In this work, we investigate a conceptually different class of priors—the continuous shrinkage priors. In particular, we consider the following prior on SNP effect sizes, which can be represented as global-local scale mixtures of normals:4\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\beta _j|\psi _j \sim N(0,\phi \psi _j),\quad \quad \psi _j \sim g,$$\end{document}βj∣ψj~N(0,ϕψj),ψj~g,where ϕ is a global scaling parameter that shares across genetic markers and controls the degree of sparseness of the model, and g is an absolutely continuous density function, in contrast to a discrete mixture of atoms or densities. By appropriately choosing the continuous mixing density g, this modeling framework can produce a variety of shapes of the prior distribution on βj. In particular, g can be designed to introduce a prior distribution on the SNP effect sizes that has a sizable amount of mass near zero to impose strong shrinkage on noise, while at the same time has heavy tails to avoid over-shrinkage of truly non-zero effects. The marker-specific local shrinkage parameter ψj can then adaptively squelch small noisy estimates towards zero, while leaving data-supported
