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Summary We develop a Bayesian methodology aimed at simultaneously estimating low-rank and row-sparse matrices in a high-dimensional multiple-response linear regression model. We consider a carefully devised shrinkage prior on the matrix of regression coefficients which obviates the need to specify a prior on the rank, and shrinks the regression matrix towards low-rank and row-sparse structures. We provide theoretical support to the proposed methodology by proving minimax optimality of the posterior mean under the prediction risk in ultra-high-dimensional settings where the number of predictors can grow subexponentially relative to the sample size. A one-step post-processing scheme induced by group lasso penalties on the rows of the estimated coefficient matrix is proposed for variable selection, with default choices of tuning parameters. We additionally provide an estimate of the rank using a novel optimization function achieving dimension reduction in the covariate space. We exhibit the performance of the proposed methodology in an extensive simulation study and a real data example.
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Dynamic Shrinkage Processes
Summary We propose a novel class of dynamic shrinkage processes for Bayesian time series and regression analysis. Building on a global–local framework of prior construction, in which continuous scale mixtures of Gaussian distributions are employed for both desirable shrinkage properties and computational tractability, we model dependence between the local scale parameters. The resulting processes inherit the desirable shrinkage behaviour of popular global–local priors, such as the horseshoe prior, but provide additional localized adaptivity, which is important for modelling time series data or regression functions with local features. We construct a computationally efficient Gibbs sampling algorithm based on a Pólya–gamma scale mixture representation of the process proposed. Using dynamic shrinkage processes, we develop a Bayesian trend filtering model that produces more accurate estimates and tighter posterior credible intervals than do competing methods, and we apply the model for irregular curve fitting of minute-by-minute Twitter central processor unit usage data. In addition, we develop an adaptive time varying parameter regression model to assess the efficacy of the Fama–French five-factor asset pricing model with momentum added as a sixth factor. Our dynamic analysis of manufacturing and healthcare industry data shows that, with the exception of the market risk, no other risk factors are significant except for brief periods.
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- Award ID(s):
- 1455172
- PAR ID:
- 10580862
- Publisher / Repository:
- Wiley
- Date Published:
- Journal Name:
- Journal of the Royal Statistical Society Series B: Statistical Methodology
- Volume:
- 81
- Issue:
- 4
- ISSN:
- 1369-7412
- Page Range / eLocation ID:
- 781 to 804
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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- Free Publicly Accessible Full Text
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Accepted Manuscript
- Journal Article:
- https://doi.org/10.1111/rssb.12325
