# Concentration of a sparse Bayesian model with Horseshoe prior in estimating high-dimensional precision matrix

arXiv: https://arxiv.org/abs/2406.14269 · HTML: https://mlsanity.com/paper/2406.14269/

## Key facts

- Cited by 8 papers in the map (data through 14 Oct 2026).
- Builds on 18 of its 45 references that are in the map.
- First posted on arXiv 20 Jun 2024.

## Builds on (most cited first, 18 of 18)

- [A General Framework for Updating Belief Distributions](https://mlsanity.com/paper/1306.6430/) (2013)
- [Inconsistency of Bayesian Inference for Misspecified Linear Models, and a Proposal for Repairing It](https://mlsanity.com/paper/1412.3730/) (2014)
- [Concentration of tempered posteriors and of their variational approximations](https://mlsanity.com/paper/1706.09293/) (2017)
- [General Bayesian Updating and the Loss-Likelihood Bootstrap](https://mlsanity.com/paper/1709.07616/) (2017)
- [Calibrating general posterior credible regions](https://mlsanity.com/paper/1509.00922/) (2015)
- [A comparison of learning rate selection methods in generalized Bayesian inference](https://mlsanity.com/paper/2012.11349/) (2020)
- [An Overview on the Estimation of Large Covariance and Precision Matrices](https://mlsanity.com/paper/1504.02995/) (2015)
- [On the Robustness to Misspecification of α-Posteriors and Their Variational Approximations](https://mlsanity.com/paper/2104.08324/) (2021)
- [The Graphical Horseshoe Estimator for Inverse Covariance Matrices](https://mlsanity.com/paper/1707.06661/) (2017)
- [Adaptation of the Tuning Parameter in General Bayesian Inference with Robust Divergence](https://mlsanity.com/paper/2106.06902/) (2021)
- [Bayesian sparse multiple regression for simultaneous rank reduction and variable selection](https://mlsanity.com/paper/1612.00877/) (2016)
- [PAC-Bayesian AUC classification and scoring](https://mlsanity.com/paper/1410.1771/) (2014)
- [Bayesian Regularization for Graphical Models with Unequal Shrinkage](https://mlsanity.com/paper/1805.02257/) (2018)
- [α-Variational Inference with Statistical Guarantees](https://mlsanity.com/paper/1710.03266/) (2017)
- [Precision Matrix Estimation under the Horseshoe-like Prior-Penalty Dual](https://mlsanity.com/paper/2104.10750/) (2021)
- [Semiparametric inference using fractional posteriors](https://mlsanity.com/paper/2301.08158/) (2023)
- [Bayesian inference in high-dimensional linear models using an empirical correlation-adaptive prior](https://mlsanity.com/paper/1810.00739/) (2018)
- [Contraction of a quasi-Bayesian model with shrinkage priors in precision matrix estimation](https://mlsanity.com/paper/2104.12060/) (2021)

## Built on by (most cited first, 8 of 8)

- [High-dimensional Bayesian Tobit regression for censored response with Horseshoe prior](https://mlsanity.com/paper/2505.08288/) (2025)
- [Handling bounded response in high dimensions: a Horseshoe prior Bayesian Beta regression approach](https://mlsanity.com/paper/2505.22211/) (2025)
- [Heavy Lasso: sparse penalized regression under heavy-tailed noise via data-augmented soft-thresholding](https://mlsanity.com/paper/2506.07790/) (2025)
- [Bayesian Pliable Lasso with Horseshoe Prior for Interaction Effects in GLMs with Missing Responses](https://mlsanity.com/paper/2509.07501/) (2025)
- [Kullback-Leibler excess risk bounds for exponential weighted aggregation in Generalized linear models](https://mlsanity.com/paper/2504.10171/) (2025)
- [Sparse classification with positive-confidence data in high dimensions](https://mlsanity.com/paper/2512.24443/) (2025)
- [Robust low-rank estimation with multiple binary responses using pairwise AUC loss](https://mlsanity.com/paper/2601.08618/) (2026)
- [Mixture-of-experts Wishart model for covariance matrices with an application to Cancer drug screening](https://mlsanity.com/paper/2602.13888/) (2026)

## Abstract (arXiv)

Precision matrices are crucial in many fields such as social networks, neuroscience, and economics, representing the edge structure of Gaussian graphical models (GGMs), where a zero in an off-diagonal position of the precision matrix indicates conditional independence between nodes. In high-dimensional settings where the dimension of the precision matrix ( p ) exceeds the sample size ( n ) and the matrix is sparse, methods like graphical Lasso, graphical SCAD, and CLIME are popular for estimating GGMs. While frequentist methods are well-studied, Bayesian approaches for (unstructured) sparse precision matrices are less explored. The graphical horseshoe estimate by citeli2019graphical, applying the global-local horseshoe prior, shows superior empirical performance, but theoretical work for sparse precision matrix estimations using shrinkage priors is limited. This paper addresses these gaps by providing concentration results for the tempered posterior with the fully specified horseshoe prior in high-dimensional settings. Moreover, we also provide novel theoretical results for model misspecification, offering a general oracle inequality for the posterior. A concise set of simulations is performed to validate our theoretical findings.