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arxiv: 2605.15639 · v1 · pith:OFR4LG76new · submitted 2026-05-15 · 📊 stat.ME · stat.CO· stat.ML

Leveraging heterogeneity for identifiability: Bayesian order-based learning of multiple DAGs

Hyunwoong Chang , Fariha Taskin This is my paper

Pith reviewed 2026-05-20 16:43 UTC · model grok-4.3

classification 📊 stat.ME stat.COstat.ML
keywords causal discoveryDAG learningheterogeneous dataBayesian inferenceorder-based scoringhigh-dimensional statisticsMetropolis-Hastings
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The pith

Heterogeneous data makes causal orderings in multiple DAGs identifiable up to two permutations.

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

The paper develops a joint order-based scoring framework for learning causal structures from multiple heterogeneous datasets. It demonstrates that differences across data sources sharpen the estimation of the shared causal ordering among variables. In the most favorable settings this ordering becomes identifiable except for two possible permutations. The authors then build a Bayesian method for Gaussian DAG models that carries theoretical consistency guarantees in high-dimensional regimes. They support the approach with an efficient Metropolis-Hastings sampler using a random-to-random proposal and illustrate it on both simulations and single-nucleus RNA sequencing data.

Core claim

By jointly scoring orderings across heterogeneous data sources the authors show that the causal ordering of variables in multiple Gaussian DAGs is identifiable up to two permutations; they supply an order-based Bayesian procedure whose posterior concentrates on the true ordering (modulo those two permutations) in the high-dimensional regime.

What carries the argument

Joint order-based scoring framework that aggregates likelihood contributions from heterogeneous datasets to reduce ordering ambiguity to at most two permutations.

If this is right

  • Causal ordering estimation becomes more accurate when data are drawn from multiple heterogeneous sources rather than a single homogeneous population.
  • In favorable heterogeneity regimes only two permutations of the ordering remain possible.
  • The proposed Bayesian procedure enjoys posterior consistency for the ordering in high-dimensional settings.
  • The random-to-random proposal yields efficient mixing for Metropolis-Hastings sampling over the space of orderings.

Where Pith is reading between the lines

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

  • The two-permutation ambiguity left by heterogeneity might be resolved in practice by adding weak domain priors or a small amount of interventional data.
  • Similar identifiability gains could appear in other multi-source causal problems whenever the sources differ in their noise levels or parameter regimes.
  • The framework suggests a route to combine observational datasets collected under distinct experimental conditions without requiring explicit alignment of the graphs themselves.

Load-bearing premise

The heterogeneous data settings supply enough variation across sources to make causal orderings identifiable up to two permutations.

What would settle it

A high-dimensional simulation in which the posterior fails to concentrate on the true ordering (up to two permutations) as sample size grows would falsify the identifiability and consistency claims.

Figures

Figures reproduced from arXiv: 2605.15639 by Fariha Taskin, Hyunwoong Chang.

Figure 1
Figure 1. Figure 1: Score landscape in S p induced by the joint score Ψ1 in Example 4. Regions labeled S1–S6 correspond to different sets of orderings, as indicated in the right panel. The blue and red regions represent the union of all orderings consistent with Markov equivalent DAGs of G (1) and G (2), respectively. The maximizers of the joint score correspond to the intersection of these regions S3 ∪ S5. Region S3 consists… view at source ↗
Figure 2
Figure 2. Figure 2: Scaled log posterior probability versus the number of iterations of 50 MCMC runs with random [PITH_FULL_IMAGE:figures/full_fig_p015_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: The green boxplots summa￾rize the average Hamming distance ∆ between the true DAGs and their es￾timates using Algorithm 1 across dif￾ferent values of U based on 50 exper￾iments. The red boxplot corresponds to the case for K = 1. Specifically, the proposed method yields the smallest ∆, demonstrating significantly higher accuracy in recovering the true DAGs. Its lower false discovery rate implies that it pro… view at source ↗
Figure 4
Figure 4. Figure 4: (Left) Scaled log posterior probability versus the number of iterations in 50 MCMC runs with [PITH_FULL_IMAGE:figures/full_fig_p018_4.png] view at source ↗
read the original abstract

We propose a joint order-based scoring framework for causal structure learning of directed acyclic graph (DAG) models under heterogeneous data settings. We show that leveraging heterogeneity improves the accuracy of causal ordering estimation. In the most favorable case, the causal ordering is identifiable up to two permutations. Building on this framework, we propose an order-based Bayesian method for Gaussian DAG models and establish its theoretical properties in the high-dimensional regime. For posterior inference over the space of orderings, we introduce a random-to-random (R2R) proposal neighborhood for the Metropolis-Hastings algorithm, which is theoretically motivated and exhibits efficient mixing behavior. Simulation studies confirm the strong empirical performance of the proposed method, and an application to single-nucleus RNA sequencing data from major depressive disorder demonstrates practical utility.

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 manuscript proposes a joint order-based scoring framework for causal structure learning of multiple DAGs from heterogeneous data. It claims that leveraging heterogeneity improves the accuracy of causal ordering estimation, with the ordering identifiable up to two permutations in the most favorable case. Building on this, the authors develop an order-based Bayesian method for Gaussian DAG models, establish high-dimensional theoretical properties including posterior consistency, and introduce a random-to-random (R2R) proposal neighborhood for Metropolis-Hastings sampling over orderings. The approach is supported by simulation studies and an application to single-nucleus RNA sequencing data from major depressive disorder.

Significance. If the identifiability and consistency results hold under the stated heterogeneity model, the work offers a principled Bayesian route to exploit multi-group or multi-environment data for improved causal ordering recovery, which is a notable advance over single-DAG order-based methods. The theoretically motivated R2R proposal and high-dimensional guarantees are strengths that could influence MCMC design in graphical models. The RNA-seq application illustrates utility in genomics, where heterogeneity is common.

major comments (2)
  1. [§3] §3 (identifiability theorem): The result that heterogeneity yields causal ordering identifiable up to two permutations is load-bearing for the joint scoring framework and all subsequent high-dimensional posterior consistency claims, yet the minimal conditions on heterogeneity (e.g., separation of precision matrices or rank conditions across groups) are not stated with explicit quantitative bounds; this leaves open whether the 'most favorable case' is generic or requires strong assumptions that may fail in typical heterogeneous settings.
  2. [§4.3] §4.3 (posterior consistency): The high-dimensional consistency proof for the order-based posterior appears to rely on the identifiability step; if the heterogeneity-induced variation is weaker than required for the two-permutation bound, the concentration rate and selection consistency arguments collapse, and the paper should provide a counter-example or sensitivity analysis under minimal heterogeneity.
minor comments (2)
  1. [Abstract] The abstract states that the R2R proposal 'exhibits efficient mixing behavior' without a specific reference to a theorem, lemma, or simulation metric (e.g., autocorrelation or effective sample size); this should be tied to a numbered result.
  2. [§2] Notation for the multi-group precision matrices and the joint score function is introduced without a clear summary table; adding one would aid readability when comparing to single-environment baselines.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their constructive comments on our manuscript. We address the two major comments point by point below, indicating where revisions will be made to improve clarity and rigor.

read point-by-point responses

  1. Referee: [§3] §3 (identifiability theorem): The result that heterogeneity yields causal ordering identifiable up to two permutations is load-bearing for the joint scoring framework and all subsequent high-dimensional posterior consistency claims, yet the minimal conditions on heterogeneity (e.g., separation of precision matrices or rank conditions across groups) are not stated with explicit quantitative bounds; this leaves open whether the 'most favorable case' is generic or requires strong assumptions that may fail in typical heterogeneous settings.

    Authors: We agree that the identifiability result is foundational and that the conditions should be stated more explicitly. In the revised manuscript we will augment Theorem 3.1 with a new quantitative assumption (Assumption 3.2) that requires the minimal eigenvalue gap between any pair of group-specific precision matrices to exceed a positive constant δ whose dependence on the maximum degree and dimension is made explicit. We will also clarify that the two-permutation identifiability holds generically once this separation condition is satisfied, while weaker heterogeneity yields identifiability only up to a larger equivalence class; this distinction will be illustrated with a small numerical example in the revised §3. revision: yes

  2. Referee: [§4.3] §4.3 (posterior consistency): The high-dimensional consistency proof for the order-based posterior appears to rely on the identifiability step; if the heterogeneity-induced variation is weaker than required for the two-permutation bound, the concentration rate and selection consistency arguments collapse, and the paper should provide a counter-example or sensitivity analysis under minimal heterogeneity.

    Authors: The consistency theorem (Theorem 4.3) is explicitly conditioned on the identifiability result of §3 holding with the two-permutation bound. When heterogeneity is weaker, the posterior still concentrates on the equivalence class of orderings consistent with the data-generating process, but the rate and exact selection consistency may degrade; we will add a remark immediately after Theorem 4.3 stating this dependence. In addition, we will include a new simulation subsection that varies the eigenvalue gap δ across a grid and reports the resulting posterior concentration behavior. A full analytic counter-example at the boundary of minimal heterogeneity would require substantial new theory that lies beyond the scope of the present revision. revision: partial

Circularity Check

0 steps flagged

No circularity: derivation chain is self-contained

full rationale

The paper introduces a joint order-based scoring framework that exploits heterogeneity across multiple DAGs to improve causal ordering estimation, with an identifiability result (up to two permutations in the most favorable case) derived from the heterogeneous data model. It then proposes an order-based Bayesian method for Gaussian DAGs, introduces a theoretically motivated R2R proposal for Metropolis-Hastings, and establishes high-dimensional posterior consistency. None of these steps reduce a prediction or central claim to a fitted parameter, self-citation, or input by construction; the identifiability and consistency results are presented as consequences of the model assumptions and heterogeneity rather than tautological renamings or load-bearing self-references. The framework therefore retains independent mathematical content.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

Based solely on the abstract, the paper relies on standard assumptions for DAG models and Gaussian distributions, with heterogeneity as a key domain assumption enabling identifiability. No explicit free parameters or invented entities are mentioned.

axioms (2)
  • domain assumption Data follows Gaussian DAG models
    The Bayesian method and theoretical properties are established specifically for Gaussian DAG models.
  • domain assumption Heterogeneous data settings allow for improved identifiability of causal orderings
    This is the central premise for the joint scoring framework and the claim of identifiability up to two permutations.

pith-pipeline@v0.9.0 · 5661 in / 1431 out tokens · 171475 ms · 2026-05-20T16:43:41.330664+00:00 · methodology

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Reference graph

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