A domain-theoretic framework for conditional probability and Bayesian updating in programming

We present a domain-theoretic framework for probabilistic programming that provides a constructive definition of conditional probability and addresses computability challenges previously identified in the literature. We introduce a novel approach based on an observable notion of events that enables computability. We examine two methods for computing conditional probabilities -- one using conditional density functions and another using trace sampling with rejection -- and prove they yield consistent results within our framework. We implement these ideas in a simple probabilistic functional language with primitives for sampling and evaluation, providing both operational and denotational semantics and proving their consistency. Our work provides a rigorous foundation for implementing conditional probability in probabilistic programming languages.