arXiv subjects

Nicholas G. Polson

Publications and source records attributed to Nicholas G. Polson.

65 records · Page 4Linked to original sources

arXiv2012

Data augmentation for non-Gaussian regression models using variance-mean mixtures

We use the theory of normal variance-mean mixtures to derive a data-augmentation scheme for a class of common regularization problems. This generalizes existing theory on normal variance mixtures for priors in regression and classification. It also allows variants of the expectation-maximization algorithm to be brought to bear on a wider range of models than previously appreciated. We demonstrate the method on several examples, including sparse quantile regression and binary logistic regression. We also show that quasi-Newton acceleration can substantially improve the speed of the algorithm without compromising its robustness.

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arXiv2012

An empirical test for Eurozone contagion using an asset-pricing model with heavy-tailed stochastic volatility

This paper proposes an empirical test of financial contagion in European equity markets during the tumultuous period of 2008-2011. Our analysis shows that traditional GARCH and Gaussian stochastic-volatility models are unable to explain two key stylized features of global markets during presumptive contagion periods: shocks to aggregate market volatility can be sudden and explosive, and they are associated with specific directional biases in the cross-section of country-level returns. Our model repairs this deficit by assuming that the random shocks to volatility are heavy-tailed and correlated cross-sectionally, both with each other and with returns. The fundamental conclusion of our analysis is that great care is needed in modeling volatility if one wishes to characterize the relationship between volatility and contagion that is predicted by economic theory. In analyzing daily data, we find evidence for significant contagion effects during the major EU crisis periods of May 2010 and August 2011, where contagion is defined as excess correlation in the residuals from a factor model incorporating global and regional market risk factors. Some of this excess correlation can be explained by quantifying the impact of shocks to aggregate volatility in the cross-section of expected returns - but only, it turns out, if one is extremely careful in accounting for the explosive nature of these shocks. We show that global markets have time-varying cross-sectional sensitivities to these shocks, and that high sensitivities strongly predict periods of financial crisis. Moreover, the pattern of temporal changes in correlation structure between volatility and returns is readily interpretable in terms of the major events of the periods in question.

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arXiv2012

Good, great, or lucky? Screening for firms with sustained superior performance using heavy-tailed priors

This paper examines historical patterns of ROA (return on assets) for a cohort of 53,038 publicly traded firms across 93 countries, measured over the past 45 years. Our goal is to screen for firms whose ROA trajectories suggest that they have systematically outperformed their peer groups over time. Such a project faces at least three statistical difficulties: adjustment for relevant covariates, massive multiplicity, and longitudinal dependence. We conclude that, once these difficulties are taken into account, demonstrably superior performance appears to be quite rare. We compare our findings with other recent management studies on the same subject, and with the popular literature on corporate success. Our methodological contribution is to propose a new class of priors for use in large-scale simultaneous testing. These priors are based on the hypergeometric inverted-beta family, and have two main attractive features: heavy tails and computational tractability. The family is a four-parameter generalization of the normal/inverted-beta prior, and is the natural conjugate prior for shrinkage coefficients in a hierarchical normal model. Our results emphasize the usefulness of these heavy-tailed priors in large multiple-testing problems, as they have a mild rate of tail decay in the marginal likelihood $m(y)$---a property long recognized to be important in testing.

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arXiv2012

Simulation-based Regularized Logistic Regression

In this paper, we develop a simulation-based framework for regularized logistic regression, exploiting two novel results for scale mixtures of normals. By carefully choosing a hierarchical model for the likelihood by one type of mixture, and implementing regularization with another, we obtain new MCMC schemes with varying efficiency depending on the data type (binary v. binomial, say) and the desired estimator (maximum likelihood, maximum a posteriori, posterior mean). Advantages of our omnibus approach include flexibility, computational efficiency, applicability in p >> n settings, uncertainty estimates, variable selection, and assessing the optimal degree of regularization. We compare our methodology to modern alternatives on both synthetic and real data. An R package called reglogit is available on CRAN.

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arXiv2011

Analyzing Risky Choices: Q-Learning for Deal-No Deal

We derive an optimal strategy in the popular Deal or No Deal game show. Q-learning quantifies the continuation value inherent in sequential decision making and we use this to analyze contestants risky choices. Given their choices and optimal strategy, we invert to find implied bounds on their levels of risk aversion. In risky decision making, previous empirical evidence has suggested that past outcomes affect future choices and that contestants have time-varying risk aversion. We demonstrate that the strategies of two players (Suzanne and Frank) from the European version of the game are consistent with constant risk aversion levels except for their last risk-seeking choice.

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arXiv2011

On the half-Cauchy prior for a global scale parameter

This paper argues that the half-Cauchy distribution should replace the inverse-Gamma distribution as a default prior for a top-level scale parameter in Bayesian hierarchical models, at least for cases where a proper prior is necessary. Our arguments involve a blend of Bayesian and frequentist reasoning, and are intended to complement the original case made by Gelman (2006) in support of the folded-t family of priors. First, we generalize the half-Cauchy prior to the wider class of hypergeometric inverted-beta priors. We derive expressions for posterior moments and marginal densities when these priors are used for a top-level normal variance in a Bayesian hierarchical model. We go on to prove a proposition that, together with the results for moments and marginals, allows us to characterize the frequentist risk of the Bayes estimators under all global-shrinkage priors in the class. These theoretical results, in turn, allow us to study the frequentist properties of the half-Cauchy prior versus a wide class of alternatives. The half-Cauchy occupies a sensible 'middle ground' within this class: it performs very well near the origin, but does not lead to drastic compromises in other parts of the parameter space. This provides an alternative, classical justification for the repeated, routine use of this prior. We also consider situations where the underlying mean vector is sparse, where we argue that the usual conjugate choice of an inverse-gamma prior is particularly inappropriate, and can lead to highly distorted posterior inferences. Finally, we briefly summarize some open issues in the specification of default priors for scale terms in hierarchical models.

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arXiv2011

Default Bayesian analysis for multi-way tables: a data-augmentation approach

This paper proposes a strategy for regularized estimation in multi-way contingency tables, which are common in meta-analyses and multi-center clinical trials. Our approach is based on data augmentation, and appeals heavily to a novel class of Polya-Gamma distributions. Our main contributions are to build up the relevant distributional theory and to demonstrate three useful features of this data-augmentation scheme. First, it leads to simple EM and Gibbs-sampling algorithms for posterior inference, circumventing the need for analytic approximations, numerical integration, Metropolis--Hastings, or variational methods. Second, it allows modelers much more flexibility when choosing priors, which have traditionally come from the Dirichlet or logistic-normal family. For example, our approach allows users to incorporate Bayesian analogues of classical penalized-likelihood techniques (e.g. the lasso or bridge) in computing regularized estimates for log-odds ratios. Finally, our data-augmentation scheme naturally suggests a default strategy for prior selection based on the logistic-Z model, which is strongly related to Jeffreys' prior for a binomial proportion. To illustrate the method we focus primarily on the particular case of a meta-analysis/multi-center study (or a JxKxN table). But the general approach encompasses many other common situations, of which we will provide examples.

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arXiv2011

Local shrinkage rules, Levy processes, and regularized regression

We use Levy processes to generate joint prior distributions, and therefore penalty functions, for a location parameter as p grows large. This generalizes the class of local-global shrinkage rules based on scale mixtures of normals, illuminates new connections among disparate methods, and leads to new results for computing posterior means and modes under a wide class of priors. We extend this framework to large-scale regularized regression problems where p>n, and provide comparisons with other methodologies.

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arXiv2010

Dynamic Trees for Learning and Design

Dynamic regression trees are an attractive option for automatic regression and classification with complicated response surfaces in on-line application settings. We create a sequential tree model whose state changes in time with the accumulation of new data, and provide particle learning algorithms that allow for the efficient on-line posterior filtering of tree-states. A major advantage of tree regression is that it allows for the use of very simple models within each partition. The model also facilitates a natural division of labor in our sequential particle-based inference: tree dynamics are defined through a few potential changes that are local to each newly arrived observation, while global uncertainty is captured by the ensemble of particles. We consider both constant and linear mean functions at the tree leaves, along with multinomial leaves for classification problems, and propose default prior specifications that allow for prediction to be integrated over all model parameters conditional on a given tree. Inference is illustrated in some standard nonparametric regression examples, as well as in the setting of sequential experiment design, including both active learning and optimization applications, and in on-line classification. We detail implementation guidelines and problem specific methodology for each of these motivating applications. Throughout, it is demonstrated that our practical approach is able to provide better results compared to commonly used methods at a fraction of the cost.

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arXiv2010

Particle Learning and Smoothing

Particle learning (PL) provides state filtering, sequential parameter learning and smoothing in a general class of state space models. Our approach extends existing particle methods by incorporating the estimation of static parameters via a fully-adapted filter that utilizes conditional sufficient statistics for parameters and/or states as particles. State smoothing in the presence of parameter uncertainty is also solved as a by-product of PL. In a number of examples, we show that PL outperforms existing particle filtering alternatives and proves to be a competitor to MCMC.

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arXiv2010

Particle learning of Gaussian process models for sequential design and optimization

We develop a simulation-based method for the online updating of Gaussian process regression and classification models. Our method exploits sequential Monte Carlo to produce a fast sequential design algorithm for these models relative to the established MCMC alternative. The latter is less ideal for sequential design since it must be restarted and iterated to convergence with the inclusion of each new design point. We illustrate some attractive ensemble aspects of our SMC approach, and show how active learning heuristics may be implemented via particles to optimize a noisy function or to explore classification boundaries online.

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