Submitted on 2020-04-06
Sparse PCA is one of the most popular tools for the dimensional reduction of high-dimensional data. Although many computational methods have been proposed for sparse PCA, Bayesian methods are still very few. In particular, there is a lack of fast and efficient algorithms for Bayesian sparse PCA. To fill this gap, we propose two efficient algorithms based on the expectation–maximization (EM) algorithm and the coordinate ascent variational inference (CAVI) algorithm—the double parameter expansion-EM (dPX-EM) and the PX-coordinate ascent variation inference (PX-CAVI) algorithms. By using a new spike-and-slab prior and applying the parameter expansion approach, we are able to avoid directly dealing with the orthogonal constraint between eigenvectors, and thus making it easier to compute the posterior. Simulation studies showed that the PX-CAVI outperforms the dPX-EM algorithm as well as other two existing methods. The corresponding R code is available on the website https://github.com/Bo-Ning/Bayesian-sparse-PCA.
Submitted on 2018-12-05
Bayesian methods provide a natural means for uncertainty quantification, that is, credible sets can be easily obtained from the posterior distribution. But is this uncertainty quantification valid in the sense that the posterior credible sets attain the nominal frequentist coverage probability? This paper investigates the frequentist validity of posterior uncertainty quantification based on a class of empirical priors in the sparse normal mean model. In particular, we show that our marginal posterior credible intervals achieve the nominal frequentist coverage probability under conditions slightly weaker than needed for selection consistency and a Bernstein--von Mises theorem for the full posterior, and numerical investigations suggest that our empirical Bayes method has superior frequentist coverage probability properties compared to other fully Bayes methods.