

Original: From G 1: Tand causal minimality (Lemma 2), we have
Revised: From G 1: Tand causal minimality (Lemma 2), we have

Original: [Equation]
Revised: [Equation]

Original: Since we have a contradiction, the full time graphs G 1: Tand G 1:
Revised: Since we have a contradiction, the full time graphs G 1: Tand G 1:

Original: Tmust be equal.
Revised: Tmust be equal.

Original: Therefore, the full time graph is identifiable from the distribution of data given states.
Revised: Therefore, the full time graph is identifiable from the distribution of data given states.

Original: Given the identifiability results, in theory, we can estimate the state-dependent TiMiNo in Eq.
Revised: Given the identifiability results, in theory, we can estimate the state-dependent TiMiNo in Eq.

Original: 2 with maximum likelihood [Zhang et al., 2015].
Revised: 2 with maximum likelihood [Zhang et al., 2015].

Original: In our implementation, we train the model with the modified VAE objective 19.
Revised: In our implementation, we train the model with the modified VAE objective 19.

Original: The empirical results show that it infers causal structures correctly; however, it is nontrivial to prove that the identifiability holds with the modified VAE objective, which has been discussed in [Geffner et al., 2022] as well and will be our future work.
Revised: The empirical results show that it infers causal structures correctly; however, it is nontrivial to prove that the identifiability holds with the modified VAE objective, which has been discussed in [Geffner et al., 2022] as well and will be our future work.

Original: A.4 IDENTIFIABILITY OF CONDITIONAL SUMMARY GRAPH
Revised: A.4 IDENTIFIABILITY OF CONDITIONAL SUMMARY GRAPH

Original: Assuming the full time graph has been extracted successfully, one can deduce the structure of the conditional summary graph by observing the edges of pairs of variables conditioned on the state variable which is the cause of the interaction.
Revised: Assuming the full time graph has been extracted successfully, one can deduce the structure of the conditional summary graph by observing the edges of pairs of variables conditioned on the state variable which is the cause of the interaction.

Original: In particular, the precise mathematical statement for this identifiability result is as follows, assuming that all the states of each element have been visited at least once in order to condition on each possible state.
Revised: In particular, the precise mathematical statement for this identifiability result is as follows, assuming that all the states of each element have been visited at least once in order to condition on each possible state.

Original: Corollary 1.
Revised: Corollary 1.

Original: Consider the state-dependent TiMINo, where only first-order Markov interactions are present, and there is no instantaneous effects.
Revised: Consider the state-dependent TiMINo, where only first-order Markov interactions are present, and there is no instantaneous effects.

Original: Then the conditional summary graph G 1: K is identifiable given the states S , if for any 1 ≤ i ≤ N and any 1 ≤ k ≤ K there exists 1 ≤ t ≤ T such that s ti = k .
Revised: Then the conditional summary graph G 1: K is identifiable given the states S , if for any 1 ≤ i ≤ N and any 1 ≤ k ≤ K there exists 1 ≤ t ≤ T such that s ti = k .

Original: Proof.
Revised: Proof.

Original: Notice that G k = {V , E k } with V = { x 1 , . . .
Revised: Notice that G k = {V , E k } with V = { x 1 , . . .

Original: , x N } , therefore identifying G k is equivalent to identifying the edge set E k .
Revised: , x N } , therefore identifying G k is equivalent to identifying the edge set E k .

Original: From the assumptions, the full time graph G 1: T is identifiable (Theorem 1).
Revised: From the assumptions, the full time graph G 1: T is identifiable (Theorem 1).

Original: Then for each variable x t +1 j we can retrieve from G 1: T the parents PA ( x t +1 j ) .
Revised: Then for each variable x t +1 j we can retrieve from G 1: T the parents PA ( x t +1 j ) .

Original: Then for each x ti ∈ PA ( x t +1 j ) , we can query its state s ti and add an edge x i → x j to the edge set E k if s ti = k .
Revised: Then for each x ti ∈ PA ( x t +1 j ) , we can query its state s ti and add an edge x i → x j to the edge set E k if s ti = k .

Original: Now for any i, k consider the i -th node in the k -th summary graph G k , since we assume that there exists 1 ≤ t ≤ T such that s ti
Revised: Now for any i, k consider the i -th node in the k -th summary graph G k , since we assume that there exists 1 ≤ t ≤ T such that s ti

Original: =
Revised: =

Original: k , this makes sure that the edges coming out of node x i at state k are added to E k .
Revised: k , this makes sure that the edges coming out of node x i at state k are added to E k .

Original: Therefore this procedure captures all possible edge interactions, and G 1: K is identifiable given identifiability of G 1: T .
Revised: Therefore this procedure captures all possible edge interactions, and G 1: K is identifiable given identifiability of G 1: T .

Original: Notice that the assumption on each variable visiting all possible states is required if one aims to obtain the full structure of the conditional summary graph.
Revised: Notice that the assumption on each variable visiting all possible states is required if one aims to obtain the full structure of the conditional summary graph.

Original: Otherwise, if the state k of a variable x i is not visited, this leaves the outgoing edges of the i -th node in G k undetermined, since there is no information to extract from the full time graph.
Revised: Otherwise, if the state k of a variable x i is not visited, this leaves the outgoing edges of the i -th node in G k undetermined, since there is no information to extract from the full time graph.

Original: B IDENTIFIABILITY EXTENSIONS FOR STATE-DEPENDENT TIMINO
Revised: B IDENTIFIABILITY EXTENSIONS FOR STATE-DEPENDENT TIMINO

Original: The first-order Markov assumption makes our definitions and identifiability analysis simpler.
Revised: The first-order Markov assumption makes our definitions and identifiability analysis simpler.

Original: However, we argue this can still be very useful because (i) it usually suffices to describe particle trajectories (position, velocity, acceleration, etc), (ii) it allows us to present the conditional summary graph as a compact representation of the causal structure, and (iii) we present a method (SDCI) which leverages this assumption for efficient causal discovery.
Revised: However, we argue this can still be very useful because (i) it usually suffices to describe particle trajectories (position, velocity, acceleration, etc), (ii) it allows us to present the conditional summary graph as a compact representation of the causal structure, and (iii) we present a method (SDCI) which leverages this assumption for efficient causal discovery.

Original: One could easily extend our identifiability results and definitions to higher order time-lagged effects.
Revised: One could easily extend our identifiability results and definitions to higher order time-lagged effects.

Original: Incorporating contemporaneous effects is also possible, but requires further assumptions to consider an identifiable functional model class (IFMOC)
Revised: Incorporating contemporaneous effects is also possible, but requires further assumptions to consider an identifiable functional model class (IFMOC)

Original: [Peters et al., 2011]
Revised: [Peters et al., 2011]

Original: (e.g. linear f s t − 1 j with non-Gaussian disturbances).
Revised: (e.g. linear f s t − 1 j with non-Gaussian disturbances).

Original: C EXTENSIONS OF SDCI C.1 STATE DEPENDENT ON OBJECT DYNAMICS
Revised: C EXTENSIONS OF SDCI C.1 STATE DEPENDENT ON OBJECT DYNAMICS

Original: Although we assume access to the states at all times and thus we require no supervision upon S , we might find practical to consider the setting where the state is in fact dependent on
Revised: Although we assume access to the states at all times and thus we require no supervision upon S , we might find practical to consider the setting where the state is in fact dependent on

Original: X .
Revised: X .

Original: Therefore, we extend our approach to model the temporal dynamics of S by defining the joint distribution p ( X , S | W ) as follows, again following the first-order Markov assumption:
Revised: Therefore, we extend our approach to model the temporal dynamics of S by defining the joint distribution p ( X , S | W ) as follows, again following the first-order Markov assumption:

Original: [Equation]
Revised: [Equation]

Original: The distribution p ψ ( x t +1 | x t , s t , W ) is defined as above.
Revised: The distribution p ψ ( x t +1 | x t , s t , W ) is defined as above.

Original: For the variable s t +1 j , we use a categorical distribution p ψ
Revised: For the variable s t +1 j , we use a categorical distribution p ψ

Original: [Equation]
Revised: [Equation]

Original: t
Revised: t

Original: [Equation]
Revised: [Equation]

Original: 17) where the logit ˜ s t +1 j ∈ R n ϵ is computed as
Revised: 17) where the logit ˜ s t +1 j ∈ R n ϵ is computed as

Original: [Equation]
Revised: [Equation]

Original: with f s as a neural network that combines information from previous observation, states and the predicted edge-type interactions to predict the next state.
Revised: with f s as a neural network that combines information from previous observation, states and the predicted edge-type interactions to predict the next state.

Original: Objective (with S observed).
Revised: Objective (with S observed).

Original: The encoder q ϕ ( W | X , S ) and the decoder & state dynamic model p ψ ( X , S | W ) are trained using a modified VAE objective:
Revised: The encoder q ϕ ( W | X , S ) and the decoder & state dynamic model p ψ ( X , S | W ) are trained using a modified VAE objective:

Original: [Equation]
Revised: [Equation]

Original: [Figure]
Revised: [Figure]

Original: where we use λ to balance the accuracy in learning the decoder for X and the state dynamics.
Revised: where we use λ to balance the accuracy in learning the decoder for X and the state dynamics.

Original: Here we used a factorised prior p ( W ) =
Revised: Here we used a factorised prior p ( W ) =

Original: (cid:81)
Revised: (cid:81)

Original: Kk
Revised: Kk

Original: =1 (cid:81) ij p k ( w ijk ) which acts as a regularizer over the inferred edge-type distribution q .
Revised: =1 (cid:81) ij p k ( w ijk ) which acts as a regularizer over the inferred edge-type distribution q .

Original: In our experiments we set this prior to be Uniform ( { 0 , . . . , n ϵ − 1 } ) , although in certain applications it may be useful to use different p k to encourage different sparsity levels for G k .
Revised: In our experiments we set this prior to be Uniform ( { 0 , . . . , n ϵ − 1 } ) , although in certain applications it may be useful to use different p k to encourage different sparsity levels for G k .

Original: C.2 HIDDEN STATE REGIME
Revised: C.2 HIDDEN STATE REGIME

Original: Often in practice only observations of X are available, i.e., the states S are hidden variables.
Revised: Often in practice only observations of X are available, i.e., the states S are hidden variables.

Original: In this case we make a factorised approximation q ϕ ( W , S | X ) =
Revised: In this case we make a factorised approximation q ϕ ( W , S | X ) =

Original: q ϕ ( W | X ) q ϕ
Revised: q ϕ ( W | X ) q ϕ

Original: ( S | X ) , where q ϕ ( W | X ) is defined in a similar way as the state fullyobserved case (see above) except for using network f ϕ ( X ) for the softmax logits.
Revised: ( S | X ) , where q ϕ ( W | X ) is defined in a similar way as the state fullyobserved case (see above) except for using network f ϕ ( X ) for the softmax logits.

Original: For the states, we consider q ϕ
Revised: For the states, we consider q ϕ

Original: ( S | X ) = (cid:81)
Revised: ( S | X ) = (cid:81)

Original: Tt =1 (cid:81) Ni =1 q ϕ ( s ti | x ti ) , with each q ϕ ( s ti | x ti ) defined as
Revised: Tt =1 (cid:81) Ni =1 q ϕ ( s ti | x ti ) , with each q ϕ ( s ti | x ti ) defined as

Original: [Equation]
Revised: [Equation]

Original: with γ < 1 as a temperature factor and ˆ f s as a neural network.
Revised: with γ < 1 as a temperature factor and ˆ f s as a neural network.

Original: For the decoding process given W ∼ q ϕ
Revised: For the decoding process given W ∼ q ϕ

Original: ( W | X ) , we do not explicitly perform the edge query step for z tij as it needs samples s tij from q ϕ ( S | X ) .
Revised: ( W | X ) , we do not explicitly perform the edge query step for z tij as it needs samples s tij from q ϕ ( S | X ) .

Original: Instead we directly compute the h tij information for the predicted interaction x i → x j at time t by marginalising out s ti :
Revised: Instead we directly compute the h tij information for the predicted interaction x i → x j at time t by marginalising out s ti :

Original: [Equation]
Revised: [Equation]

Original: and we apply similar Gumbel-softmax tricks as in the fullyobserved state case to replace 1 ( w ijk = e ) during training.
Revised: and we apply similar Gumbel-softmax tricks as in the fullyobserved state case to replace 1 ( w ijk = e ) during training.

Original: The VAE training objective in this case is similar to Eq.except that the expectations of the log-likelihood terms are taken under distribution q ϕ ( W , S | X ) , and the KL regulariser is replaced by KL (cid:0) q ϕ
Revised: The VAE training objective in this case is similar to Eq.except that the expectations of the log-likelihood terms are taken under distribution q ϕ ( W , S | X ) , and the KL regulariser is replaced by KL (cid:0) q ϕ

Original: ( W | X ) || p ( W ) (cid:1) .
Revised: ( W | X ) || p ( W ) (cid:1) .

Original: We do not include the entropy H [ q ϕ ( S | X )] in the objective as one would have done in sequential VAEs: as S are discrete, the entropy is always non-negative, meaning that removing it still results in a valid variational lower-bound.
Revised: We do not include the entropy H [ q ϕ ( S | X )] in the objective as one would have done in sequential VAEs: as S are discrete, the entropy is always non-negative, meaning that removing it still results in a valid variational lower-bound.

Original: Note that in this hidden state regime the previous theoretical guarantees on identifiability no longer hold due to violations of our assumptions.
Revised: Note that in this hidden state regime the previous theoretical guarantees on identifiability no longer hold due to violations of our assumptions.

Original: However, this setting can still be useful as a showcase for analysis of non-stationary time series and possible directions of future work.
Revised: However, this setting can still be useful as a showcase for analysis of non-stationary time series and possible directions of future work.

Original: D IMPLEMENTATION DETAILS
Revised: D IMPLEMENTATION DETAILS

Original: All the experiments are implemented in Pytorch [Paszke et al., 2019] and carried out on NVIDIA RTX 2080Ti GPUs.
Revised: All the experiments are implemented in Pytorch [Paszke et al., 2019] and carried out on NVIDIA RTX 2080Ti GPUs.

Original: D.1 ENCODER ARCHITECTURE
Revised: D.1 ENCODER ARCHITECTURE

Original: Below we provide details of the encoder architectures.
Revised: Below we provide details of the encoder architectures.

Original: SDCI-Static encoder The first design of the architecture extends directly from ACD [Löwe et al., 2020] and we refer to this model as SDCI-Static.
Revised: SDCI-Static encoder The first design of the architecture extends directly from ACD [Löwe et al., 2020] and we refer to this model as SDCI-Static.

Original: We discuss the network construction for hidden state case as an example, for which the logits ϕ ij for the distribution q ϕ
Revised: We discuss the network construction for hidden state case as an example, for which the logits ϕ ij for the distribution q ϕ

Original: ( W | X ) are obtained as follows.
Revised: ( W | X ) are obtained as follows.

Original: First, the model computes a latent embedding for each node i using the whole sequence:
Revised: First, the model computes a latent embedding for each node i using the whole sequence:

Original: [Equation]
Revised: [Equation]

Original: Then each embedding is updated using a graph neural network (GNN) that captures the correlations between nodes.
Revised: Then each embedding is updated using a graph neural network (GNN) that captures the correlations between nodes.

Original: Specifically the message passing procedure follows the two equations below:
Revised: Specifically the message passing procedure follows the two equations below:

Original: [Equation]
Revised: [Equation]

Original: t t+
Revised: t t+

Original: Pairwise concatenation
Revised: Pairwise concatenation

Original: Aggregate
Revised: Aggregate

Original: MLP + Aggr Input
Revised: MLP + Aggr Input

Original: PairwiseEmbedding t t+ t t+
Revised: PairwiseEmbedding t t+ t t+

Original: Embedding
Revised: Embedding

Original: Edge type distribution
Revised: Edge type distribution

Original: Aggregation
Revised: Aggregation

Original: TConsecutive frame grouping
Revised: TConsecutive frame grouping

Original: Finally, we obtain the softmax logit ϕ ij ∈ R K × n ϵ for every possible edge x i → x j and every possible state 1 ≤ k ≤ K :
Revised: Finally, we obtain the softmax logit ϕ ij ∈ R K × n ϵ for every possible edge x i → x j and every possible state 1 ≤ k ≤ K :

Original: [Equation]
Revised: [Equation]

Original: The above network architecture design is visualised in Figure 6.
Revised: The above network architecture design is visualised in Figure 6.

Original: according to equation 6.
Revised: according to equation 6.

Original: The details of the architecture settings follows the design in Löwe et al.
Revised: The details of the architecture settings follows the design in Löwe et al.

Original: Each embedding step f ϕ i uses two-layers of 256 dimensions and
Revised: Each embedding step f ϕ i uses two-layers of 256 dimensions and

Original: ELU [Clevert et al., 2016] activations followed by a batch normalization.
Revised: ELU [Clevert et al., 2016] activations followed by a batch normalization.

Original: f ϕ 4 uses skip connections and we modify its output size to generate a pairwise embedding for each of the
Revised: f ϕ 4 uses skip connections and we modify its output size to generate a pairwise embedding for each of the

Original: K states.
Revised: K states.

Original: For fully-observed state case, the architecture for q ϕ ( W | X , S ) follows a similar structure, except that for the first layer we use h 1 i = f ϕ 

Revised: For fully-observed state case, the architecture for q ϕ ( W | X , S ) follows a similar structure, except that for the first layer we use h 1 i = f ϕ

Original: ( concat ( x 1: T i , s 1: T i )) , where s 1: T i is a set of one-hot vectors representing the states { s ti } Tt =1 .
Revised: ( concat ( x 1: T i , s 1: T i )) , where s 1: T i is a set of one-hot vectors representing the states { s ti } Tt =1 .

Original: SDCI-Temporal encoder To preserve the temporal ordering information in the encoder, we propose another encoder design, named SDCI-Temporal, which pays attention to capturing the first-order dynamical structure in the data.
Revised: SDCI-Temporal encoder To preserve the temporal ordering information in the encoder, we propose another encoder design, named SDCI-Temporal, which pays attention to capturing the first-order dynamical structure in the data.

Original: Again using the hidden state case as an example, the encoder transformations are defined as:
Revised: Again using the hidden state case as an example, the encoder transformations are defined as:

Original: [Equation]
Revised: [Equation]

Original: Figure 7 shows the structure of the SDCI-Temporal encoder.
Revised: Figure 7 shows the structure of the SDCI-Temporal encoder.

Original: We no longer use the whole sequence at first, but concatenate consecutive frames and set it as the input to f ϕ 1 , we perform this computation from 1 to
Revised: We no longer use the whole sequence at first, but concatenate consecutive frames and set it as the input to f ϕ 1 , we perform this computation from 1 to

Original: T − 1 time-steps.
Revised: T − 1 time-steps.

Original: In the case of fully observed states, x ti and x t +1 i are replaced by concat ( x ti , one-hot ( s ti )) and concat ( x t
Revised: In the case of fully observed states, x ti and x t +1 i are replaced by concat ( x ti , one-hot ( s ti )) and concat ( x t

Original: +1 i , one-hot ( s t +1 i )) respectively as the inputs to f ϕ 1 .
Revised: +1 i , one-hot ( s t +1 i )) respectively as the inputs to f ϕ 1 .

Original: All the subsequent steps, except for the aggregator f ϕ aggr have the same structure as the SDCI-Static encoder.
Revised: All the subsequent steps, except for the aggregator f ϕ aggr have the same structure as the SDCI-Static encoder.

Original: The design of the aggregator f ϕ aggr , which aims to summarize the temporal correlations captured throughout the whole sequence, is crucial to the performance of SDCI-Temporal encoder.
Revised: The design of the aggregator f ϕ aggr , which aims to summarize the temporal correlations captured throughout the whole sequence, is crucial to the performance of SDCI-Temporal encoder.

Original: Preliminary empirical results showed that with an
Revised: Preliminary empirical results showed that with an

Original: MLP f ϕ aggr , SDCI-Temporal performs poorly for inferring causal structures from data.
Revised: MLP f ϕ aggr , SDCI-Temporal performs poorly for inferring causal structures from data.

Original: Instead we proposed a 1D CNN for aggregation, which reported better results.
Revised: Instead we proposed a 1D CNN for aggregation, which reported better results.

Original: In detail, for the main text experiments, f ϕ aggr consists of two-layer 1D
Revised: In detail, for the main text experiments, f ϕ aggr consists of two-layer 1D

Original: CNN of 256 filters of window size 5 plus a maxpool operation for the final aggregation.
Revised: CNN of 256 filters of window size 5 plus a maxpool operation for the final aggregation.

Original: Future work towards designing better aggregation schemes might consider attentive pooling [Lin et al., 2017], or simply perform an average pool.
Revised: Future work towards designing better aggregation schemes might consider attentive pooling [Lin et al., 2017], or simply perform an average pool.

Original: D.2 TRAINING SPECIFICATIONS
Revised: D.2 TRAINING SPECIFICATIONS

Original: All the models participating in the experiments of this section have been trained using the following training scheme, including ACD [Löwe et al., 2020].
Revised: All the models participating in the experiments of this section have been trained using the following training scheme, including ACD [Löwe et al., 2020].

Original: Customized decoder (linear data only)
Revised: Customized decoder (linear data only)

Original: One of our objectives in the linear data experiments is to recover the underlying world parameters { β k } n ϵ − 1 k =1 .
Revised: One of our objectives in the linear data experiments is to recover the underlying world parameters { β k } n ϵ − 1 k =1 .

Original: Therefore the decoder design imitates the message passing operation presented in
Revised: Therefore the decoder design imitates the message passing operation presented in

Original: Eq. 31, which allows us to initialize the decoder using the underlying world parameters and analyse the performance of the encoder as a separate entity from the whole model.
Revised: Eq. 31, which allows us to initialize the decoder using the underlying world parameters and analyse the performance of the encoder as a separate entity from the whole model.

Original: Hyper-parameters Following Kipf et al.
Revised: Hyper-parameters Following Kipf et al.

Original: [2018], the models are trained using ADAM optimizer [Kingma and Ba,
Revised: [2018], the models are trained using ADAM optimizer [Kingma and Ba,

Original: The learning rate of the encoder is 5 · 10 − 4 , the learning rate of the decoder is 1 · 10 − 3 for the synthetic linear data experiments and 5 · 10 − 4 for spring data experiments.
Revised: The learning rate of the encoder is 5 · 10 − 4 , the learning rate of the decoder is 1 · 10 − 3 for the synthetic linear data experiments and 5 · 10 − 4 for spring data experiments.

Original: Learning rate decay is in use with factor of 0.5 everyepochs.
Revised: Learning rate decay is in use with factor of 0.5 everyepochs.

Original: We train for 1000 epochs in the linear experiments and 500 epochs in the springs experiments, using a batch size of 128.
Revised: We train for 1000 epochs in the linear experiments and 500 epochs in the springs experiments, using a batch size of 128.

Original: The decoder is trained with teacher forcing every 10 time-steps, i.e., it receives the ground-truth as input every 10 time-steps.
Revised: The decoder is trained with teacher forcing every 10 time-steps, i.e., it receives the ground-truth as input every 10 time-steps.

Original: The temperature τ is set to 0 .
Revised: The temperature τ is set to 0 .

Original: 5 and the variance of the Gaussian distribution of the decoder is σ 2 = 5 · 10 − 5 .
Revised: 5 and the variance of the Gaussian distribution of the decoder is σ 2 = 5 · 10 − 5 .

Original: When considering the setting where we make the state dependent on the dynamics of the objects (scenarios 2 and 3), we set λ = 10 3 .
Revised: When considering the setting where we make the state dependent on the dynamics of the objects (scenarios 2 and 3), we set λ = 10 3 .

Original: For inferring the hidden states, we set the temperature γ = 0 .
Revised: For inferring the hidden states, we set the temperature γ = 0 .

Original: 1 in the linear experiments and γ = 0 .
Revised: 1 in the linear experiments and γ = 0 .

Original: in the spring data experiments.
Revised: in the spring data experiments.

Original: E EVALUATION E.1 SUMMARY GRAPH IN TDFCM
Revised: E EVALUATION E.1 SUMMARY GRAPH IN TDCM

Original: Huang et al.
Revised: Huang et al.

Original: [2015] do not specify explicitly the computation of the summary graph (or an equivalent object).
Revised: [2015] do not specify explicitly the computation of the summary graph (or an equivalent object).

Original: To allow a fair comparison, we take the same approach as in Huang et al.
Revised: To allow a fair comparison, we take the same approach as in Huang et al.

Original: [2019], where and edge from i to j is not incorporated in the summary graph if the corresponding estimated timevarying coefficient has mean and variance lower than a theshold.
Revised: [2019], where and edge from i to j is not incorporated in the summary graph if the corresponding estimated timevarying coefficient has mean and variance lower than a theshold.

Original: E.2 ACCURACY OF THE SUMMARY GRAPH
Revised: E.2 ACCURACY OF THE SUMMARY GRAPH

Original: To clarify, the evaluation of the summary graph considers correctly classifying the interaction between all pairs of nodes in both directions.
Revised: To clarify, the evaluation of the summary graph considers correctly classifying the interaction between all pairs of nodes in both directions.

Original: To exemplify, see Figure 5b.
Revised: To exemplify, see Figure 5b.

Original: In this case, we have N=3 variables, which means that our method needs to produce 6 predictions (one for every pair of edges in both directions).
Revised: In this case, we have N=3 variables, which means that our method needs to produce 6 predictions (one for every pair of edges in both directions).

Original: We then compute the accuracy of these predictions and average across all the samples in the test set.
Revised: We then compute the accuracy of these predictions and average across all the samples in the test set.

Original: E.3 COMPUTING THE SUMMARY GRAPH IN SDCI
Revised: E.3 COMPUTING THE SUMMARY GRAPH IN SDCI

Original: Notice that SDCI can extract the conditional summary graph (CSG) whereas the baselines we compare with only consider the summary graph (SG).
Revised: Notice that SDCI can extract the conditional summary graph (CSG) whereas the baselines we compare with only consider the summary graph (SG).

Original: Consequently, the only immediate way to compare the performance in capturing the causal structure among the methods we consider is to evaluate the latter.
Revised: Consequently, the only immediate way to compare the performance in capturing the causal structure among the methods we consider is to evaluate the latter.

Original: From the definition of summary graph, we deduce that one can estimate it by taking the union of the graphs in the CSG.
Revised: From the definition of summary graph, we deduce that one can estimate it by taking the union of the graphs in the CSG.

Original: This is used to compute the summary graphs of both SDCI and the ground truth structure of the generative process.
Revised: This is used to compute the summary graphs of both SDCI and the ground truth structure of the generative process.

Original: F DATASETS
Revised: F DATASETS

Original: In this section we provide detailed information about the datasets used in this work.
Revised: In this section we provide detailed information about the datasets used in this work.

Original: We generate 50000 samples of each setting for training the models.
Revised: We generate 50000 samples of each setting for training the models.

Original: Regarding testing, we compute all the metrics using 10000 samples, except when comparing with CD-NOD and SAEM,  where we usesamples since they require retraining the whole model for each sample.
Revised: Regarding testing, we compute all the metrics using 10000 samples, except when comparing with CD-NOD, SAEM, and TdCM, where we use 200 samples since they require retraining the whole model for each sample.

Original: However, notice that for both baselines we use sequences of much longer length (e.g., T = 100 or T = 1000 ) as these methods rely on longer sequences to get accurate results.
Revised: Moreover, notice that for these baselines we use sequences of much longer length (e.g., T =or T = 1000 ) as these methods rely on longer sequences to get accurate results.

Original: F.1 LINEAR DATA
Revised: F.1 LINEAR DATA

Original: The ground-truth structural equation for the data generation process at time t is
Revised: The ground-truth structural equation for the data generation process at time t is

Original: [Equation]
Revised: [Equation]

Original: where α ∈ R controls the self-connection, and ϵ ti denotes independent noise sampled at each time-step.
Revised: where α ∈ R controls the self-connection, and ϵ ti denotes independent noise sampled at each time-step.

Original: Below we provide details of the data generation process for the linear data.
Revised: Below we provide details of the data generation process for the linear data.

Original: First, we set the edge-type interactions.
Revised: First, we set the edge-type interactions.

Original: In our experiments we set α = 0 .
Revised: In our experiments we set α = 0 .

Original: 9 , β 1 = 0 .
Revised: 9 , β 1 = 0 .

Original: 5 , and β 2 = − 0 .
Revised: 5 , and β 2 = − 0 .

Original: 5 and ϵ ti ∼ U ( − 0 .
Revised: 5 and ϵ ti ∼ U ( − 0 .

Original: To generate each sample, we need to sample the initial values of the continuous variable for each element, x 0 i ∼ N (0 , 2) , and the underlying causal structure dependent on the state, G 1: K .
Revised: To generate each sample, we need to sample the initial values of the continuous variable for each element, x 0 i ∼ N (0 , 2) , and the underlying causal structure dependent on the state, G 1: K .

Original: At each time-step, it suffices to query the edge-type k for each pair of variables and apply the corresponding causal effect β k following Equation 31.
Revised: At each time-step, it suffices to query the edge-type k for each pair of variables and apply the corresponding causal effect β k following Equation 31.

Original: The edge-type is k = (cid:16) E s tj (cid:17) ji , where ( E s ) ji denotes the causal effect from j to i at state s , which has been defined at the beginning of the sequence.
Revised: The edge-type is k = (cid:16) E s tj (cid:17) ji , where ( E s ) ji denotes the causal effect from j to i at state s , which has been defined at the beginning of the sequence.

Original: For all our experiments with this dataset, we simulate N = 3 variables.
Revised: For all our experiments with this dataset, we simulate N = 3 variables.

Original: When considering hidden states in scenario class 2, we set s ti = 1 ( | x ti | > 2) (2 states).
Revised: When considering hidden states in scenario class 2, we set s ti = 1 ( | x ti | > 2) (2 states).

Original: There are potential concerns that the generated samples produced in the linear data may be unstable.
Revised: There are potential concerns that the generated samples produced in the linear data may be unstable.

Original: Still we use this data for one of the evaluations with the following reasons.
Revised: Still we use this data for one of the evaluations with the following reasons.

Original: First, they define a simulated environment where one has the ground truth and can debug and control simulation errors with ease.
Revised: First, they define a simulated environment where one has the ground truth and can debug and control simulation errors with ease.

Original: Furthermore, for one-dimensional variables x i ∈ R (which is our case), this dataset reduces to a first order
Revised: Furthermore, for one-dimensional variables x i ∈ R (which is our case), this dataset reduces to a first order

Original: Vector Autoregressive (VAR) model
Revised: Vector Autoregressive (VAR) model

Original: [Sims, 1980], which is widely used in works related to causal discovery for time series data [Gong et al., 2015].
Revised: [Sims, 1980], which is widely used in works related to causal discovery for time series data [Gong et al., 2015].

Original: The evolution of a sequence in this case can be expressed as follows:
Revised: The evolution of a sequence in this case can be expressed as follows:

Original: [Equation]
Revised: [Equation]

Original: where A is the causal transition matrix and e t is an independent noise process.
Revised: where A is the causal transition matrix and e t is an independent noise process.

Original: Regarding stability, the samples in this dataset are described by a causal transition matrix A where the diagonal elements are α and the off-diagonal elements are β k where k is the edge-type interaction.
Revised: Regarding stability, the samples in this dataset are described by a causal transition matrix A where the diagonal elements are α and the off-diagonal elements are β k where k is the edge-type interaction.

Original: For a first-order VAR to be stable, the singular values of A need to be smaller than one.
Revised: For a first-order VAR to be stable, the singular values of A need to be smaller than one.

Original: Taking into the account that each sample can obey a different underlying causal graph, one needs to check this condition for all the possible arrangements of the off-diagonal elements (since the diagonal elements are always α ).
Revised: Taking into the account that each sample can obey a different underlying causal graph, one needs to check this condition for all the possible arrangements of the off-diagonal elements (since the diagonal elements are always α ).

Original: The number of matrices that one needs to check grows rapidly for increasing number of variables, which makes the verification of this condition computationally infeasible (recall that computing the eigenvalues of a matrix has cubic cost O ( N 3 ) ).
Revised: The number of matrices that one needs to check grows rapidly for increasing number of variables, which makes the verification of this condition computationally infeasible (recall that computing the eigenvalues of a matrix has cubic cost O ( N 3 ) ).

Original: In practice, we generate random samples and keep them if the magnitude of the last observation is comparable to the initial one.
Revised: In practice, we generate random samples and keep them if the magnitude of the last observation is comparable to the initial one.

Original: We also require that the states are visited in similar proportions to satisfy our assumptions on the data.
Revised: We also require that the states are visited in similar proportions to satisfy our assumptions on the data.

Original: F.2 SPRING DATA
Revised: F.2 SPRING DATA

Original: When considering springs with directed connections, we follow the generation procedure described Kipf et al.
Revised: When considering springs with directed connections, we follow the generation procedure described Kipf et al.

Original: [2018] with a small modification where the spring interaction between a pair of particles can change over time (depending on the state).
Revised: [2018] with a small modification where the spring interaction between a pair of particles can change over time (depending on the state).

Original: In this dataset, N particles are simulated inside a 2D box where they can collide elastically with its walls.
Revised: In this dataset, N particles are simulated inside a 2D box where they can collide elastically with its walls.

Original: Each pair of variables is connected by a spring with uniform probability.
Revised: Each pair of variables is connected by a spring with uniform probability.

Original: To allow for identification of causal connections (directed edges), the connection is made unidirectional.
Revised: To allow for identification of causal connections (directed edges), the connection is made unidirectional.

Original: The springs interact via the Hooke’s law and this setting yield the following equations:
Revised: The springs interact via the Hooke’s law and this setting yield the following equations:

Original: [Equation]
Revised: [Equation]

Original: where f ij is the unidirectional interaction from particle j to particle i , δ k denotes the edge-type for each pair of variables, and r i and ˙
Revised: where f ij is the unidirectional interaction from particle j to particle i , δ k denotes the edge-type for each pair of variables, and r i and ˙

Original: r
Revised: r

Original: i denote the 2D position and velocity of each particle.
Revised: i denote the 2D position and velocity of each particle.

Original: The continuous variable x
Revised: The continuous variable x

Original: i is constructed by concatenating the position and the velocity measurements.
Revised: i is constructed by concatenating the position and the velocity measurements.

Original: Notice that the above equation defines the evolution of the continuous variable for a single time-step.
Revised: Notice that the above equation defines the evolution of the continuous variable for a single time-step.

Original: In our setting, we have that k
Revised: In our setting, we have that k

Original: = (cid:16) E s tj (cid:17) ji .
Revised: = (cid:16) E s tj (cid:17) ji .

Original: Thus, f ij will change over time, contrary to Kipf et al.
Revised: Thus, f ij will change over time, contrary to Kipf et al.

Original: Since we consider two edgetypes, we define δ 0 = 

Revised: Since we consider two edgetypes, we define δ 0 =

Original: and δ 1 = 0 . 1 .
Revised: and δ 1 = 0 . 1 .

Original: To generate samples, we first generate a random conditional summary graph G 1: K and the initial location and velocity.
Revised: To generate samples, we first generate a random conditional summary graph G 1: K and the initial location and velocity.

Original: Then, trajectories are simulated by solving the previous differential equations using leapfrog integration.
Revised: Then, trajectories are simulated by solving the previous differential equations using leapfrog integration.

Original: The step size used is 0.001 and the trajectories are obtained by sub-sampling each 100 steps.
Revised: The step size used is 0.001 and the trajectories are obtained by sub-sampling each 100 steps.

Original: In our experiments, we set N = 5 and T = 80 .
Revised: In our experiments, we set N = 5 and T = 80 .

Original: When considering hidden states in scenario class 2, we set s ti = 1 ( x ti > 0) (2 states).
Revised: When considering hidden states in scenario class 2, we set s ti = 1 ( x ti > 0) (2 states).

Original: [Table]
Revised: [Table]

Original: G ADDITIONAL RESULTS
Revised: G ADDITIONAL RESULTS

Original: In this section we report additional experiments and qualitative visualisations, which can be helpful to complement the main results from Section 4 in the main text.
Revised: In this section we report additional experiments and qualitative visualisations, which can be helpful to complement the main results from Section 4 in the main text.

Original: G.1 LINEAR DATA
Revised: G.1 LINEAR DATA

Original: Results for scenario class 1 We consider the case of scenario class 1 (see Section 1) where the states S are observed and their dynamics are independent from X .
Revised: Results for scenario class 1 We consider the case of scenario class 1 (see Section 1) where the states S are observed and their dynamics are independent from X .

Original: We compare
Revised: We compare

Original: SCDI-Static with ACD, and evaluate the effect of explicitly modeling the underlying state.
Revised: SCDI-Static with ACD, and evaluate the effect of explicitly modeling the underlying state.

Original: We further consider the case where the decoder is fixed and uses the ground-truth β k values, for which we denote as - F IXED DECODER or - F IXED
Revised: We further consider the case where the decoder is fixed and uses the ground-truth β k values, for which we denote as - F IXED DECODER or - F IXED

Original: DEC .
Revised: DEC .

Original: for short.
Revised: for short.

Original: Table 3 shows the edge-type identification accuracy and test data reconstruction mean-squared error (MSE) for a simulation with N = 3 variables, K = 2 states, and n ϵ = 2 edge-types (no-edge and β 1 ).
Revised: Table 3 shows the edge-type identification accuracy and test data reconstruction mean-squared error (MSE) for a simulation with N = 3 variables, K = 2 states, and n ϵ = 2 edge-types (no-edge and β 1 ).

Original: SDCI-Static successfully performs the task of identifying the edges.
Revised: SDCI-Static successfully performs the task of identifying the edges.

Original: In terms of the ℓ 1 error for estimating { β k } , ACD scores worse ( ∼ 10 − 3 ) compared to SDCI ( ∼ 10 − 5 ).
Revised: In terms of the ℓ 1 error for estimating { β k } , ACD scores worse ( ∼ 10 − 3 ) compared to SDCI ( ∼ 10 − 5 ).

Original: We also evaluate SDCI-Temporal on the synthetic linear data with 2 edge-types and 2 states.
Revised: We also evaluate SDCI-Temporal on the synthetic linear data with 2 edge-types and 2 states.

Original: Table 4 extends the results reported in Table 3 in the main text.
Revised: Table 4 extends the results reported in Table 3 in the main text.

Original: We see that SDCITemporal models generally return worse results than SDCIStatic models.
Revised: We see that SDCITemporal models generally return worse results than SDCIStatic models.

Original: Specifically, SDCI-Temporal performs much worse when using learned decoder, and it fails to identify the underlying causal structure thus producing inaccruate forecasting results (see test MSE).
Revised: Specifically, SDCI-Temporal performs much worse when using learned decoder, and it fails to identify the underlying causal structure thus producing inaccruate forecasting results (see test MSE).

Original: Further inspection also shows that its ℓ 1 error for estimating { β k } is considerably higher ( ∼ 10 − 2 ) than ACD and SDCI ( ∼ 10 − 3 and ∼ 10 − 5 respectively).
Revised: Further inspection also shows that its ℓ 1 error for estimating { β k } is considerably higher ( ∼ 10 − 2 ) than ACD and SDCI ( ∼ 10 − 3 and ∼ 10 − 5 respectively).

Original: Given that SDCI-Static with learned decoder returns accurate results, the failure of SDCI-Temporal is attributed to the inaccurate posterior approximation in VAE learning, which justifies improvement need for the aggregator f ϕ aggr as already discussed in Appendix D.1.
Revised: Given that SDCI-Static with learned decoder returns accurate results, the failure of SDCI-Temporal is attributed to the inaccurate posterior approximation in VAE learning, which justifies improvement need for the aggregator f ϕ aggr as already discussed in Appendix D.1.

Original: We repeat the same experiments with 3 edge-types andstates, and report the results in Table 5.
Revised: We repeat the same experiments with 3 edge-types andstates, and report the results in Table 5.

Original: As before, SCDIStatic shows superior performance in comparison to ACD and SCDI-Temporal.
Revised: As before, SCDIStatic shows superior performance in comparison to ACD and SCDI-Temporal.

Original: Notice that the latter achieves similar accuracy scores in comparison to its F IXED D ECODER setting.
Revised: Notice that the latter achieves similar accuracy scores in comparison to its F IXED D ECODER setting.

Original: In terms of the ℓ 1 error for estimating { β k } , we observe that both SDCI-Static and SDCI-Temporal provide more accurate estimations ( ∼ 10 − 2 and ∼ 10 − 3 respec-
Revised: In terms of the ℓ 1 error for estimating { β k } , we observe that both SDCI-Static and SDCI-Temporal provide more accurate estimations ( ∼ 10 − 2 and ∼ 10 − 3 respec-

Original: [Table] [Table]
Revised: [Table] [Table]

Original: tively) compared to ACD ( ∼ 10 − 1 ).
Revised: tively) compared to ACD ( ∼ 10 − 1 ).

Original: Compared to the failure of SDCI-Temporal (learned decoder) in the previous experiment, one can deduce that accurate estimation of the edge-type parameters plays a crucial role in learning to infer the underlying causal structure.
Revised: Compared to the failure of SDCI-Temporal (learned decoder) in the previous experiment, one can deduce that accurate estimation of the edge-type parameters plays a crucial role in learning to infer the underlying causal structure.

Original: This is further illustrated in Figure 8 which shows the evaluation metric curves of
Revised: This is further illustrated in Figure 8 which shows the evaluation metric curves of

Original: SDCI-Temporal during training.
Revised: SDCI-Temporal during training.

Original: Here the edge-type accuracy improves when the { β k } are estimated correctly, in which the l 1 estimation error tends towards zero.
Revised: Here the edge-type accuracy improves when the { β k } are estimated correctly, in which the l 1 estimation error tends towards zero.

Original: Scenario class 

Revised: Scenario class

Original: The results considering linear data with hidden states in scenario class 2 show that the CSG accuracy decreases when considering an additional edge-type (see Table 1).
Revised: The results considering linear data with hidden states in scenario class 2 show that the CSG accuracy decreases when considering an additional edge-type (see Table 1).

Original: However, when leaving the summary graph constant (3- EDGE CONST )
Revised: However, when leaving the summary graph constant (3- EDGE CONST )

Original: we obtain better CSG estimations.
Revised: we obtain better CSG estimations.

Original: Figure 9 shows the approximate posterior q ϕ ( s ti | x ti ) inferred by SDCI-Static in each of the 3 settings studied in this experiment along with the true underlying state function, s ti = 1 ( | x ti | > 2) .
Revised: Figure 9 shows the approximate posterior q ϕ ( s ti | x ti ) inferred by SDCI-Static in each of the 3 settings studied in this experiment along with the true underlying state function, s ti = 1 ( | x ti | > 2) .

Original: We notice that SDCI-Static achieves better results when the quality of the estimated state function is
Revised: We notice that SDCI-Static achieves better results when the quality of the estimated state function is

Original: [Figure]
Revised: [Figure]

Original: [Figure] [Figure]
Revised: [Figure] [Figure]

Original: [Table]
Revised: [Table]

Original: higher.
Revised: higher.

Original: This is expected, since querying the correct edgetype from the conditional summary graph strongly relies on accurate state estimations.
Revised: This is expected, since querying the correct edgetype from the conditional summary graph strongly relies on accurate state estimations.

Original: Therefore, the model has to learn the state distribution and only then will be able to infer accurate causal structures.
Revised: Therefore, the model has to learn the state distribution and only then will be able to infer accurate causal structures.

Original: Although they can be challenging in general, our results show that SDCI-Static is successful in performing both learning steps in simple settings.
Revised: Although they can be challenging in general, our results show that SDCI-Static is successful in performing both learning steps in simple settings.

Original: This offers a promising direction of work towards the conditional stationary setting with hidden states.
Revised: This offers a promising direction of work towards the conditional stationary setting with hidden states.

Original: Figure 10 shows samples for the linear data considering hidden states and 2 edge-types along with the corresponding causal summary graphs and summary graphs inferred by both SDCI-Static and ACD.
Revised: Figure 10 shows samples for the linear data considering hidden states and 2 edge-types along with the corresponding causal summary graphs and summary graphs inferred by both SDCI-Static and ACD.

Original: Since SDCI-Static achieves high accuracy, the majority of the graph estimations match the true causal summary graph, which results in good forecasts.
Revised: Since SDCI-Static achieves high accuracy, the majority of the graph estimations match the true causal summary graph, which results in good forecasts.

Original: We also show two cases (two last rows) where the graph estimations do not match the ground-truth, and in these cases the model does not predict the future trajectories accurately.
Revised: We also show two cases (two last rows) where the graph estimations do not match the ground-truth, and in these cases the model does not predict the future trajectories accurately.

Original: Similar to what we show in the main text, the trajectories obtained with ACD drop rapidly to 0, and the summary graph estimates are considerably worse.
Revised: Similar to what we show in the main text, the trajectories obtained with ACD drop rapidly to 0, and the summary graph estimates are considerably worse.

Original: G.2 SPRINGS DATA
Revised: G.2 SPRINGS DATA

Original: Scenario class 

Revised: Scenario class

Original: We consider the case where the particles are contained in a box, and the state of a particle transitions when it collides with the wall of the box (see Figure 1c).
Revised: We consider the case where the particles are contained in a box, and the state of a particle transitions when it collides with the wall of the box (see Figure 1c).

Original: For simplicity, we only consider K = 2 states that transition alternatively on wall collision.
Revised: For simplicity, we only consider K = 2 states that transition alternatively on wall collision.

Original: We report SDCI’s performance along with ACD in Table 6.
Revised: We report SDCI’s performance along with ACD in Table 6.

Original: Regarding edge accuracy SDCI
Revised: Regarding edge accuracy SDCI

Original: [Figure]
Revised: [Figure]

Original: graph (CSG) accuracies of the different causal discovery methods considering hidden states in the springs dataset.
Revised: graph (CSG) accuracies of the different causal discovery methods considering hidden states in the springs dataset.

Original: [Table]
Revised: [Table]

Original: performs significantly better, and ACD is limited by considering stationary dynamics only.
Revised: performs significantly better, and ACD is limited by considering stationary dynamics only.

Original: However, both methods achieve comparable test MSE metrics, indicating that that
Revised: However, both methods achieve comparable test MSE metrics, indicating that that

Original: ACD can still make decent predictions even when it fails in identifying the edge-type interactions.
Revised: ACD can still make decent predictions even when it fails in identifying the edge-type interactions.

Original: Scenario class 2 (hidden)
Revised: Scenario class 2 (hidden)

Original: We provide additional visualisations for the case with hidden states in Figure 11, where we show the predictions of both SDCI-Static and ACD as well as the corresponding conditional summary graphs and summary graphs extracted by both methods respectively.

Original: As in the linear case, SDCI-Static produces accurate causal graph estimates.

Original: Regarding time series forecasting, our method is able make reasonable predictions.

Original: Notice that to train the models, we use teacher forcing everytime-steps, which means that the learned models are less suited for long-term dynamics modelling.

Original: However, one can expect to obtain more accurate predictions by progressively reducing the teacher forcing frequency during training.

Original: Considering ACD, despite being restricted by assuming stationary time series, it still infers graph structures that allow the model to produce decent forecasts.

Original: Results for scenario class 2 We consider the hidden state regime, where the underlying state of a particle changes depending on its location in the box ( K = 2 , see Figure 1b).
Revised: We consider the hidden state regime, where the underlying state of a particle changes depending on its location in the box ( K = 2 , see Figure 1b).

Original: Results in Table 7 shows a clear advantage of SDCI over baselines in terms of summary graph identification accuracy, and its accuracy levels are closer to the levels in scenario class 3 with observed states (table 6).
Revised: Results in Table 7 shows a clear advantage of SDCI over baselines in terms of summary graph identification accuracy, and its accuracy levels are closer to the levels in scenario class 3 with observed states (table 6).

Original: Again ACD is limited by its stationary dynamics assumption, and CD-NOD fails due to its inability in handling discrete changes in the causal effects of the full time graph.
Revised: Again ACD is limited by its stationary dynamics assumption, and CD-NOD fails due to its inability in handling discrete changes in the causal effects of the full time graph.

Original: We visualize the time series forecasting results (for 50 timesteps) with SDCI and ACD given an input sequence of
Revised: We visualize the time series forecasting results (for 50 timesteps) with SDCI and ACD given an input sequence of

Original: T = 80 time-steps in Figure 12, as well as the extracted & ground truth (conditional) summary graphs in Figure 13.
Revised: T = 80 time-steps, as well as the extracted & ground truth (conditional) summary graphs in Figure 11.

Original: We see that  the conditional stationarity assumption of SDCI allows it to extract more accurategraph structures, which result in better predictions in comparison to ACD.
Revised: As in the linear case, SDCI-Static produces accurate causal graph estimates. Regarding time series forecasting, our method is able make reasonable predictions. Notice that to train the models, we use teacher forcing every 10 time-steps, which means that the learned models are less suited for long-term dynamics modelling. However, one can expect to obtain more accurate predictions by progressively reducing the teacher forcing frequency during training.

Revised: Considering ACD, despite being restricted by assuming stationary time series, it still infers graph structures that allow the model to produce decent forecasts.

Original: Overall, our method successfully decomposes the nonstationary dynamics into the conditional stationary ones while capturing the state transition dynamics.
Revised: Overall, our method successfully decomposes the nonstationary dynamics into the conditional stationary ones while capturing the state transition dynamics.

Original: [Figure]
Revised: [Figure]

Original: [Figure]

Original: [Figure]