We have developed a sophisticated statistical model for predicting the hitting performance of Major League baseball players. The Bayesian paradigm provides a principled method for balancing past performance with crucial covariates, such as player age and position. We share information across time and across players by using mixture distributions to control shrinkage for improved accuracy. We compare the performance of our model to current sabermetric methods on a held-out season (2006), and discuss both successes and limitations.

Counts or averages over arbitrary regions are often analyzed using conditionally autoregressive (CAR) models. The neighborhoods within CAR models are generally determined using only the inter-distances or boundaries between the sub-regions. To accommodate spatial variations that may depend on directions, a new class of models is developed using different weights given to neighbors in different directions. By accounting for such spatial anisotropy, the proposed model generalizes the usual CAR model that assigns equal weight to all directions. Within a fully hierarchical Bayesian framework, the posterior distributions of the parameters are derived using conjugate and non-informative priors. Efficient Markov chain Monte Carlo (MCMC) sampling algorithms are provided to generate samples from the marginal posterior distribution of the parameters. Simulation studies are presented to evaluate the performance of the estimators and are used to compare results with traditional CAR models. Finally the method is illustrated using data sets on local crime frequencies in Columbus, OH and on the elevated blood lead levels of children under the age of 72 months observed in Virginia counties for the year of 2000.

We propose a Bayesian method for multiple hypothesis testing in random effects models that uses Dirichlet process (DP) priors for a nonparametric treatment of the random e®ects distribution. We consider a general model formulation which accommodates a variety of multiple treatment conditions. A key feature of our method is the use of a product of spiked distributions, i.e., mixtures of a point-mass and continuous distributions, as the centering distribution for the DP prior. Adopting these spiked centering priors readily accommodates sharp null hypotheses and allows for the estimation of the posterior probabilities of such hypotheses. Dirichlet process mixture models naturally borrow information across objects through model-based clustering while inference on single hypotheses aver- ages over clustering uncertainty. We demonstrate via a simulation study that our method yields increased sensitivity in multiple hypothesis testing and produces a lower proportion of false discoveries than other competitive methods. While our modeling framework is general, here we present an application in the context of gene expression from microarray experiments. In our application, the modeling framework allows simultaneous inference on the parameters governing differential expression and inference on the clustering of genes. We use experimental data on the transcriptional response to oxidative stress in mouse heart muscle and compare the results from our procedure with existing nonparametric Bayesian methods that provide only a ranking of the genes by their evidence for differential expression.

This paper demonstrates the use and value of stochastic differential equations for modeling space-time data in two common settings. The first consists of point-referenced or geostatistical data where observations are collected at fixed locations and times. The second considers random point pattern data where the emergence of locations and times is random. For both cases, we employ stochastic differential equations to describe a latent process within a hierarchical model for the data. The intent is to view this latent process mechanistically and endow it with appropriate simple features and interpretable parameters. A motivating problem for the second setting is to model urban development through observed locations and times of new home construction; this gives rise to a space-time point pattern. We show that a spatio-temporal Cox process whose intensity is driven by a stochastic logistic equation is a viable mechanistic model that affords meaningful interpretation for the results of statistical inference. Other applications of stochastic logistic differential equations with space-time varying parameters include modeling population growth and product diffusion, which motivate our first, point-referenced data application. We propose a method to discretize both time and space in order to fit the model. We demonstrate the inference for the geostatistical model through a simulated dataset. Then, we fit the Cox process model to a real dataset taken from the greater Dallas metropolitan area.

A simple example is presented using standard continuous distributions with a real valued parameter in which the posterior mean is inconsistent on a dense subset of the real line.

Estimation of the number of species extant in a geographic region has been discussed in the statistical literature for more than sixty years. The focus of this work is on the use of pilot data to design future studies in this context. A Dirichlet-multinomial probability model for species frequency data is used to obtain a posterior distribution on the number of species and to learn about the distribution of species frequencies. A geometric distribution is proposed as the prior distribution for the number of species. Simulations demonstrate that this prior distribution can handle a wide range of species frequency distributions including the problematic case with many rare species and a few exceptionally abundant species. Monte Carlo methods are used along with the Dirichlet-multinomial model to perform sample size calculations from pilot data, e.g., to determine the number of additional samples required to collect a certain proportion of all the species with a pre-specified coverage probability. Simulations and real data applications are discussed.

Markov switching models can be used to study heterogeneous populations that are observed over time. This paper explores modeling the group characteristics nonparametrically, under both homogeneous and nonhomogeneous Markov switching for group probabilities. The model formulation involves a finite mixture of conditionally independent Dirichlet process mixtures, with a Markov chain defining the mixing distribution. The proposed methodology focuses on settings where the number of subpopulations is small and can be assumed to be known, and flexible modeling is required for group regressions. We develop Dirichlet process mixture prior probability models for the joint distribution of individual group responses and covariates. The implied conditional distribution of the response given the covariates is then used for inference. The modeling framework allows for both non-linearities in the resulting regression functions and non-standard shapes in the response distributions. We design a simulation-based model fitting method for full posterior inference. Furthermore, we propose a general approach for inclusion of external covariates dependent on the Markov chain but conditionally independent from the response. The methodology is applied to a problem from fisheries research involving analysis of stock-recruitment data under shifts in the ecosystem state.

Bayesian analysis is frequently confused with conjugate Bayesian analysis. This is particularly the case in the analysis of clinical trial data. Even though conjugate analysis is perceived to be simpler computationally (but see below, Berger's prior), the price to be paid is high: such analysis is not robust with respect to the prior, i.e. changing the prior may affect the conclusions without bound. Furthermore, conjugate Bayesian analysis is blind with respect to the potential conflict between the prior and the data. Robust priors, however, have bounded influence. The prior is discounted automatically when there are conflicts between prior information and data. In other words, conjugate priors may lead to a dogmatic analysis while robust priors promote self-criticism since prior and sample information are not on equal footing. The original proposal of robust priors was made by de-Finetti in the 1960's. However, the practice has not taken hold in important areas where the Bayesian approach is making definite advances such as in clinical trials where conjugate priors are ubiquitous.

We show here how the Bayesian analysis for simple binary binomial data, expressed in its exponential family form, is improved by employing Cauchy priors. This requires no undue computational cost, given the advances in computation and analytical approximations. Moreover, we introduce in the analysis of clinical trials a robust prior originally developed by J.O. Berger that we call Berger's prior. We implement specific choices of prior hyperparameters that give closed-form results when coupled with a normal log-odds likelihood. Berger's prior yields a robust analysis with no added computational complication compared to the conjugate analysis. We illustrate the results with famous textbook examples and also with a real data set and a prior obtained from a previous trial. On the formal side, we present a general and novel theorem, the "Polynomial Tails Comparison Theorem." This theorem establishes the analytical behavior of any likelihood function with tails bounded by a polynomial when used with priors with polynomial tails, such as Cauchy or Student's $t$. The advantages of the theorem are that the likelihood does not have to be a location family nor exponential family distribution and that the conditions are easily verifiable. The binomial likelihood can be handled as a direct corollary of the result. Next, we proceed to prove a striking result: the intrinsic prior to test a normal mean, obtained as an objective prior for hypothesis testing, is a limit of Berger's robust prior. This result is useful for assessments and for MCMC computations. We then generalize the theorem to prove that Berger's prior and intrinsic priors are robust with normal likelihoods. Finally, we apply the results to a large clinical trial that took place in Venezuela, using prior information based on a previous clinical trial conducted in Finland.

Our main conclusion is that introducing the existing prior information in the form of a robust prior is more justifiable simultaneously for federal agencies, researchers, and other constituents because the prior information is coherently discarded when in conflict with the sample information.