Jim Albert

Bayes Inference via Gibbs Sampling of Autoregressive Time Series Subject to Markov Mean and Variance Shifts

We examine autoregressive time series models that are subject to regime switching. These shifts are determined by the outcome of an unobserved two-state indicator variable that follows a Markov process with unknown transition probabilities. A Bayesian framework is developed in which the unobserved states, one for each time point, are treated as missing data and then analyzed via the simulation tool of Gibbs sampling. This method is expedient because the conditional posterior distribution of the parameters, given the states, and the conditional posterior distribution of the states, given the parameters, all have a form amenable to Monte Carlo sampling. The approach is straightforward and generates marginal posterior distributions for all parameters of interest. Posterior distributions of the states, future observations, and the residuals, averaged over the parameter space are also obtained. Several examples with real and artificial data sets and weak prior information illustrate the usefulness of the metho...

Bayesian Estimation of Normal Ogive Item Response Curves Using Gibbs Sampling

The problem of estimating item parameters from a two-parameter normal ogive model is considered. Gibbs sampling (Gelfand & Smith, 1990) is used to simulate draws from the joint posterior distribution of the ability and item parameters. This method gives marginal posterior density estimates for any parameter of interest; these density estimates can be used to judge the accuracy of normal approximations based on maximum likelihood estimates. This simulation technique is illustrated using data from a mathematics placement exam.

Computational Methods Using a Bayesian Hierarchical Generalized Linear Model

Abstract Suppose we observe Y 1, …, YN , where Yi has the exponential density f(yi |θi ) = exp{ϕi ([yiθi – b(θi )]}c(yi, ϕi ). The parameters of interest are not the canonical parameters θi but the means μi = b′(θi ). In the usual generalized linear model (GLM) setup, suppose the means μ1, …, μN are believed to satisfy a specific p-dimensional GLM g(μi ) = xT iβ, where the link function g and the regression coefficients {xi } are known and the regression vector β is unknown. Two problems of interest are the assessment of the goodness of fit of the GLM and the estimation of the means μi . The approach to these problems is by the use of a Bayesian two-stage prior distribution, a generalization of a model used by Lindley and Smith (1972) in the normal mean-estimation problem. At the first stage of the model, we assign independent conjugate distributions to θ1, …, θN , where the prior means of the μi satisfy the GLM. There are p + 1 unknown hyperparameters in this specification, the elements of the regression...

Bayesian Tests and Model Diagnostics in Conditionally Independent Hierarchical Models

Abstract Consider the conditionally independent hierarchical model (CIHM) in which observations yi are independently distributed from f(yi /Θ i )the parameters Θ i are independently distributed from distributions g(Θ|λ), and the hyperparameters λ are distributed according to a distribution h(λ). The posterior distribution of all parameters of the CIHM can be efficiently simulated by Markov chain Monte Carlo (MCMC) algorithms. Although these simulation algorithms have facilitated the application of CIHMs, they generally have not addressed the problem of computing quantities useful in model selection. This article explores how MCMC simulation algorithms and other related computational algorithms can be used to compute Bayes factors that are useful in criticizing a particular CIHM. In the case where the CIHM models a belief that the parameters are exchangeable or lie on a regression surface, the Bayes factor can measure the consistency of the data with the structural prior belief. Bayes factors can also be u...

Statistical thinking in sports

According to popular belief, competitive balance in national soccer competitions in Europe has decreased due to the Bosman ruling and the introduction of the Champions League. We test this hypothesis using data from 7 national competitions, for a host of indicators. We find some evidence for competitive balance having decreased in England, and weak evidence for it having decreased in Netherlands and Belgium. For Germany, France, Italy, and Spain, we find no consistent change whatsoever. We use factor analysis to examine whether our measures of competitive balance can be condensed in a limited number of factors.

College Students Conceptions of Probability

Students in an introductory statistics class were surveyed regarding their views about probability. The students were asked to assign some probabilities and give explanations for their assignments. Results from the surveys indicate that students were generally confused about the classical, frequency, and subjective notions of probabilities. Although the students were able to solve stylized classical probability problems, they were apt to assume that experimental outcomes were equally likely even when this assumption was inappropriate. In addition, the students were not comfortable specifying probabilities using the frequency and subjective views. This article discusses how this confusion about the interpretation of probability affects the teaching of an introductory statistics class and provides some activities helpful for teaching the various probability viewpoints.

Teaching Bayesian Statistics Using Sampling Methods and MINITAB

Abstract Bayesian statistics can be hard to teach at an elementary level due to the difficulty in deriving the posterior distribution for interesting nonconjugate problems. One attractive method of summarizing the posterior distribution is to directly simulate from the probability distribution of interest and then explore the simulated sample. We illustrate the use of Rubins Sampling-Importance-Resampling (SIR) algorithm to simulate posterior distributions for three inference problems. In each example, we focus on the construction of the prior distribution and then use exploratory data analysis techniques to describe the posterior samples and make inferences. The use of MINITAB macros is presented to illustrate the ease of performing this simulation on standard statistical computer programs.

A Bayesian Analysis of a Poisson Random Effects Model for Home Run Hitters

Abstract The problem of interest is to estimate the home run ability of 12 great major league players. The usual career home run statistics are the total number of home runs hit and the overall rate at which the players hit them. The observed rate provides a point estimate for a players “true” rate of hitting a home run. However, this point estimate is incomplete in that it ignores sampling errors, it includes seasons where the player has unusually good or poor performances, and it ignores the general pattern of performance of a player over his career. The observed rate statistic also does not distinguish between the peak and career performance of a given player. Given the random effects model of West (1985), one can detect aberrant seasons and estimate parameters of interest by the inspection of various posterior distributions. Posterior moments of interest are easily computed by the application of the Gibbs sampling algorithm (Gelfand and Smith 1990). A players career performance is modeled using a lo...

Streaky Hitting in Baseball

The streaky hitting patterns of all regular baseball players during the 2005 season are explored. Patterns of hits/outs, home runs and strikeouts are considered using different measures of streakiness. An adjustment method is proposed that helps in understanding the size of a streakiness measure given the players ability and number of hitting opportunities. An exchangeable model is used to estimate the hitting abilities of all players and this model is used to understand the pattern of streakiness of all players in the 2005 season. This exchangeable model that assumes that all players are consistent with constant probabilities of success appears to explain much of the observed streaky behavior. But there are some players that appear to exhibit more streakiness than one would predict from the model.

Bayesian Estimation Methods for 2 × 2 Contingency Tables Using Mixtures of Dirichlet Distributions

Abstract In the estimation of cell probabilities from a 2 × 2 table, a prior distribution is developed that can reflect prior beliefs about the cross-classification structure in the table. The posterior means using this prior shrink the classical estimates towards the association structure specified a priori by the user. Closed-form approximations to the posterior means and credible regions are developed in the special case where the two variables of the table are believed independent.