To provide further intuitions, assuming that all genetic markers are unlinked (i.e., no LD), we have D = I and thus8\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbf{E}}[\beta _j|\hat \beta _j] = \frac{1}{{1 + \phi ^{ - 1}\psi _j^{ - 1}}}\hat \beta _j = \left( {1 - \frac{1}{{1 + \phi \psi _j}}} \right)\hat \beta _j: = (1 - \tau _j)\hat \beta _j,$$\end{document}E[βj∣β^j]=11+ϕ-1ψj-1β^j=1-11+ϕψjβ^j:=(1-τj)β^j,where τj = 1/(1 + ϕψj) is the shrinkage factor for the j-th marker, which relies on both ϕ and ψj, and describes the amount of shrinkage from the marginal least squares solution towards zero; τj = 0 indicates no shrinkage while τj = 1 yields total shrinkage. Therefore, ϕ controls the overall sparsity level of the model and plays a similar role as the regularization parameter in penalized regression, while ψj adaptively modifies the amount of shrinkage for each marker. By assigning a prior on ψj, which can produce a marginal prior density on βj that has both a sharp peak at zero and heavy tails, the model can pull small effects towards zero, while asserting little influence on larger effects.
