From Bayesian Notation to Pure Racket via Discrete Measure-Theoretic Probability in λ ZFC

Bayesian practitioners build models of the world without regarding how difficult it will be to answer questions about them. When answering questions, they put off approximating as long as possible, and usually must write programs to compute converging approximations. Writing the programs is distracting, tedious and error-prone, and we wish to relieve them of it by providing languages and compilers. Their style constrains our work: the tools we provide cannot approximate early. Our approach to meeting this constraint is to 1) determine their notation’s meaning in a suitable theoretical framework; 2) generalize our interpretation in an uncomputable, exact semantics; 3) approximate the exact semantics and prove convergence; and 4) implement the approximating semantics in Racket (formerly PLT Scheme). In this way, we define languages with at least as much exactness as Bayesian practitioners have in mind, and also put off approximating as long as possible. In this paper, we demonstrate the approach using our preliminary work on discrete (countably infinite) Bayesian models.