  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.
