Reasoning about conditional probabilities in a higher-order-logic theorem prover

In the field of probabilistic analysis, the concept of conditional probability plays a major role for estimating probabilities when some partial information concerning the result of the experiment is available. This paper presents a higher-order-logic definition of conditional probability and the formal verification of some classical properties of conditional probability, such as, the total probability law and Bayes' theorem. This infrastructure, implemented in the HOL theorem prover, allows us to precisely reason about conditional probabilities for probabilistic systems within the sound core of HOL and thus proves to be quite useful for the analysis of systems used in safety-critical domains, such as space, medicine and transportation. To demonstrate the usefulness of our approach, we provide the precise probabilistic analysis of the binary asymmetric channel, a widely used concept in communication theory, within the HOL theorem prover.