We develop a robust Bayesian analysis based on heavy-tailed modeling. It is common to impose a Student-$t$ distribution to eliminate the influence of outliers. We apply it to large-scale studies in Bayesian inference, and provide 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 FDR. We suggest an adaptive method to determine it using an estimated ratio of true null genes using Storey's $q$-value method. Our methods are demonstrated on gene expression data for colorectal cancer.

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