Estimating sparse direct effects in multivariate regression with the spike-and-slab LASSO

TL;DR

This paper introduces a new sparse Gaussian chain graph model fitting method using spike-and-slab LASSO priors, with an efficient algorithm, theoretical guarantees, and real-world microbiome application.

Contribution

The paper develops an Expectation Conditional Maximization algorithm for sparse Gaussian chain graph models with adaptive penalties, improving estimation accuracy over fixed-penalty methods.

Findings

Outperforms fixed-penalty methods on simulated data

Establishes posterior contraction rate with theoretical guarantees

Successfully applied to microbiome data to estimate effects of diet and residence

Abstract

The multivariate regression interpretation of the Gaussian chain graph model simultaneously parametrizes (i) the direct effects of $p$ predictors on $q$ outcomes and (ii) the residual partial covariances between pairs of outcomes. We introduce a new method for fitting sparse Gaussian chain graph models with spike-and-slab LASSO (SSL) priors. We develop an Expectation Conditional Maximization algorithm to obtain sparse estimates of the $p \times q$ matrix of direct effects and the $q \times q$ residual precision matrix. Our algorithm iteratively solves a sequence of penalized maximum likelihood problems with self-adaptive penalties that gradually filter out negligible regression coefficients and partial covariances. Because it adaptively penalizes individual model parameters, our method is seen to outperform fixed-penalty competitors on simulated data. We establish the posterior contraction rate for our model, buttressing our method's excellent empirical performance with strong theoretical guarantees. Using our method, we estimated the direct effects of diet and residence type on the composition of the gut microbiome of elderly adults.