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.

翻译：高斯链图模型的多元回归解释同时参数化了（i）$p$个预测变量对$q$个结果变量的直接效应，以及（ii）结果变量对之间的残差偏协方差。我们提出了一种新方法，通过尖峰-板状LASSO（SSL）先验拟合稀疏高斯链图模型。我们开发了一种期望条件最大化算法，以获得直接效应的$p \times q$矩阵和残差精度$q \times q$矩阵的稀疏估计。该算法迭代求解一系列带自适应惩罚的惩罚极大似然问题，逐步过滤掉可忽略的回归系数和偏协方差。由于该方法对单个模型参数进行自适应惩罚，在模拟数据上其表现优于固定惩罚的竞争方法。我们建立了模型的后验收缩率，以强大的理论保证支撑该方法的卓越实证表现。利用该方法，我们估算了饮食和居住类型对老年人肠道微生物组组成的直接效应。