# A Family of Bayesian Cram\'er-Rao Bounds, and Consequences for Log-Concave Priors

## TL;DR

This paper introduces a broad family of Bayesian Cramér-Rao bounds applicable under minimal assumptions, including for log-concave priors, extending existing bounds and removing prior Fisher information dependence.

## Contribution

It establishes a new family of Bayesian Cramér-Rao bounds based on logarithmic Sobolev inequalities, generalizing and improving upon existing bounds like the van Trees inequality.

## Key findings

- Includes the classical Bayesian Cramér-Rao bound as a special case
- Provides bounds valid for any estimator, biased or unbiased
- Removes dependence on prior Fisher information for log-concave priors

## Abstract

Under minimal regularity assumptions, we establish a family of information-theoretic Bayesian Cram\'er-Rao bounds, indexed by probability measures that satisfy a logarithmic Sobolev inequality. This family includes as a special case the known Bayesian Cram\'er-Rao bound (or van Trees inequality), and its less widely known entropic improvement due to Efroimovich. For the setting of a log-concave prior, we obtain a Bayesian Cram\'er-Rao bound which holds for any (possibly biased) estimator and, unlike the van Trees inequality, does not depend on the Fisher information of the prior.