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

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