SMML estimators for 1-dimensional continuous data

TL;DR

This paper presents a method to compute the SMML estimator for 1D exponential families with continuous data, providing equations, solutions, and proofs of properties related to the estimator.

Contribution

It introduces a set of equations for the SMML estimator's cut-points, demonstrates solving them via Newton's method, and proves the continuity of the posterior probability.

Findings

Derived equations for SMML estimator cut-points

Applied Newton's method for solutions

Proved continuity of the posterior probability

Abstract

A method is given for calculating the strict minimum message length (SMML) estimator for 1-dimensional exponential families with continuous sufficient statistics. A set of $n$ equations are found that the $n$ cut-points of the SMML estimator must satisfy. These equations can be solved using Newton's method and this approach is used to produce new results and to replicate results that C. S. Wallace obtained using his boundary rules for the SMML estimator. A rigorous proof is also given that, despite being composed of step functions, the posterior probability corresponding to the SMML estimator is a continuous function of the data.