On the Sample Information About Parameter and Prediction

TL;DR

This paper investigates the relationship between Bayesian sample information measures for parameters and predictions, exploring the effects of dependence and providing decompositions for normal and exponential family models.

Contribution

It clarifies the relationship between parameter and predictive information measures and examines the impact of dependence on these measures in Bayesian analysis.

Findings

Joint information exceeds Lindley's measure with dependence.

Dependence increases the difference between parameter and predictive information.

Conditionally independent samples provide limited predictive information.

Abstract

The Bayesian measure of sample information about the parameter, known as Lindley's measure, is widely used in various problems such as developing prior distributions, models for the likelihood functions and optimal designs. The predictive information is defined similarly and used for model selection and optimal designs, though to a lesser extent. The parameter and predictive information measures are proper utility functions and have been also used in combination. Yet the relationship between the two measures and the effects of conditional dependence between the observable quantities on the Bayesian information measures remain unexplored. We address both issues. The relationship between the two information measures is explored through the information provided by the sample about the parameter and prediction jointly. The role of dependence is explored along with the interplay between the information measures, prior and sampling design. For the conditionally independent sequence of observable quantities, decompositions of the joint information characterize Lindley's measure as the sample information about the parameter and prediction jointly and the predictive information as part of it. For the conditionally dependent case, the joint information about parameter and prediction exceeds Lindley's measure by an amount due to the dependence. More specific results are shown for the normal linear models and a broad subfamily of the exponential family. Conditionally independent samples provide relatively little information for prediction, and the gap between the parameter and predictive information measures grows rapidly with the sample size.