Bayesian Predictive Densities Based on Latent Information Priors

TL;DR

This paper explores the construction of Bayesian predictive densities using latent information priors, aiming to optimize predictive accuracy in a parametric multinomial framework with dependent data.

Contribution

It introduces latent information priors as a new class of priors that maximize conditional mutual information, leading to minimax predictive densities.

Findings

Limits of Bayesian predictive densities form an essentially complete class.

Latent information priors are defined as priors maximizing conditional mutual information.

Minimax predictive densities are constructed from these priors.

Abstract

Construction methods for prior densities are investigated from a predictive viewpoint. Predictive densities for future observables are constructed by using observed data. The simultaneous distribution of future observables and observed data is assumed to belong to a parametric submodel of a multinomial model. Future observables and data are possibly dependent. The discrepancy of a predictive density to the true conditional density of future observables given observed data is evaluated by the Kullback-Leibler divergence. It is proved that limits of Bayesian predictive densities form an essentially complete class. Latent information priors are defined as priors maximizing the conditional mutual information between the parameter and the future observables given the observed data. Minimax predictive densities are constructed as limits of Bayesian predictive densities based on prior sequences converging to the latent information priors.