Final estimate of logarithmic fold changes The logarithmic posterior for the vector, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $\vec \beta _{i}$ \end{document}β→i, of model coefficients βir for gene i is the sum of the logarithmic likelihood of the GLM (2) and the logarithm of the prior density (10), and its maximum yields the final MAP coefficient estimates: \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$\vec\beta_{i} = {\underset{\vec\beta}{\text{arg max}}} \left(\sum_{j} \log f_{\text{NB}}\left(K_{ij};\mu_{j}(\vec\beta),\alpha_{i} \right) + \Lambda(\vec\beta) \right), $$ \end{document}β→i=arg maxβ→∑jlogfNBKij;μj(β→),αi+Λ(β→), where \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$\mu_{j}(\vec\beta) = s_{ij} e^{\sum_{r} x_{jr} \beta_{r}}, \quad \Lambda(\vec\beta) = \sum_{r} \frac{- {\beta_{r}^{2}}}{2 {\sigma_{r}^{2}}}, $$ \end{document}μj(β→)=sije∑rxjrβr,Λ(β→)=∑r−βr22σr2, and αi is the final dispersion estimate for gene i, i.e., \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $\alpha _{i}=\alpha _{i}^{\text {MAP}}$ \end{document}αi=αiMAP, except for dispersion outliers, where \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $\alpha _{i}=\alpha _{i}^{\text {gw}}$ \end{document}αi=αigw.
