Robust Sparse Bayesian Infinite Factor Models

Abstract: Most of previous works and applications of Bayesian factor model have assumed the normal likelihood regardless of its validity. We propose a Bayesian factor model for heavy-tailed high-dimensional data based on multivariate Student-$t$ likelihood to obtain better covariance estimation. We use multiplicative gamma process shrinkage prior and factor number adaptation scheme proposed in Bhattacharya & Dunson [Biometrika (2011) 291-306]. Since a naive Gibbs sampler for the proposed model suffers from slow mixing, we propose a Markov Chain Monte Carlo algorithm where fast mixing of Hamiltonian Monte Carlo is exploited for some parameters in proposed model. Simulation results illustrate the gain in performance of covariance estimation for heavy-tailed high-dimensional data. We also provide a theoretical result that the posterior of the proposed model is weakly consistent under reasonable conditions. We conclude the paper with the application of proposed factor model on breast cancer metastasis prediction given DNA signature data of cancer cell.