Sufficient Statistics and Split Idempotents in Discrete Probability Theory

TL;DR

This paper explores the concept of sufficient statistics within discrete probability theory, using string diagrams and split idempotents to provide a new categorical perspective on their fundamental properties.

Contribution

It introduces a categorical framework for understanding sufficient statistics in discrete probability, emphasizing the role of split idempotents and dagger structures.

Findings

Identifies fundamental examples of sufficient statistics in discrete probability

Reveals the role of dagger split idempotents in the structure of sufficient statistics

Shows that a sufficient statistic is a deterministic dagger epi

Abstract

A sufficient statistic is a deterministic function that captures an essential property of a probabilistic function (channel, kernel). Being a sufficient statistic can be expressed nicely in terms of string diagrams, as Tobias Fritz showed recently, in adjoint form. This reformulation highlights the role of split idempotents, in the Fisher-Neyman factorisation theorem. Examples of a sufficient statistic occur in the literature, but mostly in continuous probability. This paper demonstrates that there are also several fundamental examples of a sufficient statistic in discrete probability. They emerge after some combinatorial groundwork that reveals the relevant dagger split idempotents and shows that a sufficient statistic is a deterministic dagger epi.