The Matrix Generalized Inverse Gaussian Distribution: Properties and Applications
This work addresses a specific problem in statistical modeling and Bayesian inference for researchers and practitioners dealing with matrix distributions and latent factor models, offering incremental improvements in sampling efficiency and inference methods.
The paper tackles the lack of key properties and efficient sampling methods for the Matrix Generalized Inverse Gaussian (MGIG) distribution, showing it is unimodal with a mode solvable via an Algebraic Riccati Equation and proposing an importance sampling method that outperforms existing approaches in efficiency. It also applies this to latent factor models, developing a Collapsed Monte Carlo inference algorithm that achieves lower log loss or perplexity with fewer samples than MCMC.
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) [25], 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.