The term Λ(β), i.e., the logarithm of the density of the normal prior (up to an additive constant), can be read as a ridge penalty term, and therefore, we perform the optimization using the iteratively reweighted ridge regression algorithm [56], also known as weighted updates [57]. Specifically, the updates for a given gene are of the form \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$ \vec\beta \leftarrow \left(X^{t} W X + \vec{\lambda} I\right)^{-1} X^{t} W \vec{z}, $$ \end{document}β→←XtWX+λ→I−1XtWz→, with \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $\lambda _{r} = 1/{\sigma _{r}^{2}}$ \end{document}λr=1/σr2 and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$ z_{j} = \log \frac{\mu_{j}}{s_{j}} + \frac{K_{j} - \mu_{j}}{\mu_{j}}, $$ \end{document}zj=logμjsj+Kj−μjμj, where the current fitted values \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $\mu _{j} = s_{j} e^{\sum _{r} x_{\textit {jr}}\beta _{r}}$ \end{document}μj=sje∑rxjrβr are computed from the current estimates \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $\vec \beta $ \end{document}β→ in each iteration.
