Robust Bayesian Modeling with Adaptive Posterior FDR Control for Large-Scale Data

Controlling the false discovery rate (FDR) is a critical challenge in large-scale data analysis, particularly in the presence of outliers. A common practice involves imposing a Student-$t$ distribution to eliminate the influence of outliers. Here, we developed a robust Bayesian analysis based on heavy-tailed modeling, applied it to large-scale studies in Bayesian inference, and performed diagnoses for detecting outliers using the posterior predictive $p$-value ($ppp$). In addition, we propose an adaptive method to decide the level of the posterior false discovery rate. We demonstrated the utility of our methods using gene expression data for colorectal cancer. We suggest an adaptive method to determine it using an estimated ratio of true null genes using Storey's $q$-value method.