To make the fit robust against outliers with very high absolute LFC values, we use quantile matching: the width σr is chosen such that the (1−p) empirical quantile of the absolute value of the observed LFCs, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $\vec {\beta }^{\text {MLE}}_{r}$ \end{document}β→rMLE, matches the (1−p/2) theoretical quantile of the prior, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $N(0,{\sigma _{r}^{2}})$ \end{document}N(0,σr2), where p is set by default to 0.05. If we write the theoretical upper quantile of a normal distribution as QN(1−p) and the empirical upper quantile of the MLE LFCs as \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $Q_{|\beta _{r}|}(1 - p)$ \end{document}Q|βr|(1−p), then the prior width is calculated as: \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$ \sigma_{r} = \frac{Q_{|\beta_{r}|}(1 - p)}{Q_{N}(1 - p/2)}. $$ \end{document}σr=Q|βr|(1−p)QN(1−p/2).
