We generated the synthetic datasets rowwise, with each row a sample of a Multinomial (MN) distribution \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{{{{{\boldsymbol{y}}}}}}}_{{{{{{\boldsymbol{i}}}}}}}={{{{{\rm{MN}}}}}}({{{{{\boldsymbol{\alpha }}}}}},\bar{y})$$\end{document}yi=MN(α,ȳ), and the probability vector \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{{{{\boldsymbol{\alpha }}}}}}$$\end{document}α a softmax transformation of a multivariate normal (MVN) sample: \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{{{{\boldsymbol{\alpha }}}}}}={{{{{\rm{softmax}}}}}}({{{{{\rm{MVN}}}}}}({{{{{\boldsymbol{\mu }}}}}},\,\Sigma ))$$\end{document}α=softmax(MVN(μ,Σ)). We always used a covariance matrix of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Sigma =0.05\, I{d}_{K}$$\end{document}Σ=0.05IdK, which mimics the variances observed in the experimental data of Haber et al., while assuming no correlation between the cell types besides the compositional effects6.
