# A Note on Bayesian Model Selection for Discrete Data Using Proper Scoring Rules

## TL;DR

This paper proposes a Bayesian model selection method for discrete data using proper scoring rules, enabling the use of improper priors and demonstrating consistent model choice between Poisson and Negative Binomial models through simulations.

## Contribution

It introduces a scoring rule-based Bayesian approach for model selection with improper priors, specifically applied to discrete distributions like Poisson and Negative Binomial.

## Key findings

- The method consistently selects the correct model in simulations.
- Homogeneous scoring rules effectively handle improper priors.
- Prequential application ensures reliable model discrimination.

## Abstract

We consider the problem of choosing between parametric models for a discrete observable, taking a Bayesian approach in which the within-model prior distributions are allowed to be improper. In order to avoid the ambiguity in the marginal likelihood function in such a case, we apply a homogeneous scoring rule. For the particular case of distinguishing between Poisson and Negative Binomial models, we conduct simulations that indicate that, applied prequentially, the method will consistently select the true model.