Dispersion prior As also observed by Wu et al. [6], a log-normal prior fits the observed dispersion distribution for typical RNA-seq datasets. We solve the computational difficulty of working with a non-conjugate prior using the following argument: the logarithmic residuals from the trend fit, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $\log \alpha ^{\text {gw}}_{i} - \log \alpha _{\text {tr}}(\bar \mu _{i})$ \end{document}logαigw−logαtr(μ¯i), arise from two contributions, namely the scatter of the true logarithmic dispersions around the trend, given by the prior with variance \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $\sigma _{\text {d}}^{2}$ \end{document}σd2, and the sampling distribution of the logarithm of the dispersion estimator, with variance \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $\sigma ^{2}_{\text {lde}}$ \end{document}σlde2. The sampling distribution of a dispersion estimator is approximately a scaled χ2 distribution with m−p degrees of freedom, with m the number of samples and p the number of coefficients. The variance of the logarithm of a \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} ${\chi ^{2}_{f}}$ \end{document}χf2-distributed random variable
