Hierarchical Decompositions for the Computation of High-Dimensional Multivariate Normal Probabilities

Genton, Marc G.; Keyes, David E.; Turkiyyah, George

doi:10.1080/10618600.2017.1375936

Cited by 33 publications

(38 citation statements)

References 43 publications

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“…For instance, improved studies on the moment generating function of the unified skew-normal could facilitate direct calculation of relevant functionals without the need to sample from (β | y, X). On the same line, improving the methods for efficient evaluation of Φ n (·) in large n applications, either via data transformations, blocking methods (Chopin, 2011) or recent algorithms (Genton et al, 2018), could enlarge the range of applications which allow direct inference, prediction and model selection, without sampling from (β | y, X). Also approximations of the exact posterior, which preserve the skewness but allow analytical inference provide an interesting direction.…”

Section: Final Considerations and Future Directions Of Researchmentioning

confidence: 99%

Conjugate Bayes for probit regression via unified skew-normal distributions

Durante

2019

Biometrika

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Regression models for dichotomous data are ubiquitous in statistics. Besides being useful for inference on binary responses, these methods serve also as building blocks in more complex formulations, such as density regression, nonparametric classification and graphical models. Within the Bayesian framework, inference proceeds by updating the priors for the coefficients, typically set to be Gaussians, with the likelihood induced by probit or logit regressions for the responses. In this updating, the apparent absence of a tractable posterior has motivated a variety of computational methods, including Markov Chain Monte Carlo routines and algorithms which approximate the posterior. Despite being routinely implemented, Markov Chain Monte Carlo strategies face mixing or time-inefficiency issues in large p and small n studies, whereas approximate routines fail to capture the skewness typically observed in the posterior. This article proves that the posterior distribution for the probit coefficients has a unified skew-normal kernel, under Gaussian priors. Such a novel result allows efficient Bayesian inference for a wide class of applications, especially in large p and small-to-moderate n studies where state-of-the-art computational methods face notable issues. These advances are outlined in a genetic study, and further motivate the development of a wider class of conjugate priors for probit models along with methods to obtain independent and identically distributed samples from the unified skew-normal posterior.

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Section: Final Considerations and Future Directions Of Researchmentioning

confidence: 99%

Conjugate Bayes for probit regression via unified skew-normal distributions

Durante

2019

Biometrika

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“…The difficulty resides in the computation of high‐dimensional multivariate Gaussian distributions needed for the exponent function V and its partial derivatives

V_{τ_{i}}

. Unbiased Monte Carlo estimates of these quantities can be obtained, and Thibaud et al () and de Fondeville and Davison () suggest using crude approximations to reduce the computational time while maintaining accuracy; see also Genton et al (), who instead suggest using hierarchical matrix decompositions. Our method is not limited to these two models and could potentially be applied to any max‐stable model for which the functions V and

V_{τ_{i}}

are known and computable.…”

Section: Discussionmentioning

confidence: 99%

“…(2016) and de Fondeville and Davison (2018) suggest using crude approximations to reduce the computational time while maintaining accuracy; see also Genton et al (2018), who instead suggest using hierarchical matrix decompositions. Our method is not limited to these two models and could potentially be applied to any max-stable model for which the functions V and V i are known and computable.…”

Section: Discussionmentioning

confidence: 99%

Full likelihood inference for max‐stable data

Huser

Dombry

Ribatet

et al. 2019

Stat

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We show how to perform full likelihood inference for max‐stable multivariate distributions or processes based on a stochastic expectation–maximization algorithm, which combines statistical and computational efficiency in high dimensions. The good performance of this methodology is demonstrated by simulation based on the popular logistic and Brown–Resnick models, and it is shown to provide computational time improvements with respect to a direct computation of the likelihood. Strategies to further reduce the computational burden are also discussed.

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“…the spatial range θ 2 parameter usually can be represented by using 0.03 for weak correlation, 0.1 for medium correlation, and 0.3 for strong correlation. The smoothness θ 3 parameter, which represents the data smoothness can be represented by 0.5 for a rough process, and 1 for a smooth process [35]. The distance between any two spatial locations can be efficiently computed using Euclidian distance.…”

Section: Matérn Covariance Functionmentioning

confidence: 99%

Parallel Approximation of the Maximum Likelihood Estimation for the Prediction of Large-Scale Geostatistics Simulations

Abdulah¹,

Ltaief²,

Sun³

et al. 2018

2018 IEEE International Conference on Cluster Computing (CLUSTER)

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Maximum likelihood estimation is an important statistical technique for estimating missing data, for example in climate and environmental applications, which are usually large and feature data points that are irregularly spaced. In particular, the Gaussian log-likelihood function is the de facto model, which operates on the resulting sizable dense covariance matrix. The advent of high performance systems with advanced computing power and memory capacity have enabled full simulations only for rather small dimensional climate problems, solved at the machine precision accuracy. The challenge for high dimensional problems lies in the computation requirements of the log-likelihood function, which necessitates O(n 2 ) storage and O(n 3 ) operations, where n represents the number of given spatial locations. This prohibitive computational cost may be reduced by using approximation techniques that not only enable large-scale simulations otherwise intractable, but also maintain the accuracy and the fidelity of the spatial statistics model. In this paper, we extend the Exascale GeoStatistics software framework (i.e., ExaGeoStat 1 ) to support the Tile Low-Rank (TLR) approximation technique, which exploits the data sparsity of the dense covariance matrix by compressing the off-diagonal tiles up to a user-defined accuracy threshold. The underlying linear algebra operations may then be carried out on this data compression format, which may ultimately reduce the arithmetic complexity of the maximum likelihood estimation and the corresponding memory footprint. Performance results of TLR-based computations on shared and distributed-memory systems attain up to 13X and 5X speedups, respectively, compared to full accuracy simulations using synthetic and real datasets (up to 2M), while ensuring adequate prediction accuracy.

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Hierarchical Decompositions for the Computation of High-Dimensional Multivariate Normal Probabilities

Cited by 33 publications

References 43 publications

Conjugate Bayes for probit regression via unified skew-normal distributions

Conjugate Bayes for probit regression via unified skew-normal distributions

Full likelihood inference for max‐stable data

Parallel Approximation of the Maximum Likelihood Estimation for the Prediction of Large-Scale Geostatistics Simulations

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