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We propose a new data-augmentation strategy for fully Bayesian inference in models with binomial likelihoods. The approach appeals to a new class of Pólya–Gamma distributions, which are constructed in detail. A variety of examples are presented to show the versatility of the method, including logistic regression, negative binomial regression, nonlinear mixed-effect models, and spatial models for count data. In each case, our data-augmentation strategy leads to simple, effective methods for posterior inference that (1) circumvent the need for analytic approximations, numerical integration, or Metropolis–Hastings; and (2) outperform other known data-augmentation strategies, both in ease of use and in computational efficiency. All methods, including an efficient sampler for the Pólya–Gamma distribution, are implemented in the R package BayesLogit . Supplementary materials for this article are available online.

We propose the Bayesian bridge estimator for regularized regression and classification. Two key mixture representations for the Bayesian bridge model are developed: a scale mixture of normal distributions with respect to an »-stable random variable; a mixture of Bartlett–Fejer kernels (or triangle densities) with respect to a two-component mixture of gamma random variables. Both lead to Markov chain Monte Carlo methods for posterior simulation, and these methods turn out to have complementary domains of maximum efficiency. The first representation is a well-known result due to West and is the better choice for collinear design matrices. The second representation is new and is more efficient for orthogonal problems, largely because it avoids the need to deal with exponentially tilted stable random variables. It also provides insight into the multimodality of the joint posterior distribution, which is a feature of the bridge model that is notably absent under ridge or lasso-type priors. We prove a theorem that extends this representation to a wider class of densities representable as scale mixtures of beta distributions, and we provide an explicit inversion formula for the mixing distribution. The connections with slice sampling and scale mixtures of normal distributions are explored. On the practical side, we find that the Bayesian bridge model outperforms its classical cousin in estimation and prediction across a variety of data sets, both simulated and real. We also show that the Markov chain Monte Carlo algorithm for fitting the bridge model exhibits excellent mixing properties, particularly for the global scale parameter. This makes for a favourable contrast with analogous Markov chain Monte Carlo algorithms for other sparse Bayesian models. All methods described in this paper are implemented in the R package BayesBridge. An extensive set of simulation results is provided in two on-line supplemental files.

We consider how to use Hamiltonian Monte Carlo to sample from a distribution whose log-density is piecewise quadratic, conditioned on the sample lying on the level set of a piecewise affine, continuous function.

Abstract Optimizing root system architecture offers a promising approach to developing stress tolerant cultivars in the face of climate change, as root systems are critical for water and nutrient uptake as well as mechanical stability. However, breeding for optimal root system architecture has been hindered by the difficulty in measuring root growth in the field. Here, we describe a technology, the RootTracker (RT), which employs capacitance touch sensors to monitor in-field root growth over time. Configured in a cylindrical shutter-like fashion around a planted seed, 264 electrodes are individually charged multiple times over the course of an experiment. Signature changes in the measured capacitance and resistance readings indicate when a root has touched or grown close to an electrode. Using the RootTracker, we have measured root system dynamics of commercial maize hybrids growing in both typical Midwest field conditions and under different irrigation regimes. We observed rapid responses of root growth to water deficits and found evidence for a “priming response” in which an early water deficit causes more and deeper roots to grow at later time periods. There was genotypic variation among hybrid maize lines in their root growth in response to drought, indicating a potential to breed for root systems adapted for different environments. Competing Interest Statement All authors worked for Hi Fidelity Genetics during their contributing work. Philip N. Benfey is a cofounder of Hi Fidelity Genetics.

Yahoo! mail uses images asking users to type in letters that they see in images--to make it harder for spammers to sign up for e-mail accounts. These simple puzzles are known as CAPTCHAs (Completely Automated Public Turing tests to tell Computers and Humans Apart), and the first ones were created by researchers at Carnegie Mellon University in 2000. Here, Aboufadel et al present a method for solving the CAPTCHA used by the Holiday Inn chain of hotels. Their methods of breaking the HIPC CAPTCHA employ linear regression and rotation of axes.

Bayesian analysis of state-space models includes computing the posterior distribution of the system's parameters as well as filtering, smoothing, and predicting the system's latent states. When the latent states wander around \\(R^n\\) there are several well-known modeling components and computational tools that may be profitably combined to achieve these tasks. However, there are scenarios, like tracking an object in a video or tracking a covariance matrix of financial assets returns, when the latent states are restricted to a curve within \\(R^n\\) and these models and tools do not immediately apply. Within this constrained setting, most work has focused on filtering and less attention has been paid to the other aspects of Bayesian state-space inference, which tend to be more challenging. To that end, we present a state-space model whose latent states take values on the manifold of symmetric positive-definite matrices and for which one may easily compute the posterior distribution of the latent states and the system's parameters, in addition to filtered distributions and one-step ahead predictions. Deploying the model within the context of finance, we show how one can use realized covariance matrices as data to predict latent time-varying covariance matrices. This approach out-performs factor stochastic volatility.

Efficiently sampling from the Pólya-Gamma distribution, \\(PG(b,z)\\), is an essential element of Pólya-Gamma data augmentation. Polson et. al (2013) show how to efficiently sample from the \\(PG(1,z)\\) distribution. We build two new samplers that offer improved performance when sampling from the \\(PG(b,z)\\) distribution and \\(b\\) is not unity.

This paper presents two case studies of data sets where the main inferential goal is to characterize time-varying patterns in model structure. Both of these examples are seen to be general cases of the so-called \"partition problem,\" where auxiliary information (in this case, time) defines a partition over sample space, and where different models hold for each element of the partition. In the first case study, we identify time-varying graphical structure in the covariance matrix of asset returns from major European equity indices from 2006--2010. This structure has important implications for quantifying the notion of financial contagion, a term often mentioned in the context of the European sovereign debt crisis of this period. In the second case study, we screen a large database of historical corporate performance in order to identify specific firms with impressively good (or bad) streaks of performance.

Dynamic linear models with Gaussian observations and Gaussian states lead to closed-form formulas for posterior simulation. However, these closed-form formulas break down when the response or state evolution ceases to be Gaussian. Dynamic, generalized linear models exemplify a class of models for which this is the case, and include, amongst other models, dynamic binomial logistic regression and dynamic negative binomial regression. Finding and appraising posterior simulation techniques for these models is important since modeling temporally correlated categories or counts is useful in a variety of disciplines, including ecology, economics, epidemiology, medicine, and neuroscience. In this paper, we present one such technique, Pólya-Gamma data augmentation, and compare it against two competing methods. We find that the Pólya-Gamma approach works well for dynamic logistic regression and for dynamic negative binomial regression when the count sizes are small. Supplementary files are provided for replicating the benchmarks.