Essentially all widely used prior densities for β can be represented as scale mixtures of normals:2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p(\beta _j) = {\int} N(0,\Psi _j){\mathrm{d}}G(\Psi _j),\quad \quad j = 1,2, \cdots ,M,$$\end{document}p(βj)=∫N(0,Ψj)dG(Ψj),j=1,2,⋯,M,or equivalently, as the following hierarchical form:3\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\beta _j|\Psi _j \sim N(0,\Psi _j),\quad \quad \Psi _j \sim G,\quad \quad j = 1,2, \cdots ,M,$$\end{document}βj∣Ψj~N(0,Ψj),Ψj~G,j=1,2,⋯,M,where N(μ, σ2) is a normal distribution with mean μ and variance σ2, and G is a mixing distribution. For example, if G places all its mass at a single point, i.e., \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$G(\Psi _j) = \delta _{\sigma _\beta ^2}$$\end{document}G(Ψj)=δσβ2, where δ• is the Dirac delta measure, then marginally \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\beta _j \sim N(0,\sigma _\beta ^2)$$\end{document}βj~N(0,σβ2), and we have recovered the infinitesimal model16. To create a more flexible model of the genetic architecture, a discrete mixture of two or more point masses or densities can be used, which allows for a wider effect size distribution than a
