Generalized Power Priors for Improved Bayesian Inference with Historical Data

TL;DR

This paper introduces a generalized power prior for Bayesian inference that incorporates a divergence measure called Amari's alpha-divergence, enhancing flexibility and theoretical understanding in utilizing historical data.

Contribution

It extends the power prior framework by linking it to alpha-divergence, providing a geometric interpretation and potential for improved adaptive performance.

Findings

The generalized power posterior minimizes a linear combination of alpha-divergences.

The framework offers a geometric perspective as a generalized geodesic on probability manifolds.

The approach allows for adaptive tuning of divergence parameters for better inference.

Abstract

The power prior is a class of informative priors designed to incorporate historical data alongside current data in a Bayesian framework. It includes a power parameter that controls the influence of historical data, providing flexibility and adaptability. A key property of the power prior is that the resulting posterior minimizes a linear combination of KL divergences between two pseudo-posterior distributions: one ignoring historical data and the other fully incorporating it. We extend this framework by identifying the posterior distribution as the minimizer of a linear combination of Amari's $\alpha$-divergence, a generalization of KL divergence. We show that this generalization can lead to improved performance by allowing for the data to adapt to appropriate choices of the $\alpha$ parameter. Theoretical properties of this generalized power posterior are established, including behavior as a generalized geodesic on the Riemannian manifold of probability distributions, offering novel insights into its geometric interpretation.