The Beta-Bernoulli process and algebraic effects

TL;DR

This paper applies algebraic effects from programming language theory to analyze the Beta-Bernoulli process, revealing new insights into its structure and potential for generalization to other Bayesian models.

Contribution

It develops a complete equational theory for the Beta-Bernoulli process within the algebraic effects framework, connecting measure-theoretic and syntactic semantics.

Findings

The theory is complete with respect to measure-theoretic semantics.

It highlights the role of abstract data types and program equations.

Potential for generalizing to other stochastic processes.

Abstract

In this paper we use the framework of algebraic effects from programming language theory to analyze the Beta-Bernoulli process, a standard building block in Bayesian models. Our analysis reveals the importance of abstract data types, and two types of program equations, called commutativity and discardability. We develop an equational theory of terms that use the Beta-Bernoulli process, and show that the theory is complete with respect to the measure-theoretic semantics, and also in the syntactic sense of Post. Our analysis has a potential for being generalized to other stochastic processes relevant to Bayesian modelling, yielding new understanding of these processes from the perspective of programming.