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## Automating and evaluating reversible jump MCMC proposal distributions

### Statistics and Computing (2008-11-19) 19: 409-421 , November 19, 2008

The reversible jump Markov chain Monte Carlo (MCMC) sampler (Green in Biometrika 82:711–732, 1995) has become an invaluable device for Bayesian practitioners. However, the primary difficulty with the sampler lies with the efficient construction of transitions between competing models of possibly differing dimensionality and interpretation. We propose the use of a marginal density estimator to construct between-model proposal distributions. This provides both a step towards black-box simulation for reversible jump samplers, and a tool to examine the utility of common between-model mapping strategies. We compare the performance of our approach to well established alternatives in both time series and mixture model examples.

## Hierarchical priors for Bayesian CART shrinkage

### Statistics and Computing (2000-01-01) 10: 17-24 , January 01, 2000

The Bayesian CART (classification and regression tree) approach proposed by Chipman, George and McCulloch (1998) entails putting a prior distribution on the set of all CART models and then using stochastic search to select a model. The main thrust of this paper is to propose a new class of hierarchical priors which enhance the potential of this Bayesian approach. These priors indicate a preference for smooth local mean structure, resulting in tree models which shrink predictions from adjacent terminal node towards each other. Past methods for tree shrinkage have searched for trees without shrinking, and applied shrinkage to the identified tree only after the search. By using hierarchical priors in the stochastic search, the proposed method searches for shrunk trees that fit well and improves the tree through shrinkage of predictions.

## Computing with Fisher geodesics and extended exponential families

### Statistics and Computing (2016-01-01) 26: 325-332 , January 01, 2016

Recent progress using geometry in the design of efficient Markov chain Monte Carlo (MCMC) algorithms have shown the effectiveness of the Fisher Riemannian structure. Furthermore, the theory of the underlying geometry of spaces of statistical models has made an important breakthrough by extending the classical theory on exponential families to their closures, the so-called extended exponential families. This paper looks at the underlying geometry of the Fisher information, in particular its limiting behaviour near boundaries, which illuminates the excellent behaviour of the corresponding geometric MCMC algorithms. Further, the paper shows how Fisher geodesics in extended exponential families smoothly attach the boundaries of extended exponential families to their relative interior. We conjecture that this behaviour could be exploited for trans-dimensional MCMC algorithms.

## Bayesian inferences for receiver operating characteristic curves in the absence of a gold standard

### Journal of Agricultural, Biological, and Environmental Statistics (2006-06-01) 11: 210-229 , June 01, 2006

Sensitivity and specificity are used to characterize the accuracy of a diagnostic test. Receiver operating characteristic (ROC) analysis can be used more generally to plot the sensitivity versus (1-specificity) over all possible cutoff points. We develop an ROC analysis that can be applied to diagnostic tests with and without a gold standard. Moreover, the method can be applied to multiple correlated diagnostic tests that are used on the same individual. Simulation studies were performed to assess the discrimination ability of the no-gold-standard method compared with the situation where a gold standard exists. We used the area under the ROC curve (AUC) to quantify the diagnostic accuracy of tests and the difference between AUCs to compare their accuracies. In particular, we can estimate the prevalence of disease/infection under the no-gold-standard method. The method we proposed works well in the absence of a gold standard for correlated test data. Correlation affected the width of posterior probability intervals for these differences. The proposed method was used to analyze ELISA test scores for Johne’s disease in dairy cattle.

## On Bayesian analyses and finite mixtures for proportions

### Statistics and Computing (2001-04-01) 11: 179-190 , April 01, 2001

When the results of biological experiments are tested for a possible difference between treatment and control groups, the inference is only valid if based upon a model that fits the experimental results satisfactorily. In dominant-lethal testing, foetal death has previously been assumed to follow a variety of models, including a Poisson, Binomial, Beta-binomial and various mixture models. However, discriminating between models has always been a particularly difficult problem. In this paper, we consider the data from 6 separate dominant-lethal assay experiments and discriminate between the competing models which could be used to describe them. We adopt a Bayesian approach and illustrate how a variety of different models may be considered, using Markov chain Monte Carlo (MCMC) simulation techniques and comparing the results with the corresponding maximum likelihood analyses. We present an auxiliary variable method for determining the probability that any particular data cell is assigned to a given component in a mixture and we illustrate the value of this approach. Finally, we show how the Bayesian approach provides a natural and unique perspective on the model selection problem via reversible jump MCMC and illustrate how probabilities associated with each of the different models may be calculated for each data set. In terms of estimation we show how, by averaging over the different models, we obtain reliable and robust inference for any statistic of interest.

## Automatic choice of driving values in Monte Carlo likelihood approximation via posterior simulations

### Statistics and Computing (2003-04-01) 13: 101-109 , April 01, 2003

For models with random effects or missing data, the likelihood function is sometimes intractable analytically but amenable to Monte Carlo approximation. To get a good approximation, the parameter value that drives the simulations should be sufficiently close to the maximum likelihood estimate (MLE) which unfortunately is unknown. Introducing a working prior distribution, we express the likelihood function as a posterior expectation and approximate it using posterior simulations. If the sample size is large, the sample information is likely to outweigh the prior specification and the posterior simulations will be concentrated around the MLE automatically, leading to good approximation of the likelihood near the MLE. For smaller samples, we propose to use the current posterior as the next prior distribution to make the posterior simulations closer to the MLE and hence improve the likelihood approximation. By using the technique of data duplication, we can simulate from the sharpened posterior distribution without actually updating the prior distribution. The suggested method works well in several test cases. A more complex example involving censored spatial data is also discussed.

## Geometric Ergodicity of Metropolis-Hastings Algorithms for Conditional Simulation in Generalized Linear Mixed Models

### Methodology And Computing In Applied Probability (2001-09-01) 3: 309-327 , September 01, 2001

Conditional simulation is useful in connection with inference and prediction for a generalized linear mixed model. We consider random walk Metropolis and Langevin-Hastings algorithms for simulating the random effects given the observed data, when the joint distribution of the unobserved random effects is multivariate Gaussian. In particular we study the desirable property of geometric ergodicity, which ensures the validity of central limit theorems for Monte Carlo estimates.

## Parallelizing MCMC for Bayesian spatiotemporal geostatistical models

### Statistics and Computing (2007-12-01) 17: 323-335 , December 01, 2007

When MCMC methods for Bayesian spatiotemporal modeling are applied to large geostatistical problems, challenges arise as a consequence of memory requirements, computing costs, and convergence monitoring. This article describes the parallelization of a reparametrized and marginalized posterior sampling (RAMPS) algorithm, which is carefully designed to generate posterior samples efficiently. The algorithm is implemented using the Parallel Linear Algebra Package (PLAPACK). The scalability of the algorithm is investigated via simulation experiments that are implemented using a cluster with 25 processors. The usefulness of the method is illustrated with an application to sulfur dioxide concentration data from the Air Quality System database of the U.S. Environmental Protection Agency.

## Model based labeling for mixture models

### Statistics and Computing (2012-03-01) 22: 337-347 , March 01, 2012

Label switching is one of the fundamental problems for Bayesian mixture model analysis. Due to the permutation invariance of the mixture posterior, we can consider that the posterior of a *m*-component mixture model is a mixture distribution with *m*! symmetric components and therefore the object of labeling is to recover one of the components. In order to do labeling, we propose to first fit a symmetric *m*!-component mixture model to the Markov chain Monte Carlo (MCMC) samples and then choose the label for each sample by maximizing the corresponding classification probabilities, which are the probabilities of all possible labels for each sample. Both parametric and semi-parametric ways are proposed to fit the symmetric mixture model for the posterior. Compared to the existing labeling methods, our proposed method aims to approximate the posterior directly and provides the labeling probabilities for all possible labels and thus has a model explanation and theoretical support. In addition, we introduce a situation in which the “ideally” labeled samples are available and thus can be used to compare different labeling methods. We demonstrate the success of our new method in dealing with the label switching problem using two examples.

## Combining snow water equivalent data from multiple sources to estimate spatio-temporal trends and compare measurement systems

### Journal of Agricultural, Biological, and Environmental Statistics (2002-12-01) 7: 536-557 , December 01, 2002

Owing to the importance of snowfall to water supplies in the western United States, governmentagencies regularly collect data on snow water equivalent (the amount of water in snow) over this region. Several differentmeasurementsystem, of possibly different levels of accuracy and reliability, are in operation: snow courses, snow telemetry, aerial markers, and airborne gamma radiation. Data are available at more than 2,000 distinct sites, dating back a variable number of years (in a few cases to 1910). Historically, these data have been used primarily to generate flood forecasts, and short-term (intra-annual) predictions of streamflow and water supply. However, they also have potential for addressing the possible effects of long-term climate change on snowpack accumulations and seasonal water supplies. We presenta Bayesian spatio-temporalanalysis of the combined snow water equivalent (SWE) data from all four systems that all ows for systematic differences in accuracy and reliability. The primary objectives of our analysis are (1) to estimate the long-term temporal trend in SWE over the western U.S. and characterizehow this trend variesspatially, with quantifiable estimates of variability, and (2) to investigate whether there are systematic differences in the accuracy and reliability of the four measurement systems. We find substantial evidence of a decreasing temporal trend in SWE in the Pacific North west and northern Rockies, but no evidence of a trend in the intermountain region and southern Rockies. Our analysis also indicates that some of the systems differ significantly with respect to their accuracy and reliability.