Minimum Message Length Analysis of the Behrens–Fisher Problem

Abstract

Given two sequences of Gaussian data, the Behrens–Fisher problem is to infer whether there exists a difference between the two corresponding population means if the population variances are unknown. This paper examines the Behrens–Fisher-type problem within the minimum message length framework of inductive inference. Using a special bounding on a uniform prior over the population means, a simple Bayesian hypothesis test is derived that does not require computationally expensive numerical integration of the posterior distribution. The minimum message length procedure is then compared against well-known methods on the Behrens–Fisher hypothesis testing problem and the estimation of the common mean problem showing excellent performance in both cases. Extensions to the generalised Behrens–Fisher problem and the multivariate Behrens–Fisher problem are also discussed.