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High-Dimensional Bernstein Von-Mises Theorems for Covariance and Precision Matrices

Partha Sarkar, Kshitij Khare, Malay Ghosh, Matt P. Wand

TL;DR

The paper addresses high-dimensional Bayesian inference for covariance and precision matrices by deriving Bernstein–von Mises theorems in regimes where $p_n$ grows with $n$, showing posterior distributions concentrate around Gaussian limits. It develops dense-setting results with general priors (IW, DSIW, matrix-F) and sharp contraction rates for sparse precision in Gaussian graphical models, both under known and unknown graph structures via G-Wishart priors. The work provides finite-sample-inspired asymptotics, proves BvM with explicit limiting matrix-normal forms, and demonstrates norm-equivalence to extend results to Hellinger and Rényi divergences. Together, these results justify Bayesian uncertainty quantification for high-dimensional covariance and graphical-model estimation, enabling valid credible sets for entire matrices and their functionals.

Abstract

This paper aims to examine the characteristics of the posterior distribution of covariance/precision matrices in a "large $p$, large $n$" scenario, where $p$ represents the number of variables and $n$ is the sample size. Our analysis focuses on establishing asymptotic normality of the posterior distribution of the entire covariance/precision matrices under specific growth restrictions on $p_n$ and other mild assumptions. In particular, the limiting distribution turns out to be a symmetric matrix variate normal distribution whose parameters depend on the maximum likelihood estimate. Our results hold for a wide class of prior distributions which includes standard choices used by practitioners. Next, we consider Gaussian graphical models which induce sparsity in the precision matrix. Asymptotic normality of the corresponding posterior distribution is established under mild assumptions on the prior and true data-generating mechanism.

High-Dimensional Bernstein Von-Mises Theorems for Covariance and Precision Matrices

TL;DR

The paper addresses high-dimensional Bayesian inference for covariance and precision matrices by deriving Bernstein–von Mises theorems in regimes where grows with , showing posterior distributions concentrate around Gaussian limits. It develops dense-setting results with general priors (IW, DSIW, matrix-F) and sharp contraction rates for sparse precision in Gaussian graphical models, both under known and unknown graph structures via G-Wishart priors. The work provides finite-sample-inspired asymptotics, proves BvM with explicit limiting matrix-normal forms, and demonstrates norm-equivalence to extend results to Hellinger and Rényi divergences. Together, these results justify Bayesian uncertainty quantification for high-dimensional covariance and graphical-model estimation, enabling valid credible sets for entire matrices and their functionals.

Abstract

This paper aims to examine the characteristics of the posterior distribution of covariance/precision matrices in a "large , large " scenario, where represents the number of variables and is the sample size. Our analysis focuses on establishing asymptotic normality of the posterior distribution of the entire covariance/precision matrices under specific growth restrictions on and other mild assumptions. In particular, the limiting distribution turns out to be a symmetric matrix variate normal distribution whose parameters depend on the maximum likelihood estimate. Our results hold for a wide class of prior distributions which includes standard choices used by practitioners. Next, we consider Gaussian graphical models which induce sparsity in the precision matrix. Asymptotic normality of the corresponding posterior distribution is established under mild assumptions on the prior and true data-generating mechanism.
Paper Structure (20 sections, 18 theorems, 52 equations)

This paper contains 20 sections, 18 theorems, 52 equations.

Table of Contents

  1. Introduction
  2. Notation
  3. Preliminaries and model formulation in the dense setting
  4. Prior distributions for general covariance or precision matrices
  5. The inverse Wishart prior
  6. The diagonal scale-mixed inverse Wishart prior
  7. The matrix-$F$ prior
  8. BvM results for dense covariance or precision matrices
  9. Concentration graphical models: preliminaries
  10. Decomposable graphs
  11. Sparse symmetric matrix-normal distributions
  12. Concentration Graphical Models: model formulation and prior specification for known $G$
  13. The $G$-Wishart distribution
  14. BvM and posterior consistency results for sparse precision matrices in the known-$G$ Case
  15. Extension of BvM results when the underlying graph is unknown
  16. ...and 5 more sections

Key Result

Lemma 2.4

Suppose that, for each $n$, the prior density $\pi_1(\cdot)$ satisfies the following local log-Lipschitz condition around the sample covariance matrix $S$: there exist deterministic sequences $r_n > 0$ and $L_n > 0$ such that, with probability tending to one under $\mathbb{P}_0$, Assume further that the posterior contraction rate $\epsilon_n$ and the neighborhood radius $r_n$ satisfy $\varepsilon

Theorems & Definitions (25)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Lemma 2.4: Local log-Lipschitz prior implies flatness
  • Lemma 4.1
  • Lemma 4.2
  • Lemma 4.3
  • Theorem 4.4
  • Lemma 4.5
  • Theorem 4.6
  • ...and 15 more