By taking the posterior inclusion probability \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$P({\beta }_{m,k})$$\end{document}P(βm,k) as an approximation for the certainty of a credible effect for each \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\beta }_{m,k}$$\end{document}βm,k, its complementary \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$1-P({\beta }_{m,k})$$\end{document}1−P(βm,k) approximates the probability of a type I error. For a threshold c, we now rank all \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\beta }_{m,k}$$\end{document}βm,k by their type I error probability and obtain a set of credible effects \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$J(c)=\{{\beta }_{m,k}|1-P({\beta }_{m,k})\le c\}.$$\end{document}J(c)={βm,k∣1−P(βm,k)≤c}.Then, the approximate false discovery rate for the threshold is11\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\widehat{{{{{{\rm{FDR}}}}}}}(c)=\frac{{\sum }_{{\beta }_{m,k}\in J(c)}1-P({\beta }_{m,k})}{|J(c)|}.$$\end{document}FDR^(c)=∑βm,k∈J(c)1−P(βm,k)∣J(c)∣.
