文摘
While the Matrix Generalized Inverse Gaussian (\(\mathcal {MGIG}\)) 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 \(\mathcal {MGIG}\) is unimodal, and the mode can be obtained by solving an Algebraic Riccati Equation (ARE) equation [7]. Based on the property, we propose an importance sampling method for the \(\mathcal {MGIG}\) where the mode of the proposal distribution matches that of the target. The proposed sampling method is more efficient than existing approaches [32, 33], which use proposal distributions that may have the mode far from the \(\mathcal {MGIG}\)’s mode. Further, we illustrate that the the posterior distribution in latent factor models, such as probabilistic matrix factorization (PMF) [24], when marginalized over one latent factor has the \(\mathcal {MGIG}\) 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.