Fisher information. The effect of the zero-centered normal prior can be understood as shrinking the MAP LFC estimates based on the amount of information the experiment provides for this coefficient, and we briefly elaborate on this here. Specifically, for a given gene i, the shrinkage for an LFC βir depends on the observed Fisher information, given by \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$\mathcal{J}_{m}(\hat\beta_{ir}) = - \left[ \frac{\partial^{2}}{\partial \beta_{ir}^{2}} \ell\left(\vec\beta_{i} ; \vec{K_{i}}, \alpha_{i}\right) \right]_{\beta_{ir} = \hat\beta_{ir}}, $$ \end{document}Jm(β^ir)=−∂2∂βir2ℓβ→i;Ki→,αiβir=β^ir, where \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $\ell \left (\vec \beta _{i} ; \vec {K_{i}}, \alpha _{i}\right)$ \end{document}ℓβ→i;Ki→,αi is the logarithm of the likelihood, and partial derivatives are taken with respect to LFC βir. For a negative binomial GLM, the observed Fisher information, or peakedness of the logarithm of the profile likelihood, is influenced by a number of factors including the degrees of freedom, the estimated mean counts μij, and the gene’s dispersion estimate αi. The prior influences the MAP estimate when the density of the likelihood and the prior are multiplied to calculate
