We first note that, given variance parameters σ2, ϕ and ψj, j = 1,2,…, M, and the marginal least squares effect size estimates of the regression coefficients \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\hat{\boldsymbol{\beta}} = {\mathbf{Z}}^{\rm{T}}{\mathbf{y}}{\mathrm{/}}N$$\end{document}β^=ZTy∕N, the posterior mean of β is7\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbf{E}}[{\boldsymbol{\beta }}|\hat{\boldsymbol{\beta}} ] = ({\mathbf{D}} + {\mathbf{T}}^{ - 1})^{ - 1}\hat{\boldsymbol{\beta}} ,$$\end{document}E[β∣β^]=(D+T-1)-1β^,where T = diag{ϕψ1,ϕψ2,…, ϕψM} is a diagonal matrix, and D = ZΤZ/N is the LD matrix. It can be seen that the posterior mean is a matrix shrinkage version of the least squares estimate. In the degenerative special case where ψj ≡ 1, the model becomes Ridge regression and all effect sizes are shrunk towards zero at the same constant rate controlled by the overall shrinkage parameter ϕ. The introduction of the local shrinkage parameter ψj thus allows heterogeneity in the scales of effect sizes.
