Bayesian Conditional Auto-Regressive LASSO Models to Learn Sparse Networks with Predictors

Inferring a graphical structure with nodes for multiple responses and predictors is a fundamental statistical problem with broad applications from microbiome, ecology to genetics. While a multiresponse linear regression model seems like a straight-forward solution, we argue that treating it as a graphical model is flawed and caution should be taken because the regression coefficient matrix does not represent the adjacency matrix between response and predictor nodes that encodes the conditional dependence structure. This observation is especially important in biological settings when we have prior knowledge on the edges. Here, we propose an alternative model to the multiresponse linear regression whose solution yields a graph with edges that indeed represent conditional dependence. The solution to our model is sparse via Bayesian LASSO and is also guaranteed to be the sparse solution to Conditional Auto-Regressive (CAR) model. In addition, we propose an adaptive extension so that different shrinkage can be applied to different edges to incorporate edge-specific prior knowledge. Our model is computationally inexpensive through an efficient Gibbs sampling algorithm and can account for binary, counting and compositional responses via appropriate hierarchical structure. Finally, we apply our model to a human gut and a soil microbial composition datasets.