# Bayes Extended Estimators for Curved Exponential Families

## TL;DR

This paper introduces a simple approximation for the Bayesian predictive density in curved exponential families using projection onto finite-dimensional exponential families, with theoretical analysis and practical benefits.

## Contribution

It proposes the Bayes extended estimator as a practical approximation of the Bayesian predictive density for curved exponential families, with proven asymptotic properties.

## Key findings

- The projection asymptotically matches the plugin density with the posterior mean.
- Numerical simulations show the approximation approaches the true density as dimension increases.
- The method offers a good trade-off between risk performance and computational efficiency.

## Abstract

The Bayesian predictive density has complex representation and does not belong to any finite-dimensional statistical model except for in limited situations. In this paper, we introduce its simple approximate representation employing its projection onto a finite-dimensional exponential family. Its theoretical properties are established parallelly to those of the Bayesian predictive density when the model belongs to curved exponential families. It is also demonstrated that the projection asymptotically coincides with the plugin density with the posterior mean of the expectation parameter of the exponential family, which we refer to as the Bayes extended estimator. Information-geometric correspondence indicates that the Bayesian predictive density can be represented as the posterior mean of the infinite-dimensional exponential family. The Kullback--Leibler risk performance of the approximation is demonstrated by numerical simulations and it indicates that the posterior mean of the expectation parameter approaches the Bayesian predictive density as the dimension of the exponential family increases. It also suggests that approximation by projection onto an exponential family of reasonable size is practically advantageous with respect to risk performance and computational cost.