Replacing the do-calculus with Bayes rule

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.