Bayesian Model Averaging for Causality Estimation and its Approximation based on Gaussian Scale Mixture Distributions

TL;DR

This paper proposes a Bayesian model averaging approach for causal effect estimation in linear SCMs and introduces an approximation method using Gaussian scale mixture distributions to address computational challenges.

Contribution

It introduces a Bayesian model averaging framework for causal effect estimation and develops an efficient approximation method based on Gaussian scale mixtures.

Findings

Bayesian model averaging is optimal for causal effect estimation.

The proposed approximation reduces computational complexity.

The method effectively combines multiple causal models.

Abstract

In the estimation of the causal effect under linear Structural Causal Models (SCMs), it is common practice to first identify the causal structure, estimate the probability distributions, and then calculate the causal effect. However, if the goal is to estimate the causal effect, it is not necessary to fix a single causal structure or probability distributions. In this paper, we first show from a Bayesian perspective that it is Bayes optimal to weight (average) the causal effects estimated under each model rather than estimating the causal effect under a fixed single model. This idea is also known as Bayesian model averaging. Although the Bayesian model averaging is optimal, as the number of candidate models increases, the weighting calculations become computationally hard. We develop an approximation to the Bayes optimal estimator by using Gaussian scale mixture distributions.