A fast regression via SVD and marginalization

We describe a numerical scheme for evaluating the posterior moments of Bayesian linear regression models with partial pooling of the coefficients. The principal analytical tool of the evaluation is a change of basis from coefficient space to the space of singular vectors of the matrix of predictors....

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Bibliographic Details
Published in:Computational statistics 2022-04, Vol.37 (2), p.701-720
Main Authors: Greengard, Philip, Gelman, Andrew, Vehtari, Aki
Format: Article
Language:English
Subjects:
Algorithms
Approximation
Coefficients
Density
Economic Theory/Quantitative Economics/Mathematical Methods
Genomes
Integrals
Mathematics and Statistics
Original Paper
Probability and Statistics in Computer Science
Probability Theory and Stochastic Processes
Quadratures
Regression analysis
Regression models
Social exclusion
Software
Statistics
Vectors (mathematics)
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Summary:We describe a numerical scheme for evaluating the posterior moments of Bayesian linear regression models with partial pooling of the coefficients. The principal analytical tool of the evaluation is a change of basis from coefficient space to the space of singular vectors of the matrix of predictors. After this change of basis and an analytical integration, we reduce the problem of finding moments of a density over k + 2 dimensions, to finding moments of a 2-dimensional density, where k is the number of coefficients. Moments can then be computed using, for example, MCMC, the trapezoid rule, or adaptive Gaussian quadrature. An evaluation of the SVD of the matrix of predictors is the dominant computational cost and is performed once during the precomputation stage. We demonstrate numerical results of the algorithm.
ISSN:0943-4062
1613-9658
DOI:10.1007/s00180-021-01135-x