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arxiv: 2410.06128 · v4 · submitted 2024-10-08 · 💻 cs.LG · stat.ML

Amortized Inference of Causal Models via Conditional Fixed-Point Iterations

Divyat Mahajan , Jannes Gladrow , Agrin Hilmkil , Cheng Zhang , Meyer Scetbon This is my paper

Pith reviewed 2026-05-23 19:28 UTC · model grok-4.3

classification 💻 cs.LG stat.ML
keywords amortized inferencestructural causal modelsfixed-point iterationtransformer embeddingsout-of-distribution generalizationcausal mechanism recoveryinterventional data generation
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The pith

A single trained model can infer causal mechanisms for any structural causal model from its observational data and graph.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper develops an amortized method that trains one model to recover the functional mechanisms of structural causal models when given both the observational dataset and the causal graph. It first embeds each dataset with a transformer, then runs conditional fixed-point iterations to find the mechanisms that are consistent with the data under that graph. Because the model is shared across all training SCMs, it can be applied at test time to entirely new graphs and datasets without any parameter updates. This setup is shown to match the accuracy of models trained from scratch on each individual dataset while using far less data per new problem.

Core claim

The central claim is that conditioning the fixed-point iteration procedure on a transformer-derived dataset embedding produces an amortized estimator that recovers the true causal mechanisms of both in-distribution and out-of-distribution SCMs at the same level as dataset-specific baselines, and exceeds them when training data per SCM is limited.

What carries the argument

Conditional fixed-point iterations that take a dataset embedding produced by a transformer as an additional input to the iteration map, allowing the same trained parameters to solve for mechanisms across many different SCMs.

If this is right

  • One set of learned parameters suffices for mechanism inference on any number of new causal graphs and datasets.
  • Interventional data can be generated from novel SCMs by first inferring their mechanisms and then sampling from the resulting model.
  • Performance remains competitive even when each new dataset supplies only a few hundred observations.
  • The same architecture supports both in-distribution and out-of-distribution generalization without retraining.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • If the embedding step can be made to work for graphs with hundreds of nodes, the method could scale amortized causal discovery to domains where per-dataset training is currently prohibitive.
  • The conditional fixed-point layer could be swapped into other causal inference pipelines that already use iterative solvers, potentially amortizing those pipelines as well.
  • Because the model outputs a full SCM rather than a point estimate, downstream tasks such as policy optimization or counterfactual reasoning could reuse the same inferred mechanisms across multiple queries.

Load-bearing premise

The transformer embeddings of the observational data are rich enough to let the same fixed-point solver recover accurate mechanisms for both the training distribution of SCMs and for previously unseen graphs and data distributions.

What would settle it

Train the amortized model on a family of SCMs, then measure whether its recovered mechanisms on a held-out family of SCMs produce interventional distributions whose total variation distance to the true interventional distributions exceeds the distance achieved by a model retrained from scratch on each held-out SCM.

Figures

Figures reproduced from arXiv: 2410.06128 by Agrin Hilmkil, Cheng Zhang, Divyat Mahajan, Jannes Gladrow, Meyer Scetbon.

Figure 1
Figure 1. Figure 1: Sketch of the approach proposed in this work. Given a dataset of observations [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: In-Distribution Results. Benchmarking Cond-FiP for various evaluation tasks, with datasets sampled from RFF IN with d = 20. The y-axis denotes the RMSE, with mean and standard error over the respective test datasets. Results indicate Cond-FiP can generalize to novel in-distribution instances, with detailed results in Appendix C. DoWhy DECI FiP Cond-FiP 0.15 0.20 0.25 0.30 0.35 (a) Noise Prediction DoWhy DE… view at source ↗
Figure 3
Figure 3. Figure 3: OOD Results. Benchmarking Cond-FiP for various evaluation tasks, with datasets sampled from RFF OUT with d = 100 to test for OOD generalization. The y-axis denotes the RMSE, with mean and standard error over the respective test datasets. Results indicate Cond-FiP can generalize to novel OOD instances and larger graphs, with detailed results in Appendix C. Evaluation Tasks. We evaluate the methods on the fo… view at source ↗
Figure 4
Figure 4. Figure 4: Scarce Data Regime Results. Benchmarking Cond-FiP on the various evaluation tasks (RFF OUT and d = 100) as we reduce the test dataset size. The y-axis denotes the RMSE, with mean and standard error over the respective test datasets. Cond-FiP generalizes much better than the baselines in the low-data regime, with detailed results in Appendix E. DoWhy DECI FiP Cond-FiP 0.45 0.50 0.55 0.60 (a) Noise Predictio… view at source ↗
Figure 5
Figure 5. Figure 5: OOD Results without True Graph. Benchmarking Cond-FiP for various evaluation tasks, with datasets sampled from RFF OUT with d = 100 where the true graph G is not present in input context, rather its inferred via AVICI. The y-axis denotes the RMSE, with mean and standard error over the respective test datasets. Results indicate Cond-FiP can generalize to novel instances even in the absence of true graph, wi… view at source ↗
Figure 6
Figure 6. Figure 6: CSuite Results. Benchmarking Cond-FiP on the various evaluation tasks on the CSuite benchmark, which uses a different data simulator than the Cond-FiP’s training data simulator. The y-axis denotes the RMSE, with mean and standard error across the 9 test datasets. the encoder requires access to true noise variables, it can still be used for inference. We evaluate the quality of generated samples by comparin… view at source ↗
Figure 7
Figure 7. Figure 7: We compare Cond-FiP against the baselines for the different evaluation tasks on the [PITH_FULL_IMAGE:figures/full_fig_p035_7.png] view at source ↗
read the original abstract

Structural Causal Models (SCMs) offer a principled framework to reason about interventions and support out-of-distribution generalization, which are key goals in scientific discovery. However, the task of learning SCMs from observed data poses formidable challenges, and often requires training a separate model for each dataset. In this work, we propose an amortized inference framework that trains a single model to predict the causal mechanisms of SCMs conditioned on their observational data and causal graph. We first use a transformer-based architecture for amortized learning of dataset embeddings, and then extend the Fixed-Point Approach (FiP) to infer the causal mechanisms conditionally on their dataset embeddings. As a byproduct, our method can generate observational and interventional data from novel SCMs at inference time, without updating parameters. Empirical results show that our amortized procedure performs on par with baselines trained specifically for each dataset on both in and out-of-distribution problems, and also outperforms them in scarce data regimes.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The paper introduces an amortized inference framework for Structural Causal Models (SCMs). It trains a single model using a transformer architecture to produce dataset embeddings from observational data and the causal graph, then extends the Fixed-Point Approach (FiP) to infer causal mechanisms conditionally on those embeddings. The method supports generation of observational and interventional data from novel SCMs at inference time without parameter updates. The central empirical claim is that this single amortized model matches the performance of per-dataset baselines on both in-distribution and out-of-distribution tasks while outperforming them in scarce-data regimes.

Significance. If the empirical claims hold under rigorous evaluation, the work would represent a meaningful advance in amortized causal discovery by removing the need for dataset-specific retraining, which is a practical bottleneck. The combination of transformer embeddings with conditional FiP iterations offers a scalable route to handling multiple SCMs and supports OOD generalization and low-data performance, both of which are relevant to scientific discovery applications. No machine-checked proofs or parameter-free derivations are reported, but the reproducible-code potential of the amortized setup is a positive feature if the implementation details are released.

major comments (2)
  1. [Experiments] Experiments section: the abstract and summary assert performance parity and low-data gains, yet the provided description supplies no information on the precise baselines, error bars, data splits, or evaluation metrics used. Without these details it is impossible to determine whether the reported results actually support the central claim that a single model matches or exceeds dataset-specific training.
  2. [Method] Method section on conditional FiP: the extension of the Fixed-Point Approach to condition on transformer-derived embeddings is load-bearing for the amortized claim. The manuscript must explicitly show how the conditioning is implemented (e.g., which layers receive the embedding, how the fixed-point iteration is modified) and verify that the procedure remains convergent under this conditioning.
minor comments (2)
  1. Notation for dataset embeddings should be introduced once and used consistently; the current description alternates between “dataset embeddings” and “conditional embeddings” without a clear mapping.
  2. [Experiments] The paper should include a short table summarizing the number of SCMs, sample sizes, and graph densities used in the in-distribution, OOD, and scarce-data regimes.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive review and the recommendation of minor revision. Below we respond point-by-point to the major comments, indicating the revisions we will make.

read point-by-point responses

  1. Referee: [Experiments] Experiments section: the abstract and summary assert performance parity and low-data gains, yet the provided description supplies no information on the precise baselines, error bars, data splits, or evaluation metrics used. Without these details it is impossible to determine whether the reported results actually support the central claim that a single model matches or exceeds dataset-specific training.

    Authors: We thank the referee for this observation. While the full details appear in Section 4 and Appendix B (baselines are the per-dataset FiP and a non-amortized transformer variant; error bars are mean ± std over 5 seeds; data splits follow an 80/20 observational/interventional protocol with SCMs generated from the same distribution as training; metrics are mechanism MSE and interventional NLL), we agree the presentation can be improved for immediate accessibility. We will insert a concise 'Evaluation Protocol' paragraph at the opening of Section 4 that explicitly lists these elements and cross-references the tables. revision: yes

  2. Referee: [Method] Method section on conditional FiP: the extension of the Fixed-Point Approach to condition on transformer-derived embeddings is load-bearing for the amortized claim. The manuscript must explicitly show how the conditioning is implemented (e.g., which layers receive the embedding, how the fixed-point iteration is modified) and verify that the procedure remains convergent under this conditioning.

    Authors: We agree that the conditioning implementation should be stated more explicitly. In the revision we will expand Section 3.2 to describe that the dataset embedding is (i) concatenated to the initial node features before the first FiP iteration and (ii) supplied via cross-attention to every subsequent iteration layer. We will also add a short appendix subsection containing both a contraction-mapping argument (under the same Lipschitz assumptions used in the original FiP work) and empirical convergence curves confirming that the number of iterations required remains comparable to the unconditional baseline. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation is self-contained

full rationale

The paper presents an amortized inference method combining transformer-based dataset embeddings with an extension of the Fixed-Point Approach (FiP) to infer causal mechanisms conditionally. The core empirical claim compares performance against external per-dataset baselines on in-distribution, out-of-distribution, and scarce-data regimes, without any reduction of predictions to fitted inputs or self-citations that bear the load of the central result. No self-definitional equations, ansatz smuggling, or renaming of known results appear in the abstract or described framework; the method is presented as a standard architectural proposal with independent empirical support.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review yields no identifiable free parameters, axioms, or invented entities; full text would be needed to audit modeling assumptions such as the form of the fixed-point operator or embedding sufficiency.

pith-pipeline@v0.9.0 · 5699 in / 1022 out tokens · 22567 ms · 2026-05-23T19:28:17.696347+00:00 · methodology

discussion (0)

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Reference graph

Works this paper leans on

21 extracted references · 21 canonical work pages · 2 internal anchors

  1. [1]

    Akhound-Sadegh, J

    T. Akhound-Sadegh, J. Rector-Brooks, A. J. Bose, S. Mittal, P. Lemos, C.-H. Liu, M. Sendera, S. Ravanbakhsh, G. Gidel, Y . Bengio, et al. Iterated denoising energy matching for sampling from boltzmann densities. arXiv preprint arXiv:2402.06121,

    work page arXiv

  2. [2]

    What learning algorithm is in-context learning? Investigations with linear models

    E. Akyürek, D. Schuurmans, J. Andreas, T. Ma, and D. Zhou. What learning algorithm is in-context learning? investigations with linear models. arXiv preprint arXiv:2211.15661,

    work page internal anchor Pith review Pith/arXiv arXiv

  3. [3]

    Blöbaum, P

    P. Blöbaum, P. Götz, K. Budhathoki, A. A. Mastakouri, and D. Janzing. Dowhy-gcm: An extension of dowhy for causal inference in graphical causal models. arXiv preprint arXiv:2206.06821,

    work page arXiv

  4. [4]

    A. Dhir, M. Ashman, J. Requeima, and M. van der Wilk. A meta-learning approach to bayesian causal discovery. arXiv preprint arXiv:2412.16577,

    work page arXiv

  5. [5]

    Geffner, J

    T. Geffner, J. Antoran, A. Foster, W. Gong, C. Ma, E. Kiciman, A. Sharma, A. Lamb, M. Kukla, N. Pawlowski, et al. Deep end-to-end causal inference. arXiv preprint arXiv:2202.02195,

    work page arXiv

  6. [6]

    Gupta, C

    S. Gupta, C. Zhang, and A. Hilmkil. Learned causal method prediction. arXiv preprint arXiv:2311.03989,

    work page arXiv

  7. [7]

    Analyzing and improving the training dynamics of diffusion models

    T. Karras, M. Aittala, J. Lehtinen, J. Hellsten, T. Aila, and S. Laine. Analyzing and improving the training dynamics of diffusion models. ArXiv, abs/2312.02696,

    work page arXiv

  8. [8]

    semanticscholar.org/CorpusID:265659032

    URL https://api. semanticscholar.org/CorpusID:265659032. 11 N. R. Ke, S. Chiappa, J. Wang, A. Goyal, J. Bornschein, M. Rey, T. Weber, M. Botvinic, M. Mozer, and D. J. Rezende. Learning to induce causal structure. arXiv preprint arXiv:2204.04875,

    work page arXiv

  9. [9]

    Mittal, E

    S. Mittal, E. Elmoznino, L. Gagnon, S. Bhardwaj, D. Sridhar, and G. Lajoie. Does learning the right latent variables necessarily improve in-context learning? arXiv preprint arXiv:2405.19162,

    work page arXiv

  10. [10]

    Müller, N

    S. Müller, N. Hollmann, S. P. Arango, J. Grabocka, and F. Hutter. Transformers can do bayesian inference. arXiv preprint arXiv:2112.10510,

    work page arXiv

  11. [11]

    Sauter, S

    A. Sauter, S. Salehkaleybar, A. Plaat, and E. Acar. Activa: Amortized causal effect estimation without graphs via transformer-based variational autoencoder. arXiv preprint arXiv:2503.01290,

    work page arXiv

  12. [12]

    M. Wu, Y . Bao, R. Barzilay, and T. Jaakkola. Sample, estimate, aggregate: A recipe for causal discovery foundation models. arXiv preprint arXiv:2402.01929,

    work page arXiv

  13. [13]

    S. M. Xie, A. Raghunathan, P. Liang, and T. Ma. An explanation of in-context learning as implicit bayesian inference. arXiv preprint arXiv:2111.02080,

    work page internal anchor Pith review Pith/arXiv arXiv

  14. [14]

    Zhang, J

    J. Zhang, J. Jennings, C. Zhang, and C. Ma. Towards causal foundation model: on duality between causal inference and attention. arXiv preprint arXiv:2310.00809,

    work page arXiv

  15. [15]

    13 Appendix Table of Contents A Additional Details on Cond-FiP 15 A.1 DAG-Attention Mechanism

    URL https://proceedings.neurips.cc/paper/2018/file/ e347c51419ffb23ca3fd5050202f9c3d-Paper.pdf. 13 Appendix Table of Contents A Additional Details on Cond-FiP 15 A.1 DAG-Attention Mechanism . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 A.2 Details on Encoder Training . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 A.3 Inference wit...

    work page 2018

  16. [16]

    It provides access to a wide variety of SCMs, hence making it an excellent setting for amortized training

    to generate SCMs in our empirical study. It provides access to a wide variety of SCMs, hence making it an excellent setting for amortized training. • Graphs: We have the option to sample graphs as per the following schemes: Erods- Renyi [Erdos and Renyi, 1959], scale-free models [Barabási and Albert, 1999], Watts- Strogatz [Watts and Strogatz, 1998], and ...

    work page 1959

  17. [17]

    The models were trained for a total of 10k epochs with the Adam optimizer [Paszke et al., 2017], where we used a learning rate of 1e − 4 and a weight decay of 5e −

    Both of our transformer-based models contains 4 attention layers and each attention consists of 8 attention heads. The models were trained for a total of 10k epochs with the Adam optimizer [Paszke et al., 2017], where we used a learning rate of 1e − 4 and a weight decay of 5e −

    work page 2017

  18. [18]

    We also use the EMA implementation of [Karras et al., 2023] to train our models

    Each epoch contains ≃ 400 randomly generated datasets from the distribution PIN. We also use the EMA implementation of [Karras et al., 2023] to train our models. Memory Requirements. We trained Cond-FiP on a single L40 GPU with 48GB of memory, using an effective batch size of 8 with gradient accumulation. We outline the detailed memory computation as foll...

    work page 2023

  19. [19]

    require the knowledge of true graph (G) as part of the input context to Cond-FiP. In this section we conduct where we don’t provide the true graph in the input context, rather we infer the graph ˆG using an amortized causal discovery approach (A VICI [Lorch et al., 2022]) from the observational dataDX. We chose A VICI for this task since it can enable to ...

    work page 2022

  20. [20]

    This leads to a total of 12 experimental setting with 6 different GMM noise distribution for both the Large Backdoor and Weak Arrow datasets from the CSuite benchmark

    and N (5, 2). This leads to a total of 12 experimental setting with 6 different GMM noise distribution for both the Large Backdoor and Weak Arrow datasets from the CSuite benchmark. Results in Figure 7 demonstrate that Cond-FiP remains competitive with baselines across all tasks. Importantly, while baselines were trained from scratch for each specific gau...

    work page 2005

  21. [21]

    Since we don’t have access to the true causal mechanisms, we cannot compute RMSE for noise prediction or sample generation like we did in our experiments with synthetic benchmarks

    Note that the context dataset is to used to train the baselines and obtain dataset embedding for Cond-FiP, while the query dataset is used for evaluation of all the methods. Since we don’t have access to the true causal mechanisms, we cannot compute RMSE for noise prediction or sample generation like we did in our experiments with synthetic benchmarks. In...

    work page 2023