Sparse approximation for t-statistics

In the signal plus noise model, it is of interest to quantify the evidence that a signal is active given conditionally independent replicate observations $Y_j = X + \varepsilon_j$ on the signal $X$ at a particular site. We study the problem in which the signal distribution is sparse, and the error distribution has an unknown variance so that the null distribution of the standardized statistic is Student-$t$. The main contribution of this paper is a sparse-mixture approximation to the non-null marginal density of the $t$-ratio. This formula demonstrates the effect of low degrees of freedom on the Bayes factor, or the conditional probability that the site is active. We illustrate some differences on a HIV dataset for gene-expression data previously analyzed by Efron, 2012.