# Intrinsic Bayesian Cram'er-Rao Bound with an Application to Covariance Matrix Estimation

arXiv: https://arxiv.org/abs/2311.04748 · HTML: https://mlsanity.com/paper/2311.04748/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 3 of its 64 references that are in the map.
- First posted on arXiv 8 Nov 2023.

## Builds on (most cited first, 3 of 3)

- [The Fisher-Rao geometry of CES distributions](https://mlsanity.com/paper/2310.01032/) (2023)
- [An intrinsic Cram'er-Rao bound on Lie groups](https://mlsanity.com/paper/1506.05662/) (2015)
- [Barankin-Type Bound for Constrained Parameter Estimation](https://mlsanity.com/paper/2304.07942/) (2023)

## Built on by (most cited first, 1 of 1)

- [Spectrum-Aware Bounds on Invertibility for Privacy-Enhancing Instance Encoding](https://mlsanity.com/paper/2608.23382/) (2026)

## Abstract (arXiv)

This paper presents a new performance bound for estimation problems where the parameter to estimate lies in a Riemannian manifold (a smooth manifold endowed with a Riemannian metric) and follows a given prior distribution. In this setup, the chosen Riemannian metric induces a geometry for the parameter manifold, as well as an intrinsic notion of the estimation error measure. Performance bound for such error measure were previously obtained in the non-Bayesian case (when the unknown parameter is assumed to deterministic), and referred to as textitintrinsic Cram'er-Rao bound. The presented result then appears either as: textita) an extension of the intrinsic Cram'er-Rao bound to the Bayesian estimation framework; textitb) a generalization of the Van-Trees inequality (Bayesian Cram'er-Rao bound) that accounts for the aforementioned geometric structures. In a second part, we leverage this formalism to study the problem of covariance matrix estimation when the data follow a Gaussian distribution, and whose covariance matrix is drawn from an inverse Wishart distribution. Performance bounds for this problem are obtained for both the mean squared error (Euclidean metric) and the natural Riemannian distance for Hermitian positive definite matrices (affine invariant metric). Numerical simulation illustrate that assessing the error with the affine invariant metric is revealing of interesting properties of the maximum a posteriori and minimum mean square error estimator, which are not observed when using the Euclidean metric.