We consider the following phenotype model:5\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbf{y}} = {\mathbf{Z}}{\boldsymbol{\beta }} + {\boldsymbol{\varepsilon}} ,\quad \quad \varepsilon \sim N({\mathrm{0}},\sigma ^2{\mathbf{I}}),\quad \quad p(\sigma ^2) \propto \sigma ^{ - 2},$$\end{document}y=Zβ+ε,ε~N(0,σ2I),p(σ2)∝σ-2,where y is a vector of standardized phenotypes from N individuals, Z is an N × M matrix of standardized genotypes (each column is mean centered and has unit variance), β is a vector of effect sizes, ε is a vector of independent environmental effects, and we have assigned a non-informative scale-invariant Jeffreys prior on the residual variance σ2. In contrast to discrete mixture priors, we consider a conceptually different class of priors:6\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\beta _j \sim {\mathrm{N}}\left( {0,\frac{{\sigma ^2}}{N}\phi \psi _j} \right),\quad \quad \psi _j \sim g,$$\end{document}βj~N0,σ2Nϕψj,ψj~g,where the variance of βj scales with the residual variance and the sample size, ϕ is a global scaling parameter that is shared across all effect sizes, ψj is a local, marker-specific parameter, and g is an absolutely continuous mixing density function. This type of prior is known as global-local scale mixtures of normals.
