Censored Graphical Horseshoe: Bayesian sparse precision matrix estimation with censored and missing data

Abstract

Gaussian graphical models provide a powerful framework for studying conditional dependencies in multivariate data, with widespread applications spanning biomedical, environmental sciences, and other data-rich scientific domains. While the Graphical Horseshoe (GHS) method has emerged as a state-of-the-art Bayesian method for sparse precision matrix estimation, existing approaches assume fully observed data and thus fail in the presence of censoring or missingness, which are pervasive in real-world studies. In this paper, we develop the Censored Graphical Horseshoe (CGHS), a novel Bayesian framework that extends the GHS to censored and arbitrarily missing Gaussian data. By introducing a latent-variable representation, CGHS accommodates incomplete observations while retaining the adaptive global-local shrinkage properties of the Horseshoe prior. We derive efficient Gibbs samplers for posterior computation and establish new theoretical results on posterior behavior under censoring and missingness, filling a gap not addressed by frequentist Lasso-based methods. Through extensive simulations, we demonstrate that CGHS consistently improves estimation accuracy compared to penalized likelihood approaches. Our methods are implemented in the package GHScenmis available on Github: https://github.com/tienmt/ghscenmis .

Figures (5)

• Figure 1: Graphs from different methods on the real data set MKMEP. We see that GHSsen return a sparse networks compared to cglasso.
• Figure 2: Trace plots from the Gibbs sampler for selected parameter entries. Top row: three randomly chosen entries with true value 0. Middle row: three randomly chosen entries with true value $1$. Bottom row: three randomly chosen entries with true value 0.3. The true precision matrix is in Setting I with $p = 10, n = 100$. 10% of data being missing.
• Figure 3: ACF plots from the Gibbs sampler for some random entries as in Figure \ref{['fig_tracplot']}. Top row (3 plots): 3 random entries with true value 0. Middle row (3 plots): 3 random entries with true value $1$. Bottom row (3 plots): 3 random entries with true value 0.3. The ESS (effective sample size) are also given. 10% of data being missing.
• Figure 4: Trace plots from the Gibbs sampler for selected parameter entries. Top row: three randomly chosen entries with true value $-0.5$. Middle row: three randomly chosen entries with true value $1$. Bottom row: three randomly chosen entries with true value 0. The true precision matrix is in Setting II with $p = 10, n = 100$. 10% of data being missing.
• Figure 5: ACF plots from the Gibbs sampler for some random entries as in Figure \ref{['fig_tracplot']}. Top row: three randomly chosen entries with true value $-0.5$. Middle row: three randomly chosen entries with true value $1$.Bottom row: three randomly chosen entries with true value 0. The true precision matrix is in Setting II with $p = 10, n = 100$. The ESS (effective sample size) are also given. 10% of data being missing.

Theorems & Definitions (10)

• Theorem 1: Posterior contraction for censored data
• Theorem 2: Posterior contraction for missing data
• proof : Proof of Theorem \ref{['thm_censoredGHS']}
• proof : Proof of Theorem \ref{['thm_missing_GHS']}
• Lemma 1: Lemma 3 in mai2024concentration
• Theorem 3: Theorem 2.6 in alquier2020concentration
• Lemma 2
• proof : Proof of Lemma \ref{['lm_bounded_log_likelihood_censored']}
• Lemma 3
• proof : Proof of Lemma \ref{['lm_bound_KL_for_missing']}