  In a range of genomic applications, it is of interest to quantify the
evidence that the signal at site~$i$ is active given conditionally independent
replicate observations summarized by the sample mean and variance $(\bar Y,
s^2)$ at each site. We study the version of the problem in which the signal
distribution is sparse, and the error distribution has an unknown site-specific
variance so that the null distribution of the standardized statistic is
Student-$t$ rather than Gaussian. The main contribution of this paper is a
sparse-mixture approximation to the non-null 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).
