Intrinsic Bayesian Cramér-Rao Bound with an Application to Covariance Matrix Estimation

TL;DR

This work extends Bayesian Cramér-Rao analysis to parameters that reside on Riemannian manifolds by deriving an intrinsic Bayesian Cramér-Rao bound that accounts for curvature and log-map error. It defines a general framework with a Bayesian Fisher information matrix $F_B$ and a curvature term $R_m$, leading to a bound on the error covariance that mirrors the Euclidean Van Trees result but is geometry-aware. The authors apply the theory to covariance matrix estimation with Gaussian data and an inverse-Wishart prior, deriving two concrete bounds under Euclidean and affine-invariant metrics and showing that MAP and MMSE estimators are asymptotically efficient in the intrinsic distance but not in the Euclidean one. Numerical experiments corroborate that geometry-aware bounds reveal estimator efficiency properties that standard Euclidean CRBs miss, highlighting the practical impact of intrinsic geometric bounds for manifold-valued estimation problems.

Abstract

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 \textit{intrinsic} Cramér-Rao bound. The presented result then appears either as: \textit{a}) an extension of the intrinsic Cramér-Rao bound to the Bayesian estimation framework; \textit{b}) a generalization of the Van-Trees inequality (Bayesian Cramér-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.

Figures (6)

• Figure 1: A visual representation of a smooth manifold $\mathcal{M}$, its tangent space $T_{\boldsymbol{\theta}}\mathcal{M}$ at point $\boldsymbol{\theta}$, and two tangent vectors $\boldsymbol{X}_{\boldsymbol{\theta}}$ and $\boldsymbol{Y}_{\boldsymbol{\theta}}$. The metric $\langle\cdot,\cdot\rangle_{\boldsymbol{\theta}}$ is an an inner product on $T_{\boldsymbol{\theta}}\mathcal{M}$ and it induces the notion of length and angle.
• Figure 2: Illustration of directional derivative $\mathop{\mathrm{D}}\nolimits\boldsymbol{Y}_{\boldsymbol{\theta}}[\boldsymbol{X}_{\boldsymbol{\theta}}]$ (left) and Levi-Civita connection $\nabla_{\boldsymbol{Y}_{\boldsymbol{\theta}}}\boldsymbol{X}_{\boldsymbol{\theta}}$ (right) of a vector field $\boldsymbol{Y}_{\boldsymbol{\theta}}$ in the direction $\boldsymbol{X}_{\boldsymbol{\theta}}$ at $\boldsymbol{\theta}$. As the directional derivative, an Levi-Civita connection describes how the vector field $\boldsymbol{Y}_{\boldsymbol{\theta}}$ evolves in a given direction $\boldsymbol{X}_{\boldsymbol{\theta}}$. In addition, the affine connection takes into account the structure of the manifold (curvature, and non-constant metric).
• Figure 3: Illustration of geodesics (left), Riemannian exponential and logarithm mappings (right). The Riemannian distance $d_R(\boldsymbol{\theta},\boldsymbol{\hat{\theta}})$ is the length of the geodesic joining $\boldsymbol{\theta}$ and $\boldsymbol{\hat{\theta}}$.
• Figure 4: Euclidean Cramér-Rao bound and MSE (left) and intrinsic Cramér-Rao bound and distance w.r.t. $n$. The data size is $p=5$.
• Figure 5: Euclidean Bayesian Cramér-Rao bound and MSE (left) and intrinsic Bayesian Cramér-Rao bound and expectation of the natural distance w.r.t. $n$. The data size is $p=5$ and the number of degrees of freedom of the $\mathcal{IW}$ prior is $\nu=40$.

Theorems & Definitions (17)

• Theorem 1: Van Trees inequality vantrees2001vantrees2007
• Remark 1
• Theorem 2: intrinsic Cramér-Rao lower bound smith05
• Remark 2: From intrinsic (Riemannian) to Euclidean Cramér-Rao bound
• Theorem 3: Intrinsic Bayesian Cramér-Rao lower bound
• proof
• Corollary 1: First approximation of intrinsic Bayesian Cramér-Rao lower bound
• proof
• Corollary 2: Second Approximation of intrinsic Bayesian Cramér-Rao lower bound
• proof