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Consistent Bayesian causal discovery for structural equation models with equal error variances

Anamitra Chaudhuri, Yang Ni, Anirban Bhattacharya

Abstract

We consider the problem of recovering the true causal structure among a set of variables, generated by a linear acyclic structural equation model (SEM) with the error terms being independent, not necessarily Gaussian, and having equal variances. It is well-known that the true underlying directed acyclic graph (DAG) encoding the causal structure is uniquely identifiable under this assumption. Interestingly, in this setting, it further holds that the sum of minimum expected squared errors for every variable, while predicted by the best linear combination of its parent variables, is minimised if and only if the causal structure is represented by any supergraph of the true DAG. In this work, we propose a Bayesian DAG selection method, where the working model assumes Gaussian SEM with equal error variances, and employ independent g-priors on each set of SEM coefficients. Furthermore, we utilise the aforementioned key property to establish that the proposed method recovers the true graph consistently without any additional distributional assumption, and illustrate it with a simulation study.

Consistent Bayesian causal discovery for structural equation models with equal error variances

Abstract

We consider the problem of recovering the true causal structure among a set of variables, generated by a linear acyclic structural equation model (SEM) with the error terms being independent, not necessarily Gaussian, and having equal variances. It is well-known that the true underlying directed acyclic graph (DAG) encoding the causal structure is uniquely identifiable under this assumption. Interestingly, in this setting, it further holds that the sum of minimum expected squared errors for every variable, while predicted by the best linear combination of its parent variables, is minimised if and only if the causal structure is represented by any supergraph of the true DAG. In this work, we propose a Bayesian DAG selection method, where the working model assumes Gaussian SEM with equal error variances, and employ independent g-priors on each set of SEM coefficients. Furthermore, we utilise the aforementioned key property to establish that the proposed method recovers the true graph consistently without any additional distributional assumption, and illustrate it with a simulation study.

Paper Structure

This paper contains 8 sections, 5 theorems, 27 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Structural causal model
  3. Proposed method
  4. Consistent DAG selection
  5. Simulation study
  6. Discussion
  7. Proof of Theorem \ref{['thm:key_result']}
  8. Proof of posterior DAG selection consistency

Key Result

Theorem 1

For every $\gamma \in \Gamma^p$, let $r^\gamma := \sum_{j = 1}^p r_j^\gamma$, where In particular, when $\gamma = \gamma^*$, we denote the above quantities by $r^*_j, j \in [p]$, and let $r^* := \sum_{j = 1}^p r^*_j$. Then we have $r^\gamma \geq r^*$, where the equality holds if and only if $\gamma \supseteq \gamma^*$.

Figures (4)

  • Figure 1: DAG $\gamma^*$
  • Figure 2: DAG $\gamma^*$.
  • Figure 3: Boxplots of $\pi(\gamma^* \mid D_n)$ over 100 replications for four different sample sizes.
  • Figure 4: Histogram of $\pi(\gamma^* \mid D_n)$ over 100 replications for sample size $n = 100 \times 2^7$.

Theorems & Definitions (11)

  • Example 1
  • Theorem 1
  • Lemma 1
  • Theorem 2
  • proof
  • proof : Proof of Lemma \ref{['lem:marg_gam']}
  • Lemma 2
  • proof
  • Lemma 3
  • proof
  • ...and 1 more