# On Parametric Misspecified Bayesian Cram\'{e}r-Rao bound: An application to linear Gaussian systems

## TL;DR

This paper derives a Bayesian Cramér-Rao bound under model misspecification, introducing the concept of pseudotrue parameters, and applies it to linear Gaussian systems with closed-form solutions.

## Contribution

It provides a novel derivation of the Bayesian Cramér-Rao bound under model mismatch, including the definition of pseudotrue parameters, with specific application to linear Gaussian models.

## Key findings

- Derived closed-form Bayesian CRB under misspecification for linear Gaussian systems
- Validated the theoretical results through simulations
- Enhanced understanding of estimator performance limits under model mismatch

## Abstract

A lower bound is an important tool for predicting the performance that an estimator can achieve under a particular statistical model. Bayesian bounds are a kind of such bounds which not only utilizes the observation statistics but also includes the prior model information. In reality, however, the true model generating the data is either unknown or simplified when deriving estimators, which motivates the works to derive estimation bounds under modeling mismatch situations. This paper provides a derivation of a Bayesian Cram\'{e}r-Rao bound under model misspecification, defining important concepts such as pseudotrue parameter that were not clearly identified in previous works. The general result is particularized in linear and Gaussian problems, where closed-forms are available and results are used to validate the results.