Matrix- and vector-valued generalizations of Gamma processes

5.1 AΓ Process

Meier, Kirch, and Meyer (2020) mentions a construction less general than the HDP Matrix Gamma which is nonetheless broad and quite useful. We could think of it as the tractable HDP:

A special case of the ${\mathrm{Gamma}}_{d\times d}(\alpha ,\lambda )$ distribution is the so-called $A\mathrm{\Gamma }$ distribution, that has been considered in Pérez-Abreu and Stelzer (2014) and generalized to the Hpd setting in (Meier 2018, sec. 2.4). To elaborate, the $A\mathrm{\Gamma }(\eta ,\omega ,\mathrm{\Sigma })$ distribution is defined with the parameters $\eta >d-1,\omega >0$ and $\mathrm{\Sigma }\in$ ${\mathcal{S}}_d^{+}$ as the ${\mathrm{Gamma}}_{d\times d}({\alpha }_{\eta ,\mathrm{\Sigma }},{\lambda }_{\mathrm{\Sigma }})$ distribution, with $${\alpha }_{\eta ,\mathbf{\Sigma }}(d\mathit{U})=|\mathbf{\Sigma }{|}^{-\eta }\mathrm{tr}{\left({\mathbf{\Sigma }}^{-1}\mathit{U}\right)}^{-d\eta }\mathrm{\Gamma }(d\eta ){\tilde{\mathrm{\Gamma }}}_d(\eta {)}^{-1}|\mathit{U}{|}^{\eta -d}d\mathit{U},$$ where $\mathrm{\Gamma }$ denotes the Gamma function and ${\tilde{\mathrm{\Gamma }}}_d$ the complex multivariate Gamma function (see Mathai and Provost 2005), and ${\lambda }_{\mathbf{\Sigma }}(\mathit{U})=\mathrm{tr}\left({\mathbf{\Sigma }}^{-1}\mathit{U}\right)$. It has the advantage that for $\mathit{X}\sim A\mathrm{\Gamma }(\eta ,\omega ,\mathrm{\Sigma })$, the formulas for mean and covariance structure are explicitly known: $$\mathrm{E}\mathit{X}=\frac{\omega }{d}\mathbf{\Sigma },\mathrm{Cov}\mathit{X}=\frac{\omega }{d(\eta d+1)}(\eta {\mathit{I}}_{d^2}+\mathit{H})(\mathbf{\Sigma }\otimes \mathbf{\Sigma }),$$ where $\mathit{H}=\sum _{i,j=1}^d{\mathit{H}}_{i,j}\otimes H_{j,i}$ and ${\mathit{H}}_{i,j}$ being the matrix having a one at $(i,j)$ and zeros elsewhere, see (Meier 2018 Lemma 2.8). Thus the $A\mathrm{\Gamma }$-distribution is particularly well suited for Bayesian prior modelling if the prior knowledge is given in terms of mean and covariance structure.