A fast regression via SVD and marginalization

Greengard, Philip; Gelman, Andrew; Vehtari, Aki

doi:10.1007/s00180-021-01135-x

Cited by 2 publications

(3 citation statements)

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“…3 Overview of algorithm Greengard et al (2021) introduced a numerical method for computing with posterior unnormalized densities such as the one-group Bayesian regression model posterior…”

Section: Fast Methods For Normal-normal Modelsmentioning

confidence: 99%

“…It is well-known that due to conjugacy of normal-normal models the inner integral can be evaluated in O(k 3 ) operations after O(nk 2 ) precomputation. This can be done in various equivalent ways, for example, using the determinants of Lindley and Smith (1972) or the singular value decomposition of Greengard et al (2021). Since the innermost integral of C is smooth and low-dimensional, one natural approach for evaluating C is to compute the outer integrals, those with respect to σ, with quadrature (e.g.…”

Section: Fast Methods For Normal-normal Modelsmentioning

confidence: 99%

“…The tools we use are a generalization of the approach proposed by Greengard et al (2021) and generalize to higher-dimensional multilevel and higher-dimensional multigroup posterior distributions. Since we integrate the marginal density using a tensor product of Gaussian nodes, the cost of the integration scales like O(m d ) where m is the number of discretization nodes in each direction and d is the dimension of the marginalized integral (where d = 3 in the models of this paper).…”

Section: Public Opinion On Abortionmentioning

confidence: 99%

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Fast Methods for Posterior Inference of Two-Group Normal-Normal Models

Greengard

Hoskins²,

Margossian³

et al. 2023

Bayesian Anal.

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We describe a class of algorithms for evaluating posterior moments of certain Bayesian linear regression models with a normal likelihood and a normal prior on the regression coefficients. The proposed methods can be used for hierarchical mixed effects models with partial pooling over one group of predictors, as well as random effects models with partial pooling over two groups of predictors. We demonstrate the performance of the methods on two applications, one involving U.S. opinion polls and one involving the modeling of COVID-19 outbreaks in Israel using survey data. The algorithms involve analytical marginalization of regression coefficients followed by numerical integration of the remaining low-dimensional density. The dominant cost of the algorithms is an eigendecomposition computed once for each value of the outside parameter of integration. Our approach drastically reduces run times compared to state-of-the-art Markov chain Monte Carlo (MCMC) algorithms. The latter, in addition to being computationally expensive, can also be difficult to tune when applied to hierarchical models.

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“…3 Overview of algorithm Greengard et al (2021) introduced a numerical method for computing with posterior unnormalized densities such as the one-group Bayesian regression model posterior…”

Section: Fast Methods For Normal-normal Modelsmentioning

confidence: 99%

Section: Fast Methods For Normal-normal Modelsmentioning

confidence: 99%

Section: Public Opinion On Abortionmentioning

confidence: 99%

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