  This study proposes a novel hierarchical prior for inferring possibly
low-rank matrices measured with noise. We consider three-component matrix
factorization, as in singular value decomposition, and its fully Bayesian
inference. The proposed prior is specified by a scale mixture of exponential
distributions that has spike and slab components. The weights for the
spike/slab parts are inferred using a special prior based on a cumulative
shrinkage process. The proposed prior is designed to increasingly aggressively
push less important, or essentially redundant, singular values toward zero,
leading to more accurate estimates of low-rank matrices. To ensure the
parameter identification, we simulate posterior draws from an approximated
posterior, in which the constraints are slightly relaxed, using a No-U-Turn
sampler. By means of a set of simulation studies, we show that our proposal is
competitive with alternative prior specifications and that it does not incur
significant additional computational burden. We apply the proposed approach to
sectoral industrial production in the United States to analyze the structural
change during the Great Moderation period.
