Deep Autoregressive Models as Causal Inference Engines
Authors: Daniel Jiwoong Im
Kevin Zhang
Nakul Verma
Kyunghyun Cho
Venue: Accepted by TMLR ()

Abstract: Existing causal inference (CI) models are often restricted to data with low-dimensional confounders and singleton actions. We propose an autoregressive (AR) CI framework capable of handling complex confounders and sequential actions commonly found in modern applications. Our approach accomplishes this using sequencification, which transforms data from an underlying causal diagram into a sequence of tokens. Sequencification not only accommodates training with data generated from a large class of DAGs, but also extends existing CI capabilities to estimate multiple causal quantities using a single model. We can directly compute probabilities from interventional distributions, simplifying inference and improving outcome prediction accuracy. We demonstrate that an AR model adapted for CI is efficient and effective in various complex applications such as navigating mazes, playing chess endgames, and evaluating the impact of certain keywords on paper acceptance rates, where we consider causal queries beyond standard reinforcement learning-type questions.

Published in Transactions on Machine Learning Research (07/2025)
Deep Autoregressive Models as Causal Inference Engines
Daniel Jiwoong Im Center for Data Science New York University
jr64 edu
Kevin Zhang
ky:20005cclumbiic.co edu
Computer Science
Columbia University
Nakul Verma Computer Science
verma@@cs.couumbiiccoo
Columbia University
Kyunghyun Cho
kyunghhyun.cko &ryy.
Center for Data Science New York University
Reviewed on OpenReview: https: // openreview. net/ forum? id= uaREHPf211
Abstract
Existing causal inference (CI) models are often restricted to data with low-dimensional con- founders and singleton actions. We propose an autoregressive (AR) CT framework capable of handling complex confounders and sequential actions commonly found in modern ap- plications. Our approach accomplishes this using sequencifteation, which transforms data from an underlying causal diagram into a sequence of tokens. Sequencification not only accommodates training with data generated from large class of DAGs, but also extends existing CI capabilities to estimate multiple causal quantities using single model. We can directly compute probabilities from interventional distributions, simplifying inference and improving outcome prediction accuracy. We demonstrate that an AR model adapted for Cl efficient and effective in various complex applications such as navigating mazes, playing chess endgames, and evaluating the impact of certain keywords on paper acceptance rates, where consider causal queries beyond standard reinforcement learning-ter ttpiinggttpeee questions.
1 Introduction
Modeling causal relationships is important across various fields for informed decision-making (Holland 1986:
Cochran & Rubin
1973 Pearl 2010). However, existing causal inference (CI) algorithms are often limited
by their inability to handle sequential variable-length actions and high-dimensional covariates (Louizos et al.
2017 Kumor et al. 2021 Im al. 2021 Lu et al. 2022 Zhong al. 2022). This work aims address these
shortcomings by designing a CI engine applicable to complex data involving high-dimensional variables.
Consider medical setting where doctor prescribes a sequence of treatments the patient. In this scenario, the number of treatments that the patient undergoes is not necessarily fixed (e.g. additional procedures may be assigned depending patient response or two medications might be combined into onc). As result, the number of possible treatment sequences grows combinatorially. Moreover, the medical decision may be dependent on high-dimennsonal data describing the condition of the patient, such as text from electronic hcalth records. For successful treatment. effect estimation, we want a framework that can accommodate arbitrary-length action sequences for complex, high-dimenssonal confounding data. To tackle this problem, we propose using neural network-based autoregressive (AR) models for causal inference.

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Published in Transactions on Machine Learning Research (07/2025)
Antoregressive (AR) models are standard framework for learning conditional probability distributions and predicting values in sequential or time-series data. Neural network-based AR models are widely used in applications such as language modeling for next-token prediction and text generation (Devlin et al. 2019;
Radford et al.) 2019 Touvron et al., 2023). As demonstrated by large language models (LLMs), AR models
can capture complex relationships and scale effectively to large datasets. Recent studies (Gupla el al.) 2023
Zhang et al., 2024; Xu al.] 2024) show that fine-tuning pre-trained LLMs utilizes knowledge from an internet-scale text corpus, enhancing performance on various tasks.
We demonstrate that AR models can also be employed for causal inference by transforming observational data into a sequence following an underlying causal ordering. To achieve this, we employ sequencification for representing data based on a known causal diagram, specified as a directed acyelic graph (DAG). A neural network-basse AR model can then learn the conditional probability distributions implied by the DAG. This allows for efficient sampling of sequential actions and high-dimensional confounders, enabling Monte Carlo estimation to approximate various causal effects. Furthermore, since the AR model learns all conditional distributions in the sequence, single model trained on sequencified data can be used to compute wide range of interventional distributions.
Our methodology is designed to handle variable-length sequential actions, combinatorially large action spaces, and high-dimensional confouse These capabilities encompass a variety of common causal inference tasks, including average treatment effect (ATE) estimation, individual treatment effect (TTE) estimation, interven- tional distribution approximation, etc. To the best of our knowledge, all prior work can only accommodate fixed-length covariates and actions, and most evaluate on relatively low-dimensional data.
We conduct empirical studies across variety of exemplar tasks, such as navigating mazes, playing choss endgames, and evaluating the impact of specific keywords on paper acceplance rates. The experiments show that our framework can (1) infer causal effects involving nigh-dimensional variables, (2) generalize to unseen confounders and action sequences, and (3) leverage pre-trained LLMs to answer text-based causal questions.
2 Related work
Previous work has explored the use of deep learning or AR models for estimating causal effects. Here, we provide a brief overview of the relevant st udies.
Sequencification for statistical engines. Various machine learning fields have used linearized represen- tations for tasks. In natural language processing (NLP), linearization (which we refer to as sequencification) is used to convert a syntactic tree into a sequence for building language model-based parsers (Vinyals el al., 2015 Liu et al., |022 Sheng al.] 2023). In reinforcement learning (RL), an episode can be encoded as sequence of states, actions, and rewards. An autorcgressive model is then trained on those sequences to capture relationships among the variables (Chen et al. 2021 Janner et al.) 2021 These instances suggest that AR models trained on sequencified data can effectively learn statistical dependencies among multiple high-dimensional variables.
Language models as causal engines. Several works have used language models for high-dimensional
CI tasks (Feder et al. 2021 Egami et al.] 2022 Veitch et al.] 2020
A key challenge across these studies
Tu & Li, 2022 Gui & Veitch 2023). Techniques from NLP, such as topic models (Sridhar & Getoor 2019 Mozer et 2020), latent variable models (Keith) et al. 2020), and contextual embeddings (Veitch et al. 2020), have been used to produce low-dimensional
is satisfying positivity constraints (D' (m'Am et al. 2021
embeddings that can satisfy positivity. In addition to using NLP techniques to reduce the dimensionality
of obscrvations, natural language can also serve as proxy for observed confounders. For example, Roberts
et al. (2020) apply a text-matching algorithm using contextual embeddings and topic models to estimate causal effects based on proxy texts.
Deep learning for causal engines. Various deep neural network architectures have been proposed for CI. Representation learning for CI often incorporates a regularization term that enhances gencralization for counterfactual actions (Shalit et al.) 2016 Johansson et al., 2018 Wang & Jordan (024).. Deep latent variable CT models learn stochastic latent. variables to model potential outcomes using richer family of
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Normalizing flows have also been used for causal effect estimation. For example, Krause et al. (2023) develop doubly robust density estimator for potential outcomes by combining a muisance and target normalizing flow.
distributions (Louizos et al. 2017 Kocaoglu al. 2017 Im et al.
2021
For deep autoregressive models, Monti et al. (2020) introduce an AR flow model that learns an invertible density transformation between variables. Their approach enables direct computation of interventional and counterfactual distributions without the need for complex latent variable manipulations. [Garrido et al. (2021) use neural AR density estimators (Larochelle & Murray 2011 to model causal mechanisms and predict causal effects using Pearl's do-calculus (Pearl 2009).
Other studies examine treatment effect estimation in more complex, multi-dimensional scenarios. Frauen et al. (2025) propose a model-agnostic learner for a sequence of actions over fixed time horizon. An interest- ing line of work focuses on real-valued actions (e.g. amount of medical dosage administered) for heterogeneous dosc-response estimation using different neural-network architectures, including generative adversarial net- works Bica et al. 20000), varying coefficient. models Nie et al. (2021), and contrastive representation learning
Zhu al. 2024
Causal Generative Models. Another approach to CI models data as part of a generative process and learns a causal generative model. These methods typically parameterize relationships in a known causal diagram using neural networks. For causal effect identification and estimation, Xia et al. (023) use neural causal models trained via gradient-based optimization with a miniiiiaationnmaxiiiaaa
objective. Rahhman & Koccoolu (2024) introduce a modular learning framework for optimizing causal generative models in semi- Markovian scttings by decomposing the data distribution into e-factors (Tian & Pearl 2002). Their method use pre-trained generative models to learn individual conditional distribution components.
Existing limitations. Previous works exhibit notable weaknesses compared to our approach. First, most methods are validated only on relatively low-dimensional variables (tens of dimensions) and fixed-length actions. contrast, our AR model is designed to handle high-dimensional confounders and variable-length series of actions where the number of possible sequences grows exponentially. This makes applicable settings with high-dimensional data and arbitrarily long sequence of treatments. Second, prior studies often focus estimating specific causal effects or learning generative models by optimizing individual components of the causal structure. We instead use unifted end-to-end AR model that can estimate multiple causal queries. Third, we extend existing C1 capabilities by leveraging pre-trained language models for settings where domain knowledge is essential for accurate inference (c.g. NLP).
3 Background
This soction outlines the background knowledge of causal effect estimation and language models necessary for understanding our methodology.
3.1 Cl problem formulation
We study interactions between the following set of variables: an observable confounder X, action A, and outcome Y. Causal relationships are represented as directed acyelic graph (DAG), where edges denote direct effects (cf. Figure Ta]. By applying the backdoor adjustment formula PPaal 2009), we compute
the potential outcome resulting from an intervention on the action:
Ya = EY do(A = a]]] Σ
y.
x)p(x = =x).
(1)
aryy
The notation do(A= = a) represents an intervention on A, setting its value to Typically, the confounder X is assumed to be low-dimensional to avoid computing density estimates p(x) in high dimensions. Our goal is to model causal effects when typical assumptions in prior CT work are violated, including settings with complex confounders and combinatorially large action spaces.
We make the following set of standard assumptions causal inference: (1) unconfoundednesss, which states that Yo, Yi 1I. A X, and (2) positivity, which requires that p(a, x. y) > 0 for all triplets (a,
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X
X1
X2
X
A1
A2
Y
A
Y
A
Y
(a)
Cc)
(b)
Figure 1: Causal diagrams illustrating interventions on the action (a), partial action (b), and when conditioning on prefix of the confounders (c), where X, A, and Y denote the confoumders, actions, and outcome respectively. Bold red arrows indicate the pathways blocked by the corresponding intervention. Potential outcomes can computed using the backdoor adjustment formula for each scenario.
3.2 Language models
Language models (LMs) are designed to predict and generate text by learning linguistic patterns from a training corpus. Let W denole the vocabulary the text corpus, which includes special (start) and (end) tokens. LMs estimate the probability of a sequence of tokens = (Wi, "" ,,, WT in an autoregressive manner, where w1 (start) and wr (end). The probability of the sequence is decomposed into the product of next-token probabilities: p(w) p(w1) p(w2 wi pp(3 wi, .pp((r Wi,..., WT-1).
4 Language models as statistical inference engines
As suggested by Equation 1 CI requires accurate estimation of statistical quantities to calculate causal effects. In this section, we describe how AR model be adapted into a statistical inference engine for any DAG involving set of confounders, actions, and outcomes using sequencification.
4.1 Causal graphs
We assume the underlying causal DAG G (V,E) known, where the vertices V2 represent random variables and the edges Eş-r E & denote conditional dependencics. Morcover, the graph satisfies the Markov property, meaning the joint probability distribution can be factored into consally-cnnsiseent ordering,
M
PV V..... VM) = (V Pa(Vi)),
where Pa(V)) == EVV E E is the set of parent nodes of V,, We assume that the graph g fully specified and that all variables within G are observed.
4.2 Sequencification
Suppose Vi takes the value v, from its corresponding distribution. Let string(-) be an injective function that maps v, to a sequence of tokens: string(v;) = ((start,), w1, W2...,, Here, (start,)) is a special token indicating the beginning of the string representation for v,, and Tw, the length of string(v,) excluding the (start,) token. We define (start,) uniquely for each so that each random variable can be uniquely identified from its string representation by its corresponding initial token.
Let t (V1,, Vig...., Visz) be permutation of the random variables. We say is topological ordering if V; precedes V, in the ordering for all edges Ei+j- Let T denote the set of all topological orderings. Consider N samples drawn from the underlying causal diagram G: (v("), v .... v.()) PV V2 ... VMr) for n n= ,....,,.. N. For each sample, we construct string s(n) by concatenating the string representations of all random variables according to topological ordering (x) selected uniformly random from T.
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Formally, t0) ((2) ,..., ((N) sid. Uniform(7) and
8 s(n) = string(v, (v @ string(v (n))
( ... ... 4 string((v
(n))) e (end),
where t(n) = (V,, Vig.... V and @ denotes string concatenation. We refer to the process of converting observed samples into a sequence of tokens as sequencifteation.
Sequencification supports any data that can be transformed into a linear sequence of tokens. For example, tokens may represent specific values for numerical (e.g., binary) data or subwords for text, depending on the problem domain. Our approach also handle mixed data modalities, as different variables may have separate tokenization strategies.
4.3 Randomized topological orderings
We randomize over topological orderings consistent with the causal graph to obtain robust estimates of conditional probabilities. For instance, if a node in the graph has multiple independent parents, randomizing the order in which the parents are sequencod helps prevent the model from overfitting to any particular ordering. As a result, different samples may concatenated using distinct topological orderings. However, because the data is generated by ancestor sampling (!()) PPa(v("), (")), values that causally influence string(v;" (a) will al always appear earlier in (()).
A natural question that may arise is how to obtain a good estimate when samples are sequencified according different topological orderings. This is achieved by using the special (start;) token at the start of cach variable Vi during sequencification, which indicates ils position in the sequence (regardless of the topological ordering used).
4.4 Autoregressive statistical inference engines
After sequentification can train an AR language model, parameterized by 0, on the sequencified dataset D = ss(")s(( ((),,...,s()) by the minimizing negative log- ligliike:hood:
N
N
(())
1
log po((m)))))
1
Σ
log po (s)"*)
s///1):
(n)
(2)
N
N
1=1
where denotes the length of string and s. is the 4th token in s(n), The trained model can estimate any conditional probability on G by computing PV(V, Pa(Vi) ~ pe(v; Pa(v:)). This is efficiently done by au- toregressively traversing the sequence and calculating next-token probabilitics using Monte Carlo estimation over the topological orderings.
5 Language models as causal inference engines
In this section, we illustrate how infer causal effects by leveraging statistical quantitics from trained AR model, thereby transforming into causal inference engine. After learning the conditional distribution over G using AR model, estimate causal quantities by deriving the appropriate identification formula from the known causal diagram. Sequencification, combined with knowledge of G, allows us to estimate various causal effects.
5.1 Estimating causal quantities
We express the Cl problem as a language modeling task. Given our DAG that consists of three variables (the observed confounder X, action A, and outcome Y), we sequencify the data as follows:
(n) (tring(x(1)) @ string(a("l) @ string(y(")) @ (end).
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In our formulation, x(") and (*)) can be high-dimensional vector values and the action space can be combi- natorially large. Without loss of generality, we treat the outcome variable Y as scalar, represented using a single token.
We can use the trained AR model to compute the distribution Y after intervening on This is typically intractable when X is high-dimensional because the sample complexity of computing a non-parametric density estimate is exponential the number of dimensions. However, we can approximate the interventional distribution by sampling from (X( and applying Monte Carlo estimation.
$ Σ po - y do(A= =a)) = Eprov polv A a, (x) poly A A= x.
(3)
where (5) ~ (X).
Furthermore, we can intervene on a prefix subsoquence of A even when the action space is large. By expressing A @ A2 and marginalizing out A2, we can compute the effect of intervening on only A
p(Y = do(A1 = =a1)) = Σ>
poly A = =a, A2 =22, x)po a1, po(x)
(4)
x
a2
1
po a1, ab x*!))
(5)
5
where x(+)) p(X) and a generally intractable because the marginalization requires exponentially many operations.
p(A2 a (.)). For combinatorially large action spaces, Equation is
Similarly, we can condition on a prefix of the confounder by letting X X1 @ X2 and marginalizing out X2.
X2
s
pYY = do(A a),
po A X2, X1) X1) x1) ~
Σ S
polu A = a, x x1)
(6)
X2
where (+) ~ p(2 == The causal diagrams for these scenarios are shown in Figure Note that we can only intervene on prefixes of the action A₁ (or condition on prefixes of the confounder X1) because this ensures the marginalization step samples A2 (or respectively X2) conditioned on preceding variables.
We emphasize that all causal quantities can be computed by a single language model trained on sequenci- fied observations. Our approach enables efficient sampling and computation of conditional p(Y A a), interventional p(Y do(A = a)), partial interventional p(Y do(A = ai)), and conditional interventional p(Y do(A = a), X1 = x1) distributions, all using a unified model. This provides an end-to-end framework for computing variety of causal queries using single model.
While the partial interventional distribution p( do(Ai = =ai) is computable by effectively discarding A2, training an AR model on sequencified data that explicitly includes A and A2 is more flexible. Our approach can efficiently estimate not only partial interventional distributions but also interventions on A and A simultancously, providing unified framework for handling multiple causal queries. A similar argument holds for partially conditioning on the confounders.
6 Experiments
We demonstrate the effectiveness of our approach in estimating causal effects for sequential actions and high- dimensional confounders while also assessing robustness to distribution shifts. Our experiments showcase the ability to (1) infer potential outcomes with sequential actions and high-dimensional confounders, (2) efficientl approximate potential outcomes via Monte Carlo sampling, and (3) leverage knowledge from a pro-trained LLM. We evaluate our method across three environments: a maze setting for navigational decision-making, chess environment analyzing strategic moves in king vs. king-rook endgames, and the PeerRead dataset which examines the impact of theorem presence academic paper acceptance.

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Published in Transactions on Machine Learning Research (07/202) 2025)
START
START
END
END
Remusinnn
Poth
Path
Puth
(a)
(b)
(c)
Figure 2: Illustrations of the maze and chess experimental seltings. In the maze experiment, we address two questions: what is the potential outcome given (a) complete path, and (b) partial path? The blue path represents the intervention, gray indicates a potential remaining path after a partial intervention, and orange denotes the distance the end. The chess experiment aims determine which pieces Black should to checkmate White the quickest. In the example position shown in (c), the probability of moving the Black king is 0.5, 0.25, and 0.8 with the RCT, non-RCT1, and non-RCT2 policies respectively.
The maze experiments demonstrate that our unified AR model can estimate interventions, partial inter- ventions, and conditional interventions using Equations 3 and 6 respectively. In contrast, a traditional offline reinforcement learning (RL) model fails capture all three causal effects. In the chess experiments, highlight the effectivencss of our method in estimating effects via Monte Carlo approximation using Equa- tion 5 and its robustness to distribution shifts in the test data. Finally, the PeerRead setting demonstrates that the AR model can estimate effects in high-dimensional confounder scenarios and leverage pre-trained language models to improve text-based analysis. These diverse settings enable a comprehensive evaluation of the effectiveness and robustness of our approach for CI under variable-length action sequences and high- dimensional confounders such as text. Moreover, our framework is also competitive with existing methods on benchmark treatment effect estimation tasks, as shown in Appendix
All AR models are trained using vanilla transformer (Vaswani et al. 2017 unless otherwise specified. Additional details the model architecture and training process be found in Appendix
6.1 Maze navigation experiments
In this experiment, show that unified AR model estimate multiple types of causal queries using Equations and 0 Compared to baseline offline RL model, our approach offers greater flexibility for causal inference tasks.
We generate a synthetic maze dataset to analyze the causal effect of traversing different paths. The goal is to determine the distance to the exit after following a given path. The confounding variable X represents the starting position, the action A
sequence of moves, and the outcome denotes the distance from the final position to the exit, 2 We evaluate potential outcomes when intervening on complete path (ef. Figure 2a or partial path (cf. Figure 2bb. The obstacle positions fixed but are not known to the model.
The end position is fixed at the bottom-right corner, while the starting position is randomly selected from the open spaces with a probability proportional to its distance from the endpoint. Moves in the path are determined by selecting a direction based on the current position. Let the current square be in the ith row from the top and the jth column from the left. The next move chosen according to the probabilities:
0.8i
0.8j Pdown i i+j
Pup = Plet. 0.1,
Pright i +
Code available httpp://githhb. httpa//ccc//ssoonnn/eeeeeee http://ppp-tttttrrreessseveeeeee -ecedelllee--uuaal-- -nfereee---eennennnee 2y is the shortest, possible distance maze while avolding obstacles, not necessarily Hamming dislance.

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Model
0.7
AR (Ours)
RL (Baseline)
0.6
0.5
Error
0.4
0.3
0.2
0.1
0.0
B do(A)
B|
do(A), 1|
BB do Causal query
BB| do(A);
BB| do(A), X1 4|
Figure 3: Error distribution of potential outcome estimates for our AR model (blue) and the offline RL baseline (red). We compute the effect intervening on the complete path with and without conditioning on the starting row X1. The plot depicts the distribution of errors across all 4096 possible paths of length six, while the black dashed line represents the mean error. The AR and RL models exhibit comparable performance across all settings, with our AR model performing marginally better.
This policy encourages paths to move towards the bottom-right corner. All actions are possible at any position, however moves that would collide with obstacles or walls in the maze are treated as action. Paths are fixed to contain exactly six moves.
We train an AR model soquencified data and use a deep Q-learning (DQL) model an offline reinforce- ment learning (RL) baseline. The AR model is given only the starting position and must infer the effect of each move along the path. The DQL model follows the standard RT framework, where the current position is known after each move. In Appendix B we vary the dimensionality of the maze and the length of the path demonstrate the scalability of our framework in terms of the input and action dimension.
6.1.1 Causal inference using sequential actions
We estimate potential outcomes for all paths of length six in 4 x 4 maze using Equation Additionally, compute potential outcomes conditioned on the starting row (from the top of the maze) X1 E {1, 4} using Equation D Ground truth values are computed using the corresponding equations with the outcome outcome [Y a, x equal to the true number of additional moves required to reach the end of the maze after starting in position & and taking path a. For the RL method, we predict the potential outcome as the g- vavalue after taking the final action in the intervention.
Figure presents the error distribution for potential outcome estimates across all paths. In all settings, both models produce estimates that closely match the ground truth. Our AR model performs comparably to the offline RL baseline in terms of mean estimation error. The AR error distribution exhibits large tail, which can be attributed to differences in training setups. It is worth emphasizing that the RL model knows the exact position after each move in the path and therefore only needs to predict one slep ahead each time, whereas our AR framework only knows the starting position and has to predict six future steps. As expected, the extra knowledge available to the RL model lowers the variance. However, even without this knowledge, our model achieves slightly lower overall error. These results demonstrate that our approach can accurately predict both intervention and conditional intervention queries.

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AR (Ours)
RL (Baseline)
2.5
-2
Second two moves
Second two moves
EY do (i)) Error
-.5
-1
0.5
First two moves
First two
-0
Figure 4: Heatmap illustrating the distribution of potential outcome estimates when intervening on the first four moves in the path. Each row represents all possible actions for the first two moves, while each column represents all possible subsequent two moves. Our AR model accurately estimates potential outcomes for all four-move sequences by learning p(a2 x). The offline RT. baseline exhibits significant errors across nearly all interventions.
6.1.2 Causal inference using partial actions
addition to computing complete interventions, we can also intervene on partial actions. To illustrate this scenario, we estimate potential outcomes when intervening on the initial moves a path. Specifically, we decompose
A + A2, where A₁ represents the first four moves and A2 the remaining two moves, and compute p(Y do(A1 a1)). Ground truth potential outcomes
calculated using Equation by marginalizing over all possible remaining paths A2. For the RL method, we intervene on the first four moves, while the remaining path is determined according to the learned policy.
Figure shows the error distribution for potential outcome estimates across all possible four-move interven- tions. Our AR model accurately computes these estimates using Equation g In contrast, the oflline RL model does not learn the distribution p(a2 a1, x) but instead optimizes for best policy. As a result, it fails to predict partial interventions without modifications, such discarding information about Ag during training. This highlights the greater flexibility of our AR framework, which allows interventions subset of initial moves with single model. These experiments demonstrale that a single AR model can accurately estimate interventional, partial interventional, and conditional interventional distributions.
any
6.2 Chess endgame experiments
In this section, we evaluate the performance of our AR model with Monte Carlo sampling, following Equa- tion [5] and assess its robustness to distribution shifts. To explore these aspects a more complex two-player setting, we use a synthetic chess dataset featuring king vs. king-rook endgames, where White moves first and Black holds the rook. We demonstrate that our AR model can accurately compute causal effects and idcntify optimal action sequences by comparing potential outcome estimates. Additionally, leverage Monte Carlo sampling to refine estimates when only partial data is available and introduce distribution shift between training and testing data to assess generalization.
To formulate our causal query, we ask: on average, across all starting positions, which pieces should Black move on the first two turns to checkmate White the fastest? Our question aims to uncover a general strategy for king vs. king-rook endgames, much like how controlling the center fundamental principle in the opening. More broadly, it parallels CI scenarios with multiple initial conditions, where the objective is identify the most effective strategy across wide range of situations rather than an individual configuration.

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RCT
non-RCT,
non-RCT.
Potential outcome
(number of moves)
20
19-
Ground truth
AR exact
AR approx.
AR MC
18
MLP exact
MLP P sporox.
103
10
10
0
10a
10a
10a
10a
10
10
Number of Monte Carlo samples
Figure 5: Potential outcome estimates for rook-king. The enact model estimate uses all 223,660 valid starting positions as test samples, while the approzimate and Monte Carlo estimates use 1,000 randomly selected samples. Each Monte Carlo estimate sampling from ps (X), the AR Monte Carlo estimate approaches the AR exact estimate. In contrast, the MLP model cannot perform sampling on po (X).
repeated 1,000 times, with bars representing standard deviation. By
Each endgame comprises a two-move chess game, potentially incomplete. The covariate the initial piece positions. The action A (a1, 02, 03, a4) represents alternating White and Black moves. Since we interested in causal quantities involving 0g and 04- We assume Black plays optimally after selecting which piece to maneuver, cach action only dictates whether to move the king the rook, but not to which location. The outcome variable is the nunber of additional moves required to checkmate with optimal play Formally, we are interested in finding
consider Black's perspective,
(az, aq]==
arg min E
Ey do (a)a arg min 02.04 E rook)
Σv-p(Y = 1 02,1,,2,,,100 (z)(())
a.y
We evaluate the ground truth using the chess engine Stockfish 0
Figure 2c displays example endgame.
We construct three training datasets: one Randomized Control Trial (RCT), which selects each action uniformly at random between moving the king or the rook, and two non-RCT datasets, labeled non-RCT, and non-RCT,, with distinct action policics. The policy functions for non-RCT, and non-RCT, are defined as follows, where the Hamming distance between the kings:
0.8 if black king is in center 4 x square
Ti (1 04 = king) 16' Tz(a2, 04 = king) =
0.2
otherwise
Ji encourages the two kings be closer while me pushes the Black king towards the edge of the board, both of which are required for checkmate. We use different RCT and non-RCT data to demonstrate robustness in the presence absence of X + A and under varying action assigument mechanisms.
The testing dataset consists of all 223,660 valid starting positions. To assess ization, we use two distinct policies for White in the training and testing phases. White plays uniformly al random over non-optimal moves (unless no such legal moves are available) during training and plays optimally at test time. This introduces a distribution shift, which we use evaluate generalization and robustness to new settings. Additionally, we show that our proposed AR framework produces more accurate causal estimates by using Monte Carlo sampling from (X( when given only subset of the testing data.
general-
We compute the ground truth potential outcomes for all actions and compare them with three different model estinates: the eract model estimate using the entire test dataset, the approzimate model estimate using a subset of the test data, and the Monte Corlo model estimate, which also uses the same data subset
IIn the event draw, the outcome set due the 50-move rule.
4 Stockfish available /tttts://githhl http://aiiiiaallttttttaa///////a
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but additionally generates samples from the model. The approximate and Monte Carlo estimates reffect real-world scenarios, where obtaining a large number of RCT samples is often difficult. We train non- autoregressive multilayer perceptron (MLP) as baseline for comparison.
Figure D presents our three model estimates for the AR and MLP model. As the number of samples in the Monte Carlo approximation increases, the potential outcome estimate converges the exact estimate. Our results demonstrate that potential outcomes can be efficiently and accurately approximated with an AR model using only fraction of the test data. Additionally, there is gap between the AR exact model estimate and the ground truth, caused by the distribution shift in how White plays between the training and test data. The potential outcomes for the remaining actions are provided in Appendix C By comparing all potential outcome estimates, we can answer our causal question and conclude that, aggregated across all starting positions, moving the rook twice initially is the best strategy for Black.
6.3 PeerRead experiments
We use the PeerRead dataset (Kang et al. 2018 to estimate causal effects in a semi-realistic setting with high-dimensional text confounders. Our results demonstrate that an AR model can leverage pre-trained language models to enhance CI in text-based settings. The dataset consists of paper draft submissions to top computer science conferences, such as NeurIPS, ICML, and ICLR, along with their acceptance rejection decisions. We investigate the impact of including "theorems" on acceptance likelihood and evaluate how well our model captures this causal effect. Building on prior work, we focus on computational linguistics, machine learning, and artificial intelligence papers submitted between 2007 and 2017 (Viichh el al. 2020).
The covariate X represents the paper's abstract text, the action A is a binary variable indicating the presence of the kcyword "thoorem", and the outcome Y is binary variable indicating acceptance or rejection. Since real-world counterfactual outcomes are inaccessible, we follow prior methods by generating synthetic outcomes based the action A and the title buzziness Z (i.e., whether the title contains "deep", "neural", "embed", "adversarial net"). For example, = 1 and 1 likely correspond to a deep learning paper that includes a theorem, while = 0 and a = 1 may represent theoretical machine learning paper or a deep learning paper with a theorem but without a buzzy title.
Define T(2) as the proportion of data samples with a; = among those satisfying 2; = 2. Let 3 be a parameter controlling the level of confounding between title buzziness and the outcome. Following Veitch 0.2)))
et al. 2020), we generate outcomes using the model: Y; (ernoull((c ((T(=.)
Since evaluating any causal effect model requires counterfactual outcomes that are inaccessible in real-world data, we use a semi-synthetic setting, where the covariates are real-world data and the labels are generated according to patterns in the data, albeit synthetically. We demonstrate the correlation between title buzziness and the text in Appendix D
Following the original experimental design, we report the Average Treatment Effect on the Treated (ATT),
ATT :- p(Y 1 do(A 1), A A= p(Y = do = 0), A A= 1),
across three confounding levels: low, medium, and high. A positive ATT indicates that including a theorem increases a paper's chance of acceptance. For larger values of 8, the outcome becomes more correlated with title buzziness Z rather than the action A, so the ground truth ATT is smaller.
We compare our proposed approach to non-antoregrressive MLP baseline and Causal-BERT (C-BERT) from Veitch et al. (2020) 3 Like C-BERT, we fine-tune pre-trained BERT model on our sequencified representations for a fair comparison. Additionally, we evaluate GPT-2 (referred to as GPT), another pre- trained LLM with a comparable parameter size to BERT Radford et al. 2019).
BERT is trained using masked language modeling (MLM) and next-sentence prediction objectives. MLM randomly masks a fraction of input tokens and trains the model to predict them. GPT is trained with a next-token prediction objective. To adapt BERT for this setting, we randomly sample subsection of the abstract during training and mask the final token to fine-tune it as next-token prediction model.
C-BERTT learns causally sufficient. embeddings: low-dimensional document representations that preserve sufficient. informa- tion for causal identification and enable efficient, causal effect. estimation.
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Table I: PeerRead ATT performance across low, medium, and high confounding levels, with relative error indicated in parentheses. For Deep Autoregressive Causal Inference Engine (DARCIE) BERT and GPT models, we quantify the uncertainly in our estimates by reporting the mean and standard error of the ATT across three trials. Entries in bold and underlined indicate best performing models for each confounding level. Overall, our DARCIE-GPT model achieves the lowest relative error, and both of our models show improvement over other methods. Training DARCTE-GPT from scratch fails to identify causal effects due to its lack of understanding of the text.
Confounding level
Low (3 1)
Medium (3 = 5)
High (3 25)
Ground truth
0.062
0.059
0.028
Computed biased
0.065 (4.8%)
0.097 (68%)
0.160 (470%)
Reported biased (eitch et al. 2020
0.08 (30%)
0.15 (150%)
0.16 (471%)
MLP 30
0.05 (20%)
0.10 (70%)
0.30 (970%)
C-BERT 6
0.09 (45%)
0.07 (19%)
0.04 (42%)
C-BERT Splugin
0.10 (61%)
0.09 (53%)
0.05 (78%)
DARCIE-GPT (No pre-train)
0.001 (98%)
0,002 (97%)
0.001 (96%)
DARCIE-BER BERT (Ours)
0.039 (37%) + 0.007
0.039 (34%) + 0.005
0.016 (43%) = 0.003
DARCIE-GPT (Ours)
0.041 (34%) 0.004
0.048 (19%) + 0.002
0.025 (10%) 0.002
As shown in Table 1 our approach outperforms C-BERT and other benchmarks. The primary reason for the improvement is that we jointly learn representations and outcome predictions within a single model, whereas C-BERT uses additional architectural layers for multiple objective functions. To quantify the variance in effect estimation for proposed models, compute the mean and standard error of the ATT estimates across three trials. Note that most language models tend to have high variance due to the heavy-tailed nalure of next-loken prediction. Veitch et al. 2020) do not report uncertainties in their experiments.
Our proposed framework leverages the existing knowledge in tre-tr LLMs to accurately estimate causal quantitics. While fino-tuning GPT yiclds successful results, training the model from scratch fails to identify causal effects because the model lacks prior understanding of text. We also demonstrate that our results are robust to the size of the pre-trained language model in Appendix D By using LLMss, approach outperforms non-autoregressive causal methods and proves more effective for a wider range of CI tasks.
7 Conclusion
In this work, we introduce an AR framework for CI that handles high-dimensional confounders and combi- natorially large action spaces. Our proposed method, called sequencification, transforms data into linear sequence of tokens based on known causal diagram. By training a single AR model on sequencified data, we learn conditional distributions between variables in the graph. The framework enables efficient sampling and approximation of several interventional distributions in a unified manner.
We validate the offectiveness of our method for inferring causal effects across three diverse scttings: maze navigation, chess endgames, and academic papers. By handling sigh-dimensional confounders and combina- torially large action sequences, our work extends the capabilities of CT for wider range of applications.
Our approach has two main limitations. First, it requires the full causal graph to be known exactly, with all variables observed. A possible solution for handling unobserved values is to impute unconfounded missing data using a mask token. Second, our method supports conditioning or intervening only on variable prefixes. We leave the investigation of solutions to these limitations for future work.
Acknowledgements
This work was supported by the Institute of Information & Communications Technology Planning & Eval- uation (IITP) with a grant funded by the Ministry of Science and ICT (MSIT) of the Republic of Korea in connection with the Global AT Frontier Lab International Collaborative Research. This work was also sup- ported by the Samsung Advanced Institute of Technology (under the project Next Generation Deep Learning: From Pattern Recognition to AT) and the National Science Foundation (under NSF Award 1922658).
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A
Training details
The maze experiments conducted single NVIDIA Tesla T4. All models for the chess and PeerRead experiments were trained on single NVIDIA GeForce RTX 3090 in four and eight hours respectively.
Maze experiments. The maze dataset comprises 10,000 sequencified data points. We use vanilla trans- former with 3 layers, attention heads, and a hidden dimension of 64. For training, we use the Adam optimizer with a batch size of 64. The model is trained for 6,250 iterations, while the offine RL model is trained for 5,000 iterations.
Chess endgame experiments. We use 512-dimensional vanilla transformer with 6 layers and attention heads. The model is trained on next-token prediction task using the sequencified representation. Training runs for 200 epochs with the Adam optimizer, a batch size of 4096, and learning rate chosen to be as large as possible without overfitting.
For the training dataset, we sample 500,000 two-move chess games per dataset based on Black's policy function. The test datasot includes every game from all 223,660 legal starting positions and all four possible Black actions (king-king, king-rook, rook-king, rook-rook). We sequencify the data by assigning unique token to each square, legal king and rook move, and outcome.
PeerRead experiments. We fine-tuned our models using pre-trained BERT and GPT base model check- points
0 For BERT, we employed a two-phase training process similar to C-BERT. In the first phase, we trained BERT to generate abstracts, followed by a second phase where it learned to generate full sequences, including both actions and outcomes. GPT, having been pre-trained on next-token prediction, required only single training phase. This approach ensures a gradual refinement of the generative capabilities specifically tailored to the PeerRead corpus. For all training phases, we trained for 100 epochs using the Adam optimizer with batch size of 16. The learning rate set high possible without overfitting
Maze extra experiments
We investigate whether the outcomes produced by our model have some interpretability. To study the predictions of our model, we reuse the synthetic maze setup in Section 6.1 with slightly modified obstacle positions. The configuration of the maze is shown Figure 0
Go right
START
down?
END
Figure 6: Illustration of the maze setup. We answer causal question involving intervening on partial path: is it better to go right or down?
TBERT-Base is available http:://awgginnfccc...oo//////// http:://hugginggfccccco/// wttp:/aaccccoooaaaiaaaaaaaaaaaaa httppppaaa-ccomiiit//s.t... community/gpt?
and GPT available
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Potential outcomes
Potential outcomes
Action
Right 0.80
0458
8438
8:33
6:30
0.00
048
0480
8600
8:03
000
Action
Right
8005
6:00
0:38
0453
0.28
8:00
8630
6:80
0:0
0.83
0.01
Down
0:30
6.30
0.10
0.00
Down
8006
6.18
0.308
0.12
8.00
0.00
0
5
10
0
5
6
10
Potential outcomes MSE
Potential outcomes MSE
Action
Right
Action
Right
6.03
8.00
dlop
Down
008
Down
1008
-009
0401 0008
0.00
10
5
6 Outcome (Distance)
10
Outcome (Distance)
(a) Marginalization over paths from A2 ....... As.
(b) Monte Carlo sampling on paths from A As-
Figure 7: Potential outcome MSE between ground truth and model estimates for moving right vs, down first. Both the eaact model estimate (a) and approzimate model estimate (b) are shown.
Following the original experimental design, the objective to determine the distance the exit after following a path. The confounding variable X is the start and end positions, the action A is the sequence of moves in a path, and the outcome Y is the distance from the final position to the exit.
We consider the question: for fixed starting state, is it better to go right or down? To answer this, we compute the potential outcome of moving right or down initially. This task is prohibitive for most non- autoregressive CI models as they treat the path as a singular variable. Our model is capable of intervening on any subsequence of actions because the conditional probability given its parents is tractable.
To compute the potential outcome, we marginalize over subsequent actions as shown in Equation 4 Since the paths may be intractable without limiting the maximum length, cap the longest path to 16 moves there are total of 16 positions in the maze. To compare the exact estimates with the Monte Carlo approximations, we report both quantities derived from Equation 4 and 5 and denote them as ezact model estimale and approzimale model eslimale respectively.
Figure i shows the potential outcome estimate of moving right or down initially. The distribution is more concentrated towards smaller distances for moving down and more uniform for moving right. Thus, going down is better choice for the first move, which matches our intuition based on the maze configuration.
To demonstrate the scalability of our methodology with respect to the number of variables, the maze experiments using the same setup described in Section 6.1 Specifically, we adjust the maze dimensionality and path length for fine-gras inneggriiedd control over the covariates X and actions A. We increase the maze from 2D (i.c., 4 x 4) up to 5D (i.c., x 4 x 4 x 4 x 4), while keeping the path length fixed at six. Since d dimensions is defined by d coordinates, this directly controls the dimensionality of X. To vary the number of actions, we modify the path length from four to ten while holding the maze structure fixed at 2D. All model and training details (e.g. number of samples and number of optimization epochs) are identical to the original maze experiments.
extend
the starting position
We evaluate our model by computing the absolute error in estimating the effect of intervening on every possible path of a specified length. Figure D shows the distribution of these errors as we vary the maze dimensionality and path length. The average estimation error increases with both maze dimensionality and path length due to growing data complexity. The variance also increases significantly with longer paths and more gradually with digher-dimensional mazes. Nonetheless, the overall estimation accuracy remains high, with the majority of errors within 0.5 of the true effect across all configurations.
C Chess endgame potential outcome estimates
Table 2 compares the potential outcome values for all actions, presenting both exact model estimates and approximate model estimates. Our AR model performs similarly to the bascline across both metrics. Predic- tions from the model aligns with the ground truth answer to our counterfactual query: on average, moving the rook twice leads to the fastest checkmate.
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Fixed Maze Dimension (2D)
Fixed Path Length (6 moves)
4.0
3.5
3.0
2.5
Error
2.0
1.5
1.0
0.5
0.0
8
10
2
5
Path Length
Maze Dimension
Figure 8: Distribution of effect estimation errors for all possible interventions of the given path length. The inner box indicates the lower and upper quartiles of the distribution, while point represents the meann The overall error increases with the input complexity (i.e. higher-dimensiooall mazes and longer path sequences).
Table Chess endgame potential outcome estimates for all actions. The outcome represents the number of additional moves required for checkmate. Since Black aims to achieve checkmate as quickly possible, lower values are desired.
Potential outcome (Error %)
king-king
king-rook
rook-king
rook-rook
Ground truth
22.76
20.18
20.48
17.27
RCT
22.51
(1.1%)
19.96 (1.1%)
19.85
(3.1%)
17.08 (1.1%)
MLP exact
non-RCTi
22.40 (1.6%)
19.85 (1.6%)
20.01 (2.3%)
17.10 (1.0%)
non-RCT,
22.24 (2.3%)
19.90 (1.4%)
19.69 (3.9%)
17.14 (0.8%)
RCT
22.17 (2.6%)
19.45 (3.6%)
17.72 (13%)
16.18 (6.3%)
MLP approx.
non-RCT,
22.04 (3.2%)
19.32 (4.3%)
17.91 (13%)
16.12 (6.6%)
non-RCT2
21.86 (4.0%)
19.31 (4.3%)
17.86 (13%)
16.18 (6.3%)
RCT
22.38 (1.7%)
19.75 (2.1%)
19.96 (2.5%)
16.95 (1.9%)
AR exact
non-RCT,
22.48 (1.2%)
19.87 (1.5%)
20.08 (2.0%)
17.08 (1.1%)
Ron-RCT,
22.05 (3.1%)
19.88 (1.5%)
19.80 (3.3%)
17.17 (0.6%)
RCT
21.90 (3.8%)
19.19 (4.9%)
17.62 (14%)
15.97 (7.5%)
AR approx.
non-RCT,
22.03 (3.2%)
19.33 (4.2%)
17.95 (12%)
16.15 (6.5%)
non-RCT2
21.79 (1.3%)
19.29 (1.4%)
17.54 (14%)
16.31 (5.6%)
In Figure g we present the potential outcome graphs for the remaining three actions: king-king, king-rook, and rook-rook. The model behavior on these actions closely aligns with the observations for rook-king in Figure Across multiple interventions, Monte Carlo sampling improves potential outeome estimates when only subset of the test data available. Thus, our approach not only effectively models outcomes but also leverages Monte Carlo sampling to enhance predictions in limited data scenarios.
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RCT
non-RCT,
non- non-RCT2
Ground truth
exact
Potential outcome
22.50
apprax.
AR MC
MLPP exact
22.25
MLP approx
22.00
21.75
10
10
10
10a
10
10
10a
104
105
Number of Monte Carlo samples
(a) King-king potential outcome estimates.
RCT
non-RCT,
non-RCT2
20.2
Potential outcome
20.0
19.8
19.6
Ground truth
exact
approx.
19.4
AR MC
MLP exact
19.2
MLP apprax.
103
10a
10
10
10
10
10a
10
10
Number of Monte Carlo samples
(b) King-rook potential outcome estimates.
RCT
non-RCT,
non-RC non-RCT2
17.25
Potential outcome
17.00
16.75
16.50
AR exact
AR approx.
16.25
AR MC
16.00
MLP approx
10a
10
10
10a
10
10a
10a
10*
10
Number of Monte Carlo samples
(c) Rook-rook potential outcome estimates.
Figure 9: Polential outcome estimates for (a) king-king, (b) king-rook, and (c) rook-rook actions. Similar to the rook-king intervention, sampling from (X) enables the AR Moute Carlo estimate approach the AR exact estimate,
D PeerRead extra experiments
We examine the rale at which the ATT error converges lo study causal effect prediction in small data cases. Using the PeerRead experimental setup, we vary the training set size from few hundred papers to the full dataset (9305 samples). We repeat each estimation across three trials and report the mean and standard error of the relative ATT error in Figure 10 For our BERT and GPT models, the error decays gradually with the number of training samples.
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Published in Transactions on Machine Learning Research (07/2025)
140
DARCIE-BERT DARCIE-GPT
120
Relative ATT Error (%
100
80
60
40
20
10a
10
10a
Training samples
Figure 10: Mean and standard deviation of the relative ATT error while varying the training set size. The training data increases exponentially from hundred points to the entire dataset of 9305 samples.
We also study the impact of pre-trained language model size on effect estimation accuracy using three versions of GPT. Table 3 indicates that larger models generally yield more accurate estimates.
Furthermore, we demonstrate why our model performs well when the outcome generated from Z and A while the input is derived from X and A. To quantify the correlation between title buzziness and the abstract, we train logistic regression model to predict Z from X. For each model, we extract the final hidden layer output dense vectorized representation of the abstract. As baseline, we train separate logistic regression model using the bag-of-worde (BoW) representation of X. Table g shows strong correlation between X and Z, explaining the high accuracy of our potential outcome estimations.
Table 3: PeerRead ATT estimates for GPT small, medium,, and large, with relative error shown in parentheses.
Table 4: Accuracy and balanced accuracy of title buzziness predic- tion from the abstract using logis- tic regression.
Confounding level
# params
Low
Medium
High
Aoc.
Balanced
Ground truth
0.062
0.059
0.028
GPT-small
117M
0.050 (20%)
0.044 (25%)
0.020 (29%)
BoW
87.95%
73.27%
GPT-medium
345M
0.052 (16%)
0.053 (10%)
0.038 (36%)
BERT
88.37%
77.91%
GPT-large 774M
0.051 (18%)
0.054 (8%)
0.021 (25%)
GPT2
89.18%
81.00%
To illustrate the positive predicted ATT on the PeerRead datasot, we compare PY 1 A X) and
pY = 0, X) predicted by GPT in Figure 11 Our model consistently favors papers containing theorems regardless of title buzziness.
E
Ablation studies on IHDP
Although the experiments in Scction are designed to demonstrate the effectiveness of our AR model in scenarios involving variable-length action sequences and high-dinensional covariates, it is also competitive with existing methods on benchmark causal datasets. To illustrate this, we evaluate our model on a semi- synthetic bascline using the Infant Health and Development Program (IHDP) data. The datasct consists of 25-dimensional covariate (6 of which are continuous features), binary treatment A, and a continuous outcome Y. The evaluation metric is ATE prediction error.
20

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Published in Transactions on Machine Learning Research (07/2025)
==
0.7
==1
0.6
A
0.5
P(Y
0.4
0.3
0.3
0.4
0.5
0.6
0.7
p(Y = 1 A = 0, X)
Figure 11: Conditional outcome distributions given A 0 vs. A 1 for medium confounding data (3 5).
We compare performance with four other baseline models: a neural network (NN), random forest (RF), causal forest (CF), and CRNET (Zhu et al.) 2024). For the AR model, we use foodforward network embedding layer to handle continuous numerical values. We report the mean and standard error of the ATE error on the test across 30 different random train-test splits in Table Our method is competitive with common existing methods and outperforms CRNET while having lower variance.
Table 5: Evaluation existing causal methods and our proposed model on the IHDP benchmark. We report the mean and standard error of the absolute and relative ATE error across 30 training trials. The best performing model underlined and bold.
Model
ATE Error
ATE Relative Error (%)
NN
0.1651 + 0.0256
4.11 + 0.64%
RF
0.1171 + 0.0110
2.92 + 0.27%
CF CRNET
0.1085 + 0.0147
2.70 -: 0.37%
0.1550 + 0.0694
3.86 1.73%
DARCIE (Ours)
0,1104 + 0.0118
2.75 0.29%
21


Reviews:
[{"review_id": "4c4LCusYMD", "reviewer": "Reviewer_PG84", "summary": "", "questions": "", "limitations": "", "rating": null, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "Strengths:\n1. A unified and simple structure that handles multiple queries and scale to higher dimensional input space.\n2. It uses sequencification and sampling, which enable flexible and efficient inference.\n3. Strong empirical results demonstrating its broad use case for various tasks.\n\n\n\nWeaknesses:\n1. It requires knowledge on the actual DAG, which limits its application in real world-setting.\n2. Due to its autoregressive nature, it can only intervene on the prefixes of variable sequence.\n3. One causal inference challenge is the discrepancy between treatment and control space, and it is not well understood in the paper how robust this structure is to this challenge.", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "Yes", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "paper_topic_and_main_contributions": "", "excitement": null, "reproducibility": null, "ethical_concerns": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "summary_of_contributions": "This paper introduces a unified autoregressive (AR) model for causal inference that handles high-dimensional confounders and sequential actions. It fills the gaps left by traditional causal frameworks, which typically deal only with singleton actions.\n\nAdditionally, by transforming causal data into token sequences (sequencification), a single AR model can efficiently estimate a wide range of causal effects. Despite its simple design, the model proves effective across various tasks such as maze navigation, chess endgames, and text-based causal inference.", "claims_and_evidence_yn": "Yes", "claims_explanation": "", "audience_yn": "Yes", "audience_explanation": "", "requested_changes": "1. Though the paper depends on the actual DAG structure, it is based on the chain rule P(x1,x2,...xn)=p(x1)p(x2|x1)p(x3|x1,x2)p(...). From this perspective, it is worth asking whether the exact DAG is critical for performance. I suggest the authors test the robustness of their method by evaluating performance when the assumed DAG differs from the true causal structure.\n\n2. In Figure 3, the proposed method consistently exhibits higher variance in estimation errors compared to the baseline. It would be helpful to know whether this increased variance persists in other tasks. For example, in Table 1, could the authors include confidence intervals or standard deviations to better quantify uncertainty? \n\n3. In the last task, the probability of treatment is a function of title Z. however, the relationship between Z and X is not well demonstrated. A simple way to demonstrate this would be to report how well  X predicts  Z This would also clarify how varying the $\\beta$ parameter captures discrepancy in treatment assignment mechanisms", "broader_impact_concerns": "NA", "additional_comments": "", "extra_scores": {}, "extra_text": {}}, {"review_id": "DbH9Y2QHI2", "reviewer": "Reviewer_wJja", "summary": "", "questions": "", "limitations": "", "rating": null, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "__Strengths__\n\n* The paper successfully connects autoregressive modeling and causal inference in an innovative way with the help of causal ordering, addressing the limitation of high-dimensional inputs in traditional causal inference methods. \n* This approach unify the computation of multiple causal quantities with a single model, rather than requiring different models for different causal queries.\n* They demonstrate that the pre-trained language models improve performance on text-based causal inference tasks is promising, showing how domain knowledge captured during pre-training can enhance causal estimation.\n\n__Weakness__\n* The approach assumes complete knowledge of the causal graph with all variables observed. This is a strong assumption that limits applicability in settings where the causal structure is uncertain or where unmeasured confounding exists. \n* This approach would need relatively large sample to train and thus might be sensitive to real-world small data cases and large data cases where hidden confounders are more commonly presented.\n* The paper does not discuss much about the estimation uncertainty of proposed approach, which is crucial for reliable decision-making based on these estimates.\n* According to the Figure 3, it seems like the estimation's instability (estimation standard deviation) increases when the input becomes more complex (changes in X1), compared to the RL method (is this because of the error accumulation?). \n* There is no analysis of how sensitive the approach is to misspecification of the causal graph or to either complex graph or small data cases.\n\n__Question__\n* What's the defined causal ordering in the PeerRead experiment? It seems like a general token-wise ordering instead of phrases/concepts. Could the authors give some comments on the causal ordering in text-based/other unstructured data-based causal inference, and how the proposed approach could be applied?", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "Yes", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "paper_topic_and_main_contributions": "", "excitement": null, "reproducibility": null, "ethical_concerns": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "summary_of_contributions": "This paper introduces a new approach to causal inference by adapting autoregressive (AR) models to estimate causal effects. The authors position this work as bridging the gap between the capabilities of modern autoregressive models and causal inference, potentially enabling causal reasoning in complex, high-dimensional settings that were previously challenging for traditional causal inference methods. They propose a method called \"sequencification\" that transforms data from causal directed acyclic graphs (DAGs) into sequences of tokens, enabling a unified modeling to estimate multiple causal quantities (interventional, partial interventional, and conditional interventional distributions). They conduct empirical validation across three diverse domains: maze navigation (demonstrating the versatility of the unified model), chess endgames (showcasing Monte Carlo approximation), and academic paper acceptance prediction (leveraging pre-trained language models for text-based causal inference).", "claims_and_evidence_yn": "Yes", "claims_explanation": "", "audience_yn": "Yes", "audience_explanation": "", "requested_changes": "* The authors could consider investigating how robust the approach is to misspecification of the causal graph, more complex graph and much smaller data cases. \n* The paper would benefit from quantifying and discussing the uncertainty in the estimation, e.g. confidence intervals of the causal effects.\n* It would be helpful to provide a discussion about the method's scalability in terms of the number of variables/dimensions and relationships, which would better support the claim of handling high-dimensional input.", "broader_impact_concerns": "N/A", "additional_comments": "", "extra_scores": {}, "extra_text": {}}, {"review_id": "ufQaMOJSYF", "reviewer": "Reviewer_Fjad", "summary": "", "questions": "", "limitations": "", "rating": null, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "**Strength**\n1. The idea of leveraging language models to model the (conditional) distribution of the underlying DAG which features a sequential nature (through causal ordering) is interesting and novel.\n2. Empirical experiments showed the effectiveness of the proposed method.\n\n**Weakness**\n1. It is not well-written and structured as a paper in the field of causal inference. \n      - The introduction is too brief and merely an extension of the abstract, making it hard to follow for the general audience. For example, it does not specifically provide the causal inference task this paper is focused on (e.g. interventional distribution approximation, (individual/average) treatment effect estimation, counterfactual prediction, etc.), the motivation and practical application of the task, general ideas of AR models, why and how it can be related to your task.\n      - Sections 2 and 3 are not extensive which lack the discussions of many recent papers. For example, there is already existing literature tackling high-dimensional interventional distribution approximation [1], heterogeneous treatment effect estimation with sequential/temporal treatments [2,3], and high-dimensional heterogeneous dose-response curves estimation [4,5,6]. The authors should discuss how their method and task are related to/deviated from them. Moreover, what does it mean by a causal engine? The definition is not provided. Also, why is there a $y\\cdot$ in Equation 1? \n      - The basic assumptions for the causal inference task are not clearly stated and discussed. For example, I assume the underlying distribution should satisfy the Markov property to be factorized sequentially. The definition of positivity condition, and how the proposed method deals with it, including the difference with existing methods, should be clearly discussed rather than briefly mentioned in Section 2. The problem setting can be defined more rigorously, c.f. Veitch et al. 2020.\n      - While the description of the method is mostly clear, could you explain in more detail why the $⟨start_i⟩$ token at the start of each variable $V_i$ during sequencification helps obtain good estimates under random topological orderings? Could you provide a concrete example or pseudo code?\n\n2. As far as I understand, all the interventional distributions are generated by conditional distributions using the valid adjustment set $X$. Therefore, any method that models the conditional distributions can serve as a \"causal engine\" here. While I acknowledge that language models can be powerful in modeling such distributions, it would be nice if the author could provide an illustration of the embedding of the tokens on a toy example. For example, how is the sequence of tokens of [Y] similar or different given a fixed X and different actions A?\n\n3. Benchmarks and baseline methods. \n   - I think Veitch et al. 2020 is the most relevant literature to the current paper. Therefore, I expect the authors to conduct more extensive comparisons, e.g. including the baseline method C-ATM, and the benchmark dataset Reddit. What is the computational burden compared to Veitch et al. 2020? Also, DARCIE seems to be the name of the proposed method, what does it mean? \n   - I am not fully convinced that the Maze and Chess datasets are suitable for evaluating causal inference methods. The potential outcome of an interventional move is deterministic rather than probabilistic and the problems are originally from reinforcement learning (RL) and Markov decision processes. I wonder if other literature on causal inference uses this kind of dataset as the benchmark. I think the experiments are good examples of how causal inference ideas and methods can be applied to RL tasks, but not a proper evaluation of the proposed method on causal inference tasks. Note that the baseline method is also from RL.\n   - The author should explain why they did not include common benchmark datasets on treatment effect estimation (which the method seems to be able to deal with) such as MIMIC, TCGA [5] IHDP, News [6], and the synthesis dataset approach they used. Please clearly state the specific tasks the proposed method can deal with. \n\n4. Ablation studies. One of the key features of the proposed method is randomized topological orderings, the author could provide more analysis on how it contributes to the method's performance.\n\n**References:**\n\n[1] Melnychuk, Valentyn, Dennis Frauen, and Stefan Feuerriegel. \"Normalizing flows for interventional density estimation.\" International Conference on Machine Learning. PMLR, (2023).\n\n[2] Frauen, Dennis, Konstantin Hess, and Stefan Feuerriegel. \"Model-agnostic meta-learners for estimating heterogeneous treatment effects over time.\" arXiv preprint arXiv:2407.05287 (2024).\n\n[3] Kacprzyk, Krzysztof, et al. \"ODE Discovery for Longitudinal Heterogeneous Treatment Effects Inference.\" The Twelfth International Conference on Learning Representations (2024).\n\n[4] Zhu, Minqin, et al. \"Contrastive balancing representation learning for heterogeneous dose-response curves estimation.\" Proceedings of the AAAI conference on artificial intelligence. Vol. 38. No. 15. (2024).\n\n[5] Bica, Ioana, James Jordon, and Mihaela van der Schaar. \"Estimating the effects of continuous-valued interventions using generative adversarial networks.\" Advances in Neural Information Processing Systems 33 (2020): 16434-16445.\n\n[6] Nie, Lizhen, et al. \"Vcnet and functional targeted regularization for learning causal effects of continuous treatments.\" arXiv preprint arXiv:2103.07861 (2021).", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "No", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "paper_topic_and_main_contributions": "", "excitement": null, "reproducibility": null, "ethical_concerns": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "summary_of_contributions": "The paper proposes a novel causal inference (CI) framework leveraging autoregressive (AR) models—specifically language models—to estimate causal effects from data transformed into token sequences via sequencification. The approach supports high-dimensional confounders and combinatorially large action spaces, and uses a single trained model to answer multiple types of causal queries, such as interventional and conditional distributions. The framework is validated on synthetic maze navigation and chess endgame tasks, as well as the real-world PeerRead dataset.", "claims_and_evidence_yn": "No", "claims_explanation": "", "audience_yn": "Yes", "audience_explanation": "", "requested_changes": "See weakness. In particular, the assumptions, motivations, the specific task the method deals with, and related works should be clearly stated and well-structured. More benchmarks and baseline methods should be provided with motivations discussed.", "broader_impact_concerns": "NA", "additional_comments": "", "extra_scores": {}, "extra_text": {}}]

Author rebuttal:

Decision: Accept (poster)