  The concept of causality has a controversial history. The question of whether
it is possible to represent and address causal problems with probability
theory, or if fundamentally new mathematics such as the do calculus is required
has been hotly debated, e.g. Pearl (2001) states "the building blocks of our
scientific and everyday knowledge are elementary facts such as "mud does not
cause rain" and "symptoms do not cause disease" and those facts, strangely
enough, cannot be expressed in the vocabulary of probability calculus". This
has lead to a dichotomy between advocates of causal graphical modeling and the
do calculus, and researchers applying Bayesian methods. In this paper we
demonstrate that, while it is critical to explicitly model our assumptions on
the impact of intervening in a system, provided we do so, estimating causal
effects can be done entirely within the standard Bayesian paradigm. The
invariance assumptions underlying causal graphical models can be encoded in
ordinary Probabilistic graphical models, allowing causal estimation with
Bayesian statistics, equivalent to the do calculus. Elucidating the connections
between these approaches is a key step toward enabling the insights provided by
each to be combined to solve real problems.
