  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.
