Inference Maps for Regular Conditional Probabilities

A categorical foundation for Bayesian probability

Given a data measurement a posterior probability can be computed.Given two measurable spaces and with countably generated a perfect prior probability measure on and a sampling distribution there is a corresponding inferencemap which is unique up to a set of measure zero.Thus, given a data measurement a posterior probability can be computed.This procedure is iterative: with each updated probability we obtain a new joint distribution which in turn yields a new inference map and the process repeats with eachadditional measurement.The main result uses an existence theorem for regular conditional probabilities by faden which holds in more generality than the setting of polish spaces.This less stringent setting then allows for non-trivial decision rules (eilenberg--moore algebras) on finite (as well as non finite) spaces and also provides for a common framework for decision theory and bayesian probability.