Normalized Maximum Likelihood with Luckiness for Multivariate Normal Distributions

TL;DR

This paper develops a closed-form expression for the luckiness-normalized maximum likelihood (LNML) for multivariate normal distributions, addressing the limitations of NML in such cases and enhancing statistical modeling and coding theory.

Contribution

It introduces a novel closed-form solution for LNML specifically tailored to multivariate normal distributions, expanding the applicability of NML-based methods.

Findings

Closed-form LNML for multivariate normal distributions derived

Addresses NML's non-existence issue for simple families

Enhances statistical coding and model selection techniques

Abstract

The normalized maximum likelihood (NML) is one of the most important distribution in coding theory and statistics. NML is the unique solution (if exists) to the pointwise minimax regret problem. However, NML is not defined even for simple family of distributions such as the normal distributions. Since there does not exist any meaningful minimax-regret distribution for such case, it is pointed out that NML with luckiness (LNML) can be employed as an alternative to NML. In this paper, we develop the closed form of LNMLs for multivariate normal distributions.