# Jeffreys priors for mixture estimation: properties and alternatives

## TL;DR

This paper investigates Jeffreys priors for mixture models, highlighting their properties, challenges like impropriety, and proposing their use as default priors for mixture weights in overfitted models.

## Contribution

It provides a detailed analysis of Jeffreys priors in mixture estimation, including their properties, limitations, and a novel application as default priors for mixture weights.

## Key findings

- Jeffreys priors are often improper and lack closed-form expressions.
- The posterior distributions with Jeffreys priors are mostly improper.
- Jeffreys priors for mixture weights are conservative regarding the number of components.

## Abstract

While Jeffreys priors usually are well-defined for the parameters of mixtures of distributions, they are not available in closed form. Furthermore, they often are improper priors. Hence, they have never been used to draw inference on the mixture parameters. The implementation and the properties of Jeffreys priors in several mixture settings are studied. It is shown that the associated posterior distributions most often are improper. Nevertheless, the Jeffreys prior for the mixture weights conditionally on the parameters of the mixture components will be shown to have the property of conservativeness with respect to the number of components, in case of overfitted mixture and it can be therefore used as a default priors in this context.