# A Bayesian Solution to the M-Bias Problem

## TL;DR

This paper introduces a Bayesian approach to address the M-bias problem in causal inference, demonstrating how Bayesian models can incorporate causal relationships and resolve controversies about variable adjustment.

## Contribution

It provides a Bayesian solution to M-bias that aligns with Pearl's approach while conditioning on all variables, bridging causal graphical models and Bayesian statistics.

## Key findings

- Bayesian method replicates Pearl's solution to M-bias.
- Conditioning on all variables can mitigate M-bias issues.
- The approach shows causal relationships can be represented within Bayesian models.

## Abstract

It is common practice in using regression type models for inferring causal effects, that inferring the correct causal relationship requires extra covariates are included or ``adjusted for''. Without performing this adjustment erroneous causal effects can be inferred. Given this phenomenon it is common practice to include as many covariates as possible, however such advice comes unstuck in the presence of M-bias. M-Bias is a problem in causal inference where the correct estimation of treatment effects requires that certain variables are not adjusted for i.e. are simply neglected from inclusion in the model. This issue caused a storm of controversy in 2009 when Rubin, Pearl and others disagreed about if it could be problematic to include additional variables in models when inferring causal effects. This paper makes two contributions to this issue. Firstly we provide a Bayesian solution to the M-Bias problem. The solution replicates Pearl's solution, but consistent with Rubin's advice we condition on all variables. Secondly the fact that we are able to offer a solution to this problem in Bayesian terms shows that it is indeed possible to represent causal relationships within the Bayesian paradigm, albeit in an extended space. We make several remarks on the similarities and differences between causal graphical models which implement the do-calculus and probabilistic graphical models which enable Bayesian statistics. We hope this work will stimulate more research on unifying Pearl's causal calculus using causal graphical models with traditional Bayesian statistics and probabilistic graphical models.