Post-Processing Posteriors Over Precision Matrices to Produce Sparse Graph Estimates

Bashir, Amir; Carvalho, Carlos M.; Hahn, P. Richard; Jones, MaryPat

doi:10.1214/18-ba1139

Cited by 24 publications

(24 citation statements)

References 21 publications

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“…The decision analysis approach to variable selection has historical roots 35 with modern development 36,37 . Related approaches, often termed posterior summarization , have been developed for multiple linear regression and logistic regression, 36 nonlinear regressions, 38 time‐varying parameter models, 39 functional regression, 40 graphical models, 41 and seeming unrelated regressions 42 . However, existing methods face several important limitations.…”

Section: Methodsmentioning

confidence: 99%

Bayesian variable selection for understanding mixtures in environmental exposures

Kowal

Bravo

Leong

et al. 2021

Statistics in Medicine

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Social and environmental stressors are crucial factors in child development. However, there exists a multitude of measurable social and environmental factors—the effects of which may be cumulative, interactive, or null. Using a comprehensive cohort of children in North Carolina, we study the impact of social and environmental variables on 4th end‐of‐grade exam scores in reading and mathematics. To identify the essential factors that predict these educational outcomes, we design new tools for Bayesian linear variable selection using decision analysis. We extract a predictive optimal subset of explanatory variables by coupling a loss function with a novel model‐based penalization scheme, which leads to coherent Bayesian decision analysis and empirically improves variable selection, estimation, and prediction on simulated data. The Bayesian linear model propagates uncertainty quantification to all predictive evaluations, which is important for interpretable and robust model comparisons. These predictive comparisons are conducted out‐of‐sample with a customized approximation algorithm that avoids computationally intensive model refitting. We apply our variable selection techniques to identify the joint collection of social and environmental stressors—and their interactions—that offer clear and quantifiable improvements in prediction of reading and mathematics exam scores.

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Section: Methodsmentioning

confidence: 99%

Bayesian variable selection for understanding mixtures in environmental exposures

Kowal

Bravo

Leong

et al. 2021

Statistics in Medicine

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“…To determine the penalty parameter, we follow Friedman, Hastie, and Tibshirani (2019) and use

ρ_{i j} = \frac{ϖ}{false| {\overset{s}{true}}_{i j}^{*} {false|}^{\frac{κ}{2}}},

where

false| {\overset{s}{true}}_{i j}^{*} false|

denotes the absolute size of the ( i , j )th element of

{\overset{boldS}{true}}^{- 1}

and ϖ is a scalar penalty parameter whereas κ ≥ 1 controls the penalty on small precision parameters. Equation () nests the specification stipulated in Bashir et al (2019) if we set

κ = 1

{\overset{s}{true}}_{i j}

to an initial estimate of the ( i , j )th element of the precision matrix, and cross‐validate ϖ .…”

Section: Achieving Sparsity In Var Modelsmentioning

confidence: 99%

“…As a potential remedy, we propose postprocessing the estimates of the precision matrix Σ −1 (i.e., the inverse of Σ ). Friedman, Hastie, and Tibshirani (2008) and, more recently, Bashir, Carvalho, Hahn, and Jones (2019) propose methods to ex‐post sparsify precision matrices using the graphical lasso. We follow this literature and specify a loss function similar to Equation () that aims to strike a balance between model fit and parsimony.…”

Section: Achieving Sparsity In Var Modelsmentioning

confidence: 99%

Combining shrinkage and sparsity in conjugate vector autoregressive models

Hauzenberger

Huber

Onorante

2021

J of Applied Econometrics

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Summary Conjugate priors allow for fast inference in large dimensional vector autoregressive (VAR) models. But at the same time, they introduce the restriction that each equation features the same set of explanatory variables. This paper proposes a straightforward means of postprocessing posterior estimates of a conjugate Bayesian VAR to effectively perform equation‐specific covariate selection. Compared with existing techniques using shrinkage alone, our approach combines shrinkage and sparsity in both the VAR coefficients and the error variance–covariance matrices, greatly reducing estimation uncertainty in large dimensions while maintaining computational tractability. We illustrate our approach by means of two applications. The first application uses synthetic data to investigate the properties of the model across different data‐generating processes, and the second application analyzes the predictive gains from sparsification in a forecasting exercise for U.S. data.

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“…The posterior projection method has been suggested for various settings including [12], [17], [26] and [8], and was investigated in general aspects by [27]. The idea of transforming posterior samples was also used for the inference on covariance or precision matrices in [23] and [1].…”

Section: Consider the Multivariate Linear Regression Modelmentioning

confidence: 99%

Estimation of conditional mean operator under the bandable covariance structure

Lee

2022

Electron. J. Statist.

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We consider high-dimensional multivariate linear regression models, where the joint distribution of covariates and response variables is a multivariate normal distribution with a bandable covariance matrix. The main goal of this paper is to estimate the regression coefficient matrix, which is a function of the bandable covariance matrix. Although the tapering estimator of covariance has the minimax optimal convergence rate for the class of bandable covariances, we show that it is sub-optimal for the regression coefficient; that is, a minimax estimator for the class of bandable covariances may not be a minimax estimator for its functionals. We propose the blockwise tapering estimator of the regression coefficient, which has the minimax optimal convergence rate for the regression coefficient under the bandable covariance assumption. We also propose a Bayesian procedure called the blockwise tapering post-processed posterior of the regression coefficient and show that the proposed Bayesian procedure has the minimax optimal convergence rate for the regression coefficient under the bandable covariance assumption. We show that the proposed methods outperform the existing methods via numerical studies.

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Post-Processing Posteriors Over Precision Matrices to Produce Sparse Graph Estimates

Cited by 24 publications

References 21 publications

Bayesian variable selection for understanding mixtures in environmental exposures

Bayesian variable selection for understanding mixtures in environmental exposures

Combining shrinkage and sparsity in conjugate vector autoregressive models

Estimation of conditional mean operator under the bandable covariance structure

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