Bayesian Bandwidth Test and Selection for High-dimensional Banded Precision Matrices

Lee, Kyoungjae; Lin, Lizhen

doi:10.1214/19-ba1167

Cited by 2 publications

(2 citation statements)

References 39 publications

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“…The two become comparable only for near-bandable matrices with exponentially decreasing decay functions, in which case the rates are essentially equivalent. Lee and Lin [72] proposed a prior distribution that is tailored to estimate the bandwidth of large bandable precision matrices. They established strong model selection consistency for the bandwidth parameter along with the consistency of Bayes factors.…”

Section: Banding and Other Special Sparsity Patternsmentioning

confidence: 99%

Bayesian inference in high-dimensional models

Banerjee,

Castillo,

Ghosal

2021

Preprint

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Models with dimension more than the available sample size are now commonly used in various applications. A sensible inference is possible using a lowerdimensional structure. In regression problems with a large number of predictors, the model is often assumed to be sparse, with only a few predictors active. Interdependence between a large number of variables is succinctly described by a graphical model, where variables are represented by nodes on a graph and an edge between two nodes is used to indicate their conditional dependence given other variables. Many procedures for making inferences in the high-dimensional setting, typically using penalty functions to induce sparsity in the solution obtained by minimizing a loss function, were developed. Bayesian methods have been proposed for such problems more recently, where the prior takes care of the sparsity structure. These methods have the natural ability to also automatically quantify the uncertainty of the inference through the posterior distribution. Theoretical studies of Bayesian procedures in high-dimension have been carried out recently. Questions that arise are, whether the posterior distribution contracts near the true value of the parameter at the minimax optimal rate, whether the correct lower-dimensional structure is discovered with high posterior probability, and whether a credible region has adequate frequentist coverage. In this paper, we review these properties of Bayesian and related methods for several high-dimensional models such as many normal means problem, linear regression, generalized linear models, Gaussian and non-Gaussian graphical models. Effective computational approaches are also discussed.

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Section: Banding and Other Special Sparsity Patternsmentioning

confidence: 99%

Bayesian inference in high-dimensional models

Banerjee,

Castillo,

Ghosal

2021

Preprint

View full text Add to dashboard Cite

show abstract

“…Under their model each variable can be dependent on distant variables, so it is not suitable for local dependence structure. Lee and Lin (2020) suggested a prior distribution for banded Cholesky factors. Although bandwidth selection consistency as well as consistency of Bayes factors were established in high-dimensional settings, again a common bandwidth k is assumed in their model.…”

Section: Introductionmentioning

confidence: 99%

Scalable Bayesian High-dimensional Local Dependence Learning

Lee

Lin

2023

Bayesian Anal.

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In this work, we propose a scalable Bayesian procedure for learning the local dependence structure in a high-dimensional model where the variables possess a natural ordering. The ordering of variables can be indexed by time, the vicinities of spatial locations, and so on, with the natural assumption that variables far apart tend to have weak correlations. Applications of such models abound in a variety of fields such as finance, genome associations analysis and spatial modeling. We adopt a flexible framework under which each variable is dependent on its neighbors or predecessors, and the neighborhood size can vary for each variable. It is of great interest to reveal this local dependence structure by estimating the covariance or precision matrix while yielding a consistent estimate of the varying neighborhood size for each variable. The existing literature on banded covariance matrix estimation, which assumes a fixed bandwidth cannot be adapted for this general setup. We employ the modified Cholesky decomposition for the precision matrix and design a flexible prior for this model through appropriate priors on the neighborhood sizes and Cholesky factors. The posterior contraction rates of the Cholesky factor are derived which are nearly or exactly minimax optimal, and our procedure leads to consistent estimates of the neighborhood size for all the variables. Another appealing feature of our procedure is its scalability to models with large numbers of variables due to efficient posterior inference without resorting to MCMC algorithms. Numerical comparisons are carried out with competitive methods, and applications are considered for some real datasets.

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Bayesian Bandwidth Test and Selection for High-dimensional Banded Precision Matrices

Cited by 2 publications

References 39 publications

Bayesian inference in high-dimensional models

Bayesian inference in high-dimensional models

Scalable Bayesian High-dimensional Local Dependence Learning

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