arXiv subjects

Alex Lenkoski

Publications and source records attributed to Alex Lenkoski.

26 records · Page 2Linked to original sources

arXiv2013

A Direct Sampler for G-Wishart Variates

The G-Wishart distribution is the conjugate prior for precision matrices that encode the conditional independencies of a Gaussian graphical model. While the distribution has received considerable attention, posterior inference has proven computationally challenging, in part due to the lack of a direct sampler. In this note, we rectify this situation. The existence of a direct sampler offers a host of new possibilities for the use of G-Wishart variates. We discuss one such development by outlining a new transdimensional model search algorithm--which we term double reversible jump--that leverages this sampler to avoid normalizing constant calculation when comparing graphical models. We conclude with two short studies meant to investigate our algorithm's validity.

stat.CO↗

arXiv2012

A Multivariate Graphical Stochastic Volatility Model

The Gaussian Graphical Model (GGM) is a popular tool for incorporating sparsity into joint multivariate distributions. The G-Wishart distribution, a conjugate prior for precision matrices satisfying general GGM constraints, has now been in existence for over a decade. However, due to the lack of a direct sampler, its use has been limited in hierarchical Bayesian contexts, relegating mixing over the class of GGMs mostly to situations involving standard Gaussian likelihoods. Recent work, however, has developed methods that couple model and parameter moves, first through reversible jump methods and later by direct evaluation of conditional Bayes factors and subsequent resampling. Further, methods for avoiding prior normalizing constant calculations--a serious bottleneck and source of numerical instability--have been proposed. We review and clarify these developments and then propose a new methodology for GGM comparison that blends many recent themes. Theoretical developments and computational timing experiments reveal an algorithm that has limited computational demands and dramatically improves on computing times of existing methods. We conclude by developing a parsimonious multivariate stochastic volatility model that embeds GGM uncertainty in a larger hierarchical framework. The method is shown to be capable of adapting to the extreme swings in market volatility experienced in 2008 after the collapse of Lehman Brothers, offering considerable improvement in posterior predictive distribution calibration.

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arXiv2012

Tobit Bayesian Model Averaging and the Determinants of Foreign Direct Investment

We develop a fully Bayesian, computationally efficient framework for incorporating model uncertainty into Type II Tobit models and apply this to the investigation of the determinants of Foreign Direct Investment (FDI). While direct evaluation of modelprobabilities is intractable in this setting, we show that by using conditional Bayes Factors, which nest model moves inside a Gibbs sampler, we are able to incorporate model uncertainty in a straight-forward fashion. We conclude with a study of global FDI flows between 1988-2000.

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arXiv2012

Instrumental Variable Bayesian Model Averaging via Conditional Bayes Factors

We develop a method to perform model averaging in two-stage linear regression systems subject to endogeneity. Our method extends an existing Gibbs sampler for instrumental variables to incorporate a component of model uncertainty. Direct evaluation of model probabilities is intractable in this setting. We show that by nesting model moves inside the Gibbs sampler, model comparison can be performed via conditional Bayes factors, leading to straightforward calculations. This new Gibbs sampler is only slightly more involved than the original algorithm and exhibits no evidence of mixing difficulties. We conclude with a study of two different modeling challenges: incorporating uncertainty into the determinants of macroeconomic growth, and estimating a demand function by instrumenting wholesale on retail prices.

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arXiv2012

Multivariate probabilistic forecasting using Bayesian model averaging and copulas

We propose a method for post-processing an ensemble of multivariate forecasts in order to obtain a joint predictive distribution of weather. Our method utilizes existing univariate post-processing techniques, in this case ensemble Bayesian model averaging (BMA), to obtain estimated marginal distributions. However, implementing these methods individually offers no information regarding the joint distribution. To correct this, we propose the use of a Gaussian copula, which offers a simple procedure for recovering the dependence that is lost in the estimation of the ensemble BMA marginals. Our method is applied to 48-h forecasts of a set of five weather quantities using the 8-member University of Washington mesoscale ensemble. We show that our method recovers many well-understood dependencies between weather quantities and subsequently improves calibration and sharpness over both the raw ensemble and a method which does not incorporate joint distributional information.

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arXiv2011

Copula Gaussian graphical models and their application to modeling functional disability data

We propose a comprehensive Bayesian approach for graphical model determination in observational studies that can accommodate binary, ordinal or continuous variables simultaneously. Our new models are called copula Gaussian graphical models (CGGMs) and embed graphical model selection inside a semiparametric Gaussian copula. The domain of applicability of our methods is very broad and encompasses many studies from social science and economics. We illustrate the use of the copula Gaussian graphical models in the analysis of a 16-dimensional functional disability contingency table.

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arXiv2010

Bayesian inference for general Gaussian graphical models with application to multivariate lattice data

We introduce efficient Markov chain Monte Carlo methods for inference and model determination in multivariate and matrix-variate Gaussian graphical models. Our framework is based on the G-Wishart prior for the precision matrix associated with graphs that can be decomposable or non-decomposable. We extend our sampling algorithms to a novel class of conditionally autoregressive models for sparse estimation in multivariate lattice data, with a special emphasis on the analysis of spatial data. These models embed a great deal of flexibility in estimating both the correlation structure across outcomes and the spatial correlation structure, thereby allowing for adaptive smoothing and spatial autocorrelation parameters. Our methods are illustrated using simulated and real-world examples, including an application to cancer mortality surveillance.

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arXiv2010

Sparse covariance estimation in heterogeneous samples

Standard Gaussian graphical models (GGMs) implicitly assume that the conditional independence among variables is common to all observations in the sample. However, in practice, observations are usually collected form heterogeneous populations where such assumption is not satisfied, leading in turn to nonlinear relationships among variables. To tackle these problems we explore mixtures of GGMs; in particular, we consider both infinite mixture models of GGMs and infinite hidden Markov models with GGM emission distributions. Such models allow us to divide a heterogeneous population into homogenous groups, with each cluster having its own conditional independence structure. The main advantage of considering infinite mixtures is that they allow us easily to estimate the number of number of subpopulations in the sample. As an illustration, we study the trends in exchange rate fluctuations in the pre-Euro era. This example demonstrates that the models are very flexible while providing extremely interesting interesting insights into real-life applications.

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