While the Matrix Generalized Inverse Gaussian (MGIG) distribution arises naturally in some settings as a distribution over symmetric positive semi-definite matrices, certain key properties of the distribution and effective ways of sampling from the distribution have not been carefully studied. In this paper, we show that the MGIG is unimodal, and the mode can be obtained by solving an Algebraic Riccati Equation (ARE) equation, Based on the property, we propose an importance sampling method for the MGIG where the mode of the proposal distribution matches that of the target. The proposed sampling method is more efficient than existing approaches, which use proposal distributions that may have the mode far from the MGIG's mode. Further, we illustrate that the the posterior distribution in latent factor models, such as probabilistic matrix factorization (PMF) , when marginalized over one latent factor has the MGIG distribution. The characterization leads to a novel Collapsed Monte Carlo (CMC) inference algorithm for such latent factor models. We illustrate that CMC has a lower log loss or perplexity than MCMC, and needs fewer samples.
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