Generalized Power Priors for Improved Bayesian Inference with Historical Data
This work provides an incremental improvement for researchers in Bayesian statistics by offering a more flexible method for data integration.
The paper tackles the problem of incorporating historical data into Bayesian inference by extending the power prior framework to use Amari's α-divergence instead of KL divergence, resulting in improved performance through adaptive parameter choices.
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 $α$-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 $α$ 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.