LDpred-inf is a special case of LDpred when all variants are assumed to be causal (i.e., π = 1). Under this infinitesimal model, the posterior mean effect sizes in the \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\ell$$\end{document}ℓ-th LD window have a closed-form approximation:13\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbf{E}}[{\boldsymbol{\beta}} _\ell |\hat{\boldsymbol{\beta}} _\ell ,{\mathbf{D}}_\ell ] \approx \left( {{\mathbf{D}}_\ell + \frac{M}{{Nh_g^2}}{\mathbf{I}}} \right)^{ - 1}\hat{\boldsymbol{\beta}} _\ell ,$$\end{document}E[βℓ∣β^ℓ,Dℓ]≈Dℓ+MNhg2I-1β^ℓ,where \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\hat{\boldsymbol{\beta}} _\ell$$\end{document}β^ℓ is a vector of marginal least squares effect size estimates, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbf{D}}_\ell$$\end{document}Dℓ is the LD matrix that can be estimated from an external reference panel, I is an identity matrix, and it has been assumed that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$h_\ell ^2$$\end{document}hℓ2, the heritability explained by SNPs in the \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\ell$$\end{document}ℓ-th LD window, is small such that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$1 - h_\ell ^2 \approx 1$$\end{document}1-hℓ2≈1. In this work, we use an LD radius of M/3000 to approximate local LD patterns, as suggested in Vilhjalmsson et al.4
