pith. sign in

arxiv: 2605.15353 · v1 · pith:AK6DJQZInew · submitted 2026-05-14 · 💻 cs.LG · cs.AI· q-bio.MN· q-bio.QM

PACER: Acyclic Causal Discovery from Large-Scale Interventional Data

Ramon Vi\~nas Torn\'e , S\'ilvia F\`abregas Salazar , Soyon Park , Ivo Alexander Ban , Artyom Gadetsky , Nikita Doikov , Maria Brbi\'c This is my paper

Pith reviewed 2026-05-19 15:39 UTC · model grok-4.3

classification 💻 cs.LG cs.AIq-bio.MNq-bio.QM
keywords causal discoverydirected acyclic graphsinterventional dataacyclicity by constructionpermutation modeledge probabilitiesgenetic perturbationsscalable optimization
0
0 comments X p. Extension
pith:AK6DJQZI Add to your LaTeX paper What is a Pith Number? 
\usepackage{pith}
\pithnumber{AK6DJQZI}

Prints a linked pith:AK6DJQZI badge after your title and writes the identifier into PDF metadata. Compiles on arXiv with no extra files. Learn more

The pith

PACER guarantees acyclicity by jointly modeling variable permutations and edge probabilities for direct optimization over valid causal structures.

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

The paper introduces PACER to address limitations in causal discovery from high-dimensional interventional data. It builds a distribution over directed acyclic graphs by combining a model of variable orderings with edge probabilities, which automatically excludes cycles during learning. This replaces soft penalties that allow invalid graphs and cause instability or slow convergence. A reader would care because the approach supports both observational and interventional data in one framework, includes structural priors, and delivers closed-form likelihoods for linear-Gaussian cases that cut computation time dramatically. The result is a method that scales to thousands of variables while matching or beating existing performance on biological benchmarks.

Core claim

PACER parameterizes a distribution over DAGs through a joint model of variable permutations and edge probabilities, enabling direct optimization over valid causal structures without surrogate penalties. The framework supports a unified likelihood-based treatment of observational and interventional data, flexible conditional density models, and the incorporation of structural prior knowledge. For linear-Gaussian mechanisms, closed-form expressions for the expected interventional log-likelihood and its gradients are derived, producing substantial computational gains.

What carries the argument

The joint model of variable permutations and edge probabilities. It carries the argument by constructing only acyclic graphs from the start, allowing gradient-based optimization directly on valid structures instead of post-hoc corrections.

If this is right

  • Optimization stays inside the space of valid DAGs at every step, removing the need for acyclicity penalties and their associated numerical issues.
  • Closed-form likelihood and gradient expressions for linear-Gaussian models produce up to two orders of magnitude faster run times than penalty-based differentiable methods.
  • The same framework handles observational data, interventional data, and domain-specific structural priors in a single likelihood objective.
  • The method scales to networks with thousands of variables while matching or exceeding prior accuracy on protein-signaling and large-scale genetic-perturbation benchmarks.

Where Pith is reading between the lines

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

  • If the permutation-edge parameterization remains efficient for non-Gaussian conditional densities, the same construction could support causal discovery in mixed-type or count-valued data without changing the acyclicity guarantee.
  • The explicit separation of ordering and edge decisions may make it straightforward to inject partial expert knowledge as soft constraints on the permutation distribution.
  • Speed improvements of this magnitude could shift causal analysis from offline batch processing to iterative experimental design loops in high-throughput biology.

Load-bearing premise

The joint model over permutations and edge probabilities is assumed to reach every possible directed acyclic graph with enough probability mass that the optimizer does not systematically miss high-quality structures.

What would settle it

Run PACER on a small graph with a known ground-truth DAG and full interventional data; if the recovered structure has strictly lower interventional likelihood than the true DAG or than an exhaustive enumeration of all valid orderings, the coverage assumption does not hold.

Figures

Figures reproduced from arXiv: 2605.15353 by Artyom Gadetsky, Ivo Alexander Ban, Maria Brbi\'c, Nikita Doikov, Ramon Vi\~nas Torn\'e, S\'ilvia F\`abregas Salazar, Soyon Park.

Figure 1
Figure 1. Figure 1: Overview of the framework. PACER models a topological ordering of variables using a Plackett-Luce distribution. Nodes with higher weight are more likely to precede nodes with lower weight in downstream DAGs. Samples from this distribution induce complete DAGs, which are further filtered via samples from independent, edge-specific Bernoulli distributions. This defines our Bernoulli-Plackett￾Luce distributio… view at source ↗
Figure 3
Figure 3. Figure 3: Runtime scaling with increasing numbers of variables d ≤ 50. We report wall-clock training time for varying number of variables. Missing data points indicate runs that exceeded the 6 hour time limit. Error bars denote the standard deviation over three random seeds. 100 variables without timing out, both methods suffer from sharply increasing SID and substantially poorer scalability compared to PACER (Appen… view at source ↗
Figure 4
Figure 4. Figure 4: Performance on the Perturb-CITE-seq dataset. Interventional negative log-likelihood (I-NLL) and interventional mean absolute error (I-MAE) evaluated on held-out interventions across three experimental conditions: control, co-culture, and IFN-γ stimulation. Lower values indicate better performance. Sampling units are held-out perturbations. Boxes depict distribution quartiles, with the center line correspon… view at source ↗
Figure 5
Figure 5. Figure 5: Kendall τ between inferred and ground-truth partial orderings over training steps (N = 500 nodes, ρ = 0.3). The analytic variant (red) converges rapidly to τ ≈ 1.0; the REINFORCE variant (teal) converges more gradually but reaches comparable accuracy. No divergence or oscillation is observed. Gradient variance across graph sizes (with and without control variate) [PITH_FULL_IMAGE:figures/full_fig_p017_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Gradient variance of REINFORCE with control variate across graph sizes N ∈ {10, 50, 100, 500} (likelihood-free objective). Variance decreases monotonically for all N and converges to similar magnitudes, demonstrating stable scaling with graph size [PITH_FULL_IMAGE:figures/full_fig_p018_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Gradient variance of REINFORCE without control variate across number of nodes N ∈ {10, 50, 100, 500}. Variance is orders of magnitude larger than with control variate (cf [PITH_FULL_IMAGE:figures/full_fig_p018_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Gradient variance (mean ± std across seeds) for N=200 nodes and different numbers of Monte Carlo samples. Increasing the number of Monte Carlo samples consistently reduces both mean, variance, and cross-seed spread [PITH_FULL_IMAGE:figures/full_fig_p019_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: Gradient variance during training on the Replogle RPE1 dataset. Left: Analytic estimator (log scale): decreases monotonically to near zero over 20,000 steps. Center: REINFORCE with control variate for varying numbers of Monte Carlo samples; larger sample counts yield lower variance. Right: REINFORCE without control variate: variance is orders of magnitude higher, underscoring the necessity of variance redu… view at source ↗
Figure 10
Figure 10. Figure 10: Performance across simulations on interventional dataset with increasing numbers of variables d ≤ 50. SHD, SID, precision, and recall are reported under interventional settings with a fixed edge density of 0.05. Lower values are better for SHD, while higher values are better for precision and recall. Error bars indicate the standard deviation over three random datasets. Missing data points indicate runs t… view at source ↗
Figure 11
Figure 11. Figure 11: Performance of PACER across simulations on interventional linear dataset with increasing numbers of variables d ≤ 1000. Runtime, SHD, precision, and recall are reported under interventional settings with a fixed connectivity of 4. Lower values are better for SHD, while higher values are better for precision, and recall. Error bars indicate the standard deviation over three random datasets [PITH_FULL_IMAG… view at source ↗
Figure 12
Figure 12. Figure 12: Sensitivity of PACER to the sparsity hyperparameter λ on linear datasets. SHD and SID are shown for λ values ranging from 10−7 − 102 for three random seeds. 10 6 10 4 10 2 10 0 10 2 Lambda 0 20 40 60 80 100 120 SHD data_p10_e10_n10000_nnadd_struct data_p10_e40_n10000_nnadd_struct data_p20_e20_n10000_nnadd_struct data_p20_e80_n10000_nnadd_struct 10 6 10 4 10 2 10 0 10 2 Lambda 0 50 100 150 200 250 300 SID … view at source ↗
Figure 13
Figure 13. Figure 13: Sensitivity of PACER to the sparsity hyperparameter λ on additive noise model (ANM) datasets. SHD and SID are shown for λ values ranging from 10−7 − 102 for three random seeds. 33 [PITH_FULL_IMAGE:figures/full_fig_p033_13.png] view at source ↗
Figure 14
Figure 14. Figure 14: Sensitivity of PACER to the sparsity hyperparameter λ on nonlinear non-additive (NN) datasets. SHD and SID are shown for λ values ranging from 10−7 − 102 for three random seeds. 10 6 10 4 10 2 10 0 10 2 Lambda 0 5 10 15 20 SHD data_p10_e10_n10000_linear_struct data_p10_e10_n10000_nn_struct data_p10_e10_n10000_nnadd_struct 10 6 10 4 10 2 10 0 10 2 Lambda 0 5 10 15 20 25 30 35 40 SID [PITH_FULL_IMAGE:figur… view at source ↗
Figure 15
Figure 15. Figure 15: Sensitivity of PACER to the sparsity hyperparameter λ on different causal mechanism datasets with same number of nodes and edges. SHD and SID are shown for λ values ranging from 10−7 − 102 for three random seeds. J.4. Robustness to imperfect and off-target interventions We evaluate PACER’s robustness under two practically motivated deviations from the perfect-intervention assumption, directly inspired by … view at source ↗
Figure 16
Figure 16. Figure 16: Sensitivity of PACER to off-target interventions. SHD and SID are shown for three random seeds [PITH_FULL_IMAGE:figures/full_fig_p035_16.png] view at source ↗
Figure 17
Figure 17. Figure 17: Sensitivity of PACER to imperfect interventions. SHD and SID are shown for three random seeds. J.5. Hyperparameter sensitivity analysis We conduct a thorough sensitivity analysis over all major hyperparameters on a synthetic dataset. We systematically vary batch size, learning rate, number of MC samples K, number of layers, and hidden dimension on the linear Gaussian dataset: • Hidden dimension. Increasin… view at source ↗
Figure 18
Figure 18. Figure 18: Sensitivity of PACER to various hyperparameter settings. SHD and SID are shown for three random seeds. 35 [PITH_FULL_IMAGE:figures/full_fig_p035_18.png] view at source ↗
Figure 19
Figure 19. Figure 19: Kendall Tau coefficient calculated between ground-truth and inferred partial orderings by training step. We compare PACER to the differentiable DAG sampling approach of DP-DAG (Charpentier et al., 2022), including both top￾k and Sinkhorn-based methods (Charpentier et al., 2022). To isolate the efficacy of the DAG parameterization (independent of specific likelihood modeling or architectural choices) we co… view at source ↗
Figure 20
Figure 20. Figure 20: Interventional network inferred by PACER on flow cytometry data from (Sachs et al., 2005). PACER correctly recovers many canonical signaling interactions. N. Perturb-CITE-seq dataset The Perturb-CITE-seq dataset (Frangieh et al., 2021) comprises gene expression profiles from 218,331 melanoma cells subjected to CRISPR-based perturbations targeting 248 genes. The data was generated to identify gene regulato… view at source ↗
Figure 21
Figure 21. Figure 21: Execution times (minutes, log scale) for DCDFG, NOTEARS, NOTEARS-LR and PACER across Control, Cocult and IFN conditions. Points represent runs, lines represent median time. P. Extensions to the intervention model The objective in Equation 3 treats interventions as stochastic perfect interventions with known targets: for each regime r, the variables in Ir are excluded from the likelihood sum, and a single … view at source ↗
read the original abstract

Inferring the structure of directed acyclic graphs (DAGs) from data is a central challenge in causal discovery, particularly in modern high-dimensional settings where large-scale interventional data are increasingly available. While interventional data can improve identifiability, existing methods remain limited by soft acyclicity constraints, leading to optimization over invalid cyclic graphs, numerical instability, and reduced scalability. We introduce PACER (Perturbation-driven Acyclic Causal Edge Recovery), a scalable framework for causal discovery that guarantees acyclicity by construction. PACER parameterizes a distribution over DAGs through a joint model of variable permutations and edge probabilities, enabling direct optimization over valid causal structures without surrogate penalties. The framework supports a unified likelihood-based treatment of observational and interventional data, flexible conditional density models, and the incorporation of structural prior knowledge. For linear-Gaussian mechanisms, we derive closed-form expressions for the expected interventional log-likelihood and its gradients, yielding substantial computational gains. Empirically, PACER matches or exceeds state-of-the-art methods on protein signaling and large-scale genetic perturbation benchmarks, while scaling efficiently to networks with thousands of variables and achieving up to two orders of magnitude speedups over penalty-based differentiable approaches. These results demonstrate that exact and scalable causal discovery from high-dimensional perturbation data is achievable through principled search space design.

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 PACER, a framework for causal discovery from large-scale interventional data that guarantees acyclicity by construction via a joint model over variable permutations and edge probabilities. This enables direct optimization over valid DAGs without surrogate penalties, supports unified likelihood treatment of observational and interventional data with flexible conditional models and structural priors, derives closed-form expressions for expected interventional log-likelihood and gradients in the linear-Gaussian case, and reports matching or exceeding SOTA performance with up to two orders of magnitude speedups on protein signaling and genetic perturbation benchmarks while scaling to thousands of variables.

Significance. If the parameterization covers the DAG space without bias and the closed-form derivations are correct, this would be a significant contribution to causal discovery by replacing soft acyclicity constraints with an exact search-space design. The explicit independence from benchmark-specific tuning and the reported scalability gains address key bottlenecks in high-dimensional interventional settings common in biology and genetics.

major comments (2)
  1. [§3.2] §3.2 (joint permutation-edge model): The central guarantee that the parameterization represents any DAG with positive probability and without systematic bias is load-bearing for both the acyclicity claim and the performance advantages over penalty-based methods. The manuscript does not provide an explicit proof or coverage argument showing that every DAG receives positive mass, particularly when structural priors are incorporated or when non-linear conditional densities are used; this directly impacts whether direct optimization can recover the true structure.
  2. [§4] §4 (closed-form derivations): The claimed closed-form expressions for the expected interventional log-likelihood and its gradients in the linear-Gaussian case are presented as yielding substantial computational gains, yet the manuscript provides neither the full derivation steps nor an accompanying error analysis or numerical verification. This absence undermines verification of the reported speedups and scalability to thousands of variables.
minor comments (2)
  1. [Table 1, Figure 3] Table 1 and Figure 3: axis labels and units for runtime and SHD metrics should be stated explicitly to allow direct comparison with prior work.
  2. [§2.1] Notation in §2.1: the distinction between the permutation distribution and the conditional edge probability should be clarified with an explicit example for a small graph.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We are grateful to the referee for their insightful comments and for recognizing the potential significance of our work. We respond to each major comment in turn and indicate the changes we will implement in the revised manuscript.

read point-by-point responses

  1. Referee: [§3.2] §3.2 (joint permutation-edge model): The central guarantee that the parameterization represents any DAG with positive probability and without systematic bias is load-bearing for both the acyclicity claim and the performance advantages over penalty-based methods. The manuscript does not provide an explicit proof or coverage argument showing that every DAG receives positive mass, particularly when structural priors are incorporated or when non-linear conditional densities are used; this directly impacts whether direct optimization can recover the true structure.

    Authors: We agree that an explicit coverage argument strengthens the rigor of the acyclicity guarantee. In the revision we will insert a new subsection in §3.2 that formally proves every DAG receives positive probability under the joint permutation-edge model. The argument relies on the fact that any DAG admits at least one topological order; for that order the edge-probability parameters can be set strictly positive on the realized edges. Structural priors enter multiplicatively and, by construction, assign positive mass to every valid DAG, preserving coverage. The parameterization of the distribution over DAGs is independent of the choice of conditional density (linear or nonlinear), so coverage holds uniformly. We will also note that this positive-mass property ensures the true structure lies in the support of the model, permitting recovery by direct optimization. revision: yes

  2. Referee: [§4] §4 (closed-form derivations): The claimed closed-form expressions for the expected interventional log-likelihood and its gradients in the linear-Gaussian case are presented as yielding substantial computational gains, yet the manuscript provides neither the full derivation steps nor an accompanying error analysis or numerical verification. This absence undermines verification of the reported speedups and scalability to thousands of variables.

    Authors: We acknowledge that the main text omitted the full algebraic steps for brevity. In the revised supplementary material we will provide a complete, line-by-line derivation of the closed-form expected interventional log-likelihood and its gradients for the linear-Gaussian case. We will also add a short numerical verification subsection that compares the closed-form values against Monte-Carlo estimates on small synthetic graphs, together with a brief error analysis that bounds the discrepancy between the analytic and sampled quantities. These additions will allow independent verification of the claimed computational gains. revision: yes

Circularity Check

0 steps flagged

No significant circularity in PACER's parameterization and derivations

full rationale

The paper's core contribution is an explicit joint parameterization over variable permutations and edge probabilities that enforces acyclicity by design, allowing direct optimization over valid DAGs without penalty terms. This is a constructive modeling choice for the search space rather than a self-referential definition or reduction of outputs to inputs. Closed-form expressions for expected interventional log-likelihood and gradients under linear-Gaussian assumptions are derived from standard probabilistic rules applied to the model, yielding computational gains that are independent of fitted parameters or benchmarks. Empirical matching or exceeding of state-of-the-art on protein and genetic perturbation datasets serves as external validation, not a circular prediction. No load-bearing steps rely on self-citations, uniqueness theorems from prior author work, or renaming of known results; the framework is self-contained against the described assumptions.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The framework rests on standard assumptions about DAGs and conditional independence but introduces a new joint distribution over permutations and edges whose coverage properties are taken as given.

axioms (1)
  • domain assumption The joint model over permutations and edge probabilities can represent any DAG with positive probability and supports efficient sampling and optimization.
    Invoked when stating that the parameterization enables direct optimization over valid causal structures.

pith-pipeline@v0.9.0 · 5804 in / 1243 out tokens · 35648 ms · 2026-05-19T15:39:11.264185+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

What do these tags mean?
matches
The paper's claim is directly supported by a theorem in the formal canon.
supports
The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends
The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses
The paper appears to rely on the theorem as machinery.
contradicts
The paper's claim conflicts with a theorem or certificate in the canon.
unclear
Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

12 extracted references · 12 canonical work pages

  1. [1]

    O., and Saez-Rodriguez, J

    Badia-i Mompel, P., Casals-Franch, R., Wessels, L., M¨uller- Dott, S., Trimbour, R., Yang, Y ., Ramirez Flores, R. O., and Saez-Rodriguez, J. Comparison and evaluation of methods to infer gene regulatory networks from multi- modal single-cell data.bioRxiv, pp. 2024–12,

    work page 2024

  2. [2]

    orders of magnitude larger and grows strongly with N (≈90 for N=500 vs

    and increases strongly with N, confirming the critical role of the control variate. orders of magnitude larger and grows strongly with N (≈90 for N=500 vs. ≈0.002 with control variate at convergence), confirming that baseline subtraction is critical for stable large-scale training. Gradient variance across Monte Carlo samples.Figure 8 shows gradient varia...

    work page 2022

  3. [3]

    To complete the proof, we use the formulas,Var(aij) =E[a ij](1−E[aij]) and Cov(aij, akj) =E[a ij]E[akj] eθj eθi+eθj +eθk , that were established in Theorem C.4

    It remains to compute the last term in equation 23, which is the variance: EM ′∼M(θ,P) h ⟨Σ−1 j (µj − ¯µj),µ j − ¯µj⟩ i =E M ′∼M(θ,P) ⟨Σ−1 j nP i=1 (aij −E[a ij])wijxi, nP i=1 (aij −E[a ij])wijxi⟩ = nP i=1 Var(aij)w2 ij⟨Σ−1 j xi,x i⟩+ nP i=1 nP k=1,k̸=i Cov(aij, akj)wijwkj ⟨Σ−1 j xi,x k⟩. To complete the proof, we use the formulas,Var(aij) =E[a ij](1−E[ai...

    work page 2003

  4. [4]

    NOTEARS (Zheng et al., 2018), NOTEARS-LR (Fang et al., 2023), DCDI (Brouillard et al.,

    sorts variables by increasing marginal variance and uses parent search to infer DAG structure. NOTEARS (Zheng et al., 2018), NOTEARS-LR (Fang et al., 2023), DCDI (Brouillard et al.,

    work page 2018

  5. [5]

    Baselines implementations.GS, GES, GIES, IAMB, MMPC, GRaSP, BOSS, and LiNGAM are benchmarked using the implementation of the Causal Discovery Toolbox (Kalainathan et al., 2020)

    is a method that improved stability and efficiency of DCD-based methods using an alternative acyclicity constraint. Baselines implementations.GS, GES, GIES, IAMB, MMPC, GRaSP, BOSS, and LiNGAM are benchmarked using the implementation of the Causal Discovery Toolbox (Kalainathan et al., 2020). We use the original implementation of NO-TEARS (Zheng et al., 2...

    work page 2020

  6. [6]

    protein signaling dataset, a curated ground-truth causal graph is available. We therefore use standard structure recovery metrics that directly compare the inferred graph to the ground truth: • Structural Hamming Distance (SHD).SHD counts the minimum number of edge additions, deletions, and reversals required to transform the inferred graph into the groun...

    work page 2015

  7. [7]

    Error bars indicate the standard deviation over three random datasets

    Lower values are better for SHD, while higher values are better for precision, and recall. Error bars indicate the standard deviation over three random datasets. Table 7.Results for linear dataset with perfect interventions. Lower SHD and SID indicate better performance. Best scores are bolded and second best scores are underlined. Method 10 nodes,e=110 n...

    work page 2022

  8. [8]

    The dataset contains 7467 measurements across 11 proteins

    data from causallearn (Zheng et al., 2024). The dataset contains 7467 measurements across 11 proteins. GS, GES, GIES, IAMB, MMPC, GRaSP, BOSS, and LiNGAM are benchmarked using the implementation of the Causal Discovery Toolbox (Kalainathan et al.,

    work page 2024

  9. [9]

    We use the original implementation of NO-TEARS (Zheng et al., 2018)

    with default parameters. We use the original implementation of NO-TEARS (Zheng et al., 2018). In terms of the interventional scenario, we use the (Sachs et al.,

    work page 2018

  10. [10]

    We use the IGSP, GIES, CAM, DCDI-G, and DCDI-DSF results from Appendix C1 of DCDI 36 PACER: Acyclic Causal Discovery from Large-Scale Interventional Data (Brouillard et al., 2020)

    data processed by DCDI (Brouillard et al., 2020), containing 5846 measurements for the same 11 proteins across 6 interventional regimes. We use the IGSP, GIES, CAM, DCDI-G, and DCDI-DSF results from Appendix C1 of DCDI 36 PACER: Acyclic Causal Discovery from Large-Scale Interventional Data (Brouillard et al., 2020). For both settings and all experiments t...

    work page 2020

  11. [11]

    comprises gene expression profiles from 218,331 melanoma cells subjected to CRISPR-based perturbations targeting 248 genes. The data was generated to identify gene regulatory programs underlying resistance or sensitivity to T cell-mediated killing, with the goal of uncovering potential therapeutic targets in cancer. We treat the three experimental conditi...

    work page 2022

  12. [12]

    We introduce per-regime conditional parameters Ω(r) j for each intervened node j∈ I r, and include those nodes 38 PACER: Acyclic Causal Discovery from Large-Scale Interventional Data in the likelihood sum: fsoft(M ′,Ω,{Ω (r)}) = RX r=1 EX∼P (r) data "X j /∈Ir logp j Ω(Xj |M ′ j, X−j) + X j∈Ir logp j Ω(r) j (Xj |M ′ j, X−j) # . This places soft interventio...

    work page 2020