Adaptive Bayesian inference for high-dimensional sparse factor models
High-dimensional factor models are e cient and powerful tools for modeling the dependency structure of high-dimensional data which have wide applications in genomics study, nance and so on. One of the challenging and crucial tasks involved in the inference of a high dimensional factor model is determining the factor dimensionality. Although a number of frequentist estimators of the factor dimensionality have been proposed and shown to be consistent, theoretical properties of a Bayesian estimator such as concentration properties of the posterior distribution are rarely studied despite its practical usefulness. To ll this gap, we propose a novel prior distribution for a high dimensional factor model and thoroughly investigate theoretical properties of the posterior distribution. Under the proposed prior, we show that the posterior distribution asymptotically concentrates at the true factor dimensionality, and more importantly, the posterior consistency is adaptive to the unknown sparsity level of the true loading matrix. We also prove that the proposed prior attains the optimal detection rate of the factor dimensionality. Moreover, we obtain a near optimal posterior contraction rate of the covariance matrix. Numerical studies are conducted and showed superiority of the proposed method compared with other competitors.
This is a joint work with Ilsang Ohn and Lizhen Lin at University of Notre Dame, USA.