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.

翻译：在信号加噪声模型中，给定某一位点上信号$X$的条件独立重复观测$Y_j = X + \varepsilon_j$，量化信号活跃的证据具有重要研究意义。本文研究信号分布稀疏且误差分布方差未知的问题，此时标准化统计量的零分布为Student-$t$分布。本文的主要贡献在于提出了$t$比率非零边际密度的稀疏混合近似公式。该公式揭示了低自由度对贝叶斯因子（即该位点活跃的条件概率）的影响。我们通过在Efron（2012）先前分析的HIV基因表达数据集上的应用，展示了其中的一些差异。