Matrix- and vector-valued generalizations of Gamma processes

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 this of it as the tractable HDP.:

A special case of the \(\operatorname{Gamma}_{d \times d}(\alpha, \lambda)\) distribution is the so-called \(A \Gamma\) 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 \(A \Gamma(\eta, \omega, \Sigma)\) distribution is defined with the parameters \(\eta>d-1, \omega>0\) and \(\Sigma \in\) \(\mathcal{S}_{d}^{+}\) as the \(\operatorname{Gamma}_{d \times d}\left(\alpha_{\eta, \Sigma}, \lambda_{\Sigma}\right)\) distribution, with \[ \alpha_{\eta, \boldsymbol{\Sigma}}(d \boldsymbol{U})=|\boldsymbol{\Sigma}|^{-\eta} \operatorname{tr}\left(\boldsymbol{\Sigma}^{-1} \boldsymbol{U}\right)^{-d \eta} \Gamma(d \eta) \tilde{\Gamma}_{d}(\eta)^{-1}|\boldsymbol{U}|^{\eta-d} d \boldsymbol{U}, \] where \(\Gamma\) denotes the Gamma function and \(\tilde{\Gamma}_{d}\) the complex multivariate Gamma function (see Mathai and Provost 2005), and \(\lambda_{\boldsymbol{\Sigma}}(\boldsymbol{U})=\operatorname{tr}\left(\boldsymbol{\Sigma}^{-1} \boldsymbol{U}\right)\). It has the advantage that for \(\boldsymbol{X} \sim A \Gamma(\eta, \omega, \Sigma)\), the formulas for mean and covariance structure are explicitly known: \[ \mathrm{E} \boldsymbol{X}=\frac{\omega}{d} \boldsymbol{\Sigma}, \quad \operatorname{Cov} \boldsymbol{X}=\frac{\omega}{d(\eta d+1)}\left(\eta \boldsymbol{I}_{d^{2}}+\boldsymbol{H}\right)(\boldsymbol{\Sigma} \otimes \boldsymbol{\Sigma}), \] where \(\boldsymbol{H}=\sum_{i, j=1}^{d} \boldsymbol{H}_{i, j} \otimes H_{j, i}\) and \(\boldsymbol{H}_{i, j}\) being the matrix having a one at \((i, j)\) and zeros elsewhere, see (Meier 2018 Lemma 2.8). Thus the \(A\Gamma\)-distribution is particularly well suited for Bayesian prior modeling if the prior knowledge is given in terms of mean and covariance structure.