... YEAR SO ----- 1. Nolan Ryan 1973 383 2. Sandy Koufax 1965 382 3. Randy Johnson 2001 372 4. No... more ... YEAR SO ----- 1. Nolan Ryan 1973 383 2. Sandy Koufax 1965 382 3. Randy Johnson 2001 372 4. Nolan Ryan 1974 367 5. Randy Johnson 1999 364 6. Rube Waddell 1904 349 7. BobFeller 1946 348 8. Randy Johnson 2000 347 9. Nolan Ryan 1977 341 ...
Http Dx Doi Org 10 1080 01621459 1983 10478032, Mar 12, 2012
Page 1. Bayesian Estimation Methods for 2 x 2 Contingency Tables Using Mixtures of Dirichlet Dist... more Page 1. Bayesian Estimation Methods for 2 x 2 Contingency Tables Using Mixtures of Dirichlet Distributions JAMES H. ALBERT and ARJUN K. GUPTA* In the estimation of cell probabilities from a 2 x 2 table, a prior distribution ...
In the simultaneous estimation of multinomial proportions, two estimators are developed which inc... more In the simultaneous estimation of multinomial proportions, two estimators are developed which incorporate prior means and a prior parameter which reflects the accuracy of the prior means. These estimators possess substantially smaller risk than the standard estimator in a region of the parameter space and are much more robust than the conjugate Bayes estimator with respect to parameter values far from the prior mean. When vague prior information is available, these estimators and confidence regions derived from them appear to be attractive alternatives to the procedures based on the standard estimator.
This paper reviews the past and current interest in using Bayesian thinking to introduce statisti... more This paper reviews the past and current interest in using Bayesian thinking to introduce statistical inference. Rationale for using a Bayesian approach is described and particular methods are described that make it easier to understand Bayes' rule. Several older and modern introductory statistics books are reviewed that use a Bayesian perspective. It is argued that a Bayesian perspective is very helpful in teaching a course in statistical literacy.
ABSTRACT Students in an introductory statistics class were surveyed regarding their views about p... more ABSTRACT 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.
Page 1. Teaching Bayesian Statistics Using Sampling Methods and MINITAB JAMES H. ALBERT* Bayesian... more Page 1. Teaching Bayesian Statistics Using Sampling Methods and MINITAB JAMES H. ALBERT* Bayesian statistics can be hard to teach at an elementary level due to the difficulty in deriving the posterior dis-tribution for interesting nonconjugate problems. ...
Mike Schmidt was one the great homerun hitters in major league baseball. Schmidt's hitting perfor... more Mike Schmidt was one the great homerun hitters in major league baseball. Schmidt's hitting performance was notable for its consistency over a span of 14 years. The issue of consistency is investigated by exploring a dataset which lists the date for each homerun Schmidt hit during his career. This exploration suggests that the homerun ability of this hitter did vary within a given season. To see if the observed patterns correspond to real e ects, simulated homerun data is produced assuming a coin tossing model with a constant probability of success. By comparison of Schmidt's data with this simulation distribution, it is seen that Schmidt's homerun hitting behavior does exhibit some inconsistency. These patterns show how Schmidt matured as a baseball hitter over his career.
Page 1. Bayesian Methods for Binomial Data With Applications to a Nonresponse Problem JAMES H. AL... more Page 1. Bayesian Methods for Binomial Data With Applications to a Nonresponse Problem JAMES H. ALBERT and ARJUN K. GUPTA* In this article, Bayesian estimation methods for several binomial probabilities are studied ...
A simulation-based approach is proposed for approximating a Bayesian analysis. Parameters and dat... more A simulation-based approach is proposed for approximating a Bayesian analysis. Parameters and data are simulated from a Bayesian model and inference about a parameter is performed by exploring the set of simulated parameter values conditional on a set of values of a simulated statistic. The approach is used to learn about parameters of a streaky model on the basis of a statistic used to measure streakiness. The method is illustrated to detect streakiness in baseball hitting data and basketball shooting data.
Http Dx Doi Org 10 1080 01621459 1997 10474046, Feb 17, 2012
ABSTRACT Consider the conditionally independent hierarchical model (CIHM) where observations y i ... more ABSTRACT Consider the conditionally independent hierarchical model (CIHM) where observations y i are independently distributed from f(y i j` i ), the parameters ` i are independently distributed from distributions g(`j), and the hyperparameters are distributed according to a distribution h(). The posterior distribution of all parameters of the CIHM can be efficiently simulated by Monte Carlo Markov Chain (MCMC) algorithms. Although these simulation algorithms have facilitated the application of CIHM's, they generally have not addressed the problem of computing quantities useful in model selection. This paper 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 ...
ABSTRACT During a broadcast of a baseball game, a fan hears how baseball hitters perform in vario... more ABSTRACT During a broadcast of a baseball game, a fan hears how baseball hitters perform in various situations, such as at home and on the road, on grass and on turf, in clutch situations, and ahead and behind in the count. From this discussion by the media, fans get the misleading impression that much of the variability in players' hitting performance can be explained by one or more of these situational variables. For example, an announcer may state that a particular player struck out because he was behind in the count and was facing a left-handed pitcher. In baseball one can now investigate the effect of various situations, as hitting data is recorded in very fine detail. This article looks at the hitting performance of major league regulars during the 1992 baseball season to see which situational variables are “real” in the sense that they explain a significant amount of the variation in hitting of the group of players. Bayesian hierarchical models are used in measuring the size of a particular situational effect and in identifying players whose hitting performance is very different in a particular situation. Important situational variables are identified together with outstanding players who make the most of a given situation.
The problem of interest is to estimate the home run ability of 12 great major league players. The... more The problem of interest is to estimate the home run ability of 12 great major league players. The usual ca-reer 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 player's ...
This paper considers the tting, criticism and comparison of three ordinal regression models { the... more This paper considers the tting, criticism and comparison of three ordinal regression models { the cumulative, sequential and two-step models. E cient algorithms based on Markov chain Monte Carlo methods are developed for each model. In the case of the cumulative model, a new Metropolis-Hastings procedure to sample the cut points is proposed. This procedure relies on a simple transformation of the cut-points that leaves the transformed cut-points unordered. For comparing these models, we develop a coherent approach based on marginal likelihoods and Bayes factors. To help in the assignment of prior distributions to regression parameters and the cut-points, di erent methods for forming and representing prior beliefs are provided. One set of methods is based on the idea of a training sample and a prior imaginary sample. Another method is based on the direct assessment of distributions on the multinomial response, followed by change of variable to a distribution on the parameters of the model. All of the ideas are illustrated in two data sets, one from
... YEAR SO ----- 1. Nolan Ryan 1973 383 2. Sandy Koufax 1965 382 3. Randy Johnson 2001 372 4. No... more ... YEAR SO ----- 1. Nolan Ryan 1973 383 2. Sandy Koufax 1965 382 3. Randy Johnson 2001 372 4. Nolan Ryan 1974 367 5. Randy Johnson 1999 364 6. Rube Waddell 1904 349 7. BobFeller 1946 348 8. Randy Johnson 2000 347 9. Nolan Ryan 1977 341 ...
Http Dx Doi Org 10 1080 01621459 1983 10478032, Mar 12, 2012
Page 1. Bayesian Estimation Methods for 2 x 2 Contingency Tables Using Mixtures of Dirichlet Dist... more Page 1. Bayesian Estimation Methods for 2 x 2 Contingency Tables Using Mixtures of Dirichlet Distributions JAMES H. ALBERT and ARJUN K. GUPTA* In the estimation of cell probabilities from a 2 x 2 table, a prior distribution ...
In the simultaneous estimation of multinomial proportions, two estimators are developed which inc... more In the simultaneous estimation of multinomial proportions, two estimators are developed which incorporate prior means and a prior parameter which reflects the accuracy of the prior means. These estimators possess substantially smaller risk than the standard estimator in a region of the parameter space and are much more robust than the conjugate Bayes estimator with respect to parameter values far from the prior mean. When vague prior information is available, these estimators and confidence regions derived from them appear to be attractive alternatives to the procedures based on the standard estimator.
This paper reviews the past and current interest in using Bayesian thinking to introduce statisti... more This paper reviews the past and current interest in using Bayesian thinking to introduce statistical inference. Rationale for using a Bayesian approach is described and particular methods are described that make it easier to understand Bayes' rule. Several older and modern introductory statistics books are reviewed that use a Bayesian perspective. It is argued that a Bayesian perspective is very helpful in teaching a course in statistical literacy.
ABSTRACT Students in an introductory statistics class were surveyed regarding their views about p... more ABSTRACT 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.
Page 1. Teaching Bayesian Statistics Using Sampling Methods and MINITAB JAMES H. ALBERT* Bayesian... more Page 1. Teaching Bayesian Statistics Using Sampling Methods and MINITAB JAMES H. ALBERT* Bayesian statistics can be hard to teach at an elementary level due to the difficulty in deriving the posterior dis-tribution for interesting nonconjugate problems. ...
Mike Schmidt was one the great homerun hitters in major league baseball. Schmidt's hitting perfor... more Mike Schmidt was one the great homerun hitters in major league baseball. Schmidt's hitting performance was notable for its consistency over a span of 14 years. The issue of consistency is investigated by exploring a dataset which lists the date for each homerun Schmidt hit during his career. This exploration suggests that the homerun ability of this hitter did vary within a given season. To see if the observed patterns correspond to real e ects, simulated homerun data is produced assuming a coin tossing model with a constant probability of success. By comparison of Schmidt's data with this simulation distribution, it is seen that Schmidt's homerun hitting behavior does exhibit some inconsistency. These patterns show how Schmidt matured as a baseball hitter over his career.
Page 1. Bayesian Methods for Binomial Data With Applications to a Nonresponse Problem JAMES H. AL... more Page 1. Bayesian Methods for Binomial Data With Applications to a Nonresponse Problem JAMES H. ALBERT and ARJUN K. GUPTA* In this article, Bayesian estimation methods for several binomial probabilities are studied ...
A simulation-based approach is proposed for approximating a Bayesian analysis. Parameters and dat... more A simulation-based approach is proposed for approximating a Bayesian analysis. Parameters and data are simulated from a Bayesian model and inference about a parameter is performed by exploring the set of simulated parameter values conditional on a set of values of a simulated statistic. The approach is used to learn about parameters of a streaky model on the basis of a statistic used to measure streakiness. The method is illustrated to detect streakiness in baseball hitting data and basketball shooting data.
Http Dx Doi Org 10 1080 01621459 1997 10474046, Feb 17, 2012
ABSTRACT Consider the conditionally independent hierarchical model (CIHM) where observations y i ... more ABSTRACT Consider the conditionally independent hierarchical model (CIHM) where observations y i are independently distributed from f(y i j` i ), the parameters ` i are independently distributed from distributions g(`j), and the hyperparameters are distributed according to a distribution h(). The posterior distribution of all parameters of the CIHM can be efficiently simulated by Monte Carlo Markov Chain (MCMC) algorithms. Although these simulation algorithms have facilitated the application of CIHM's, they generally have not addressed the problem of computing quantities useful in model selection. This paper 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 ...
ABSTRACT During a broadcast of a baseball game, a fan hears how baseball hitters perform in vario... more ABSTRACT During a broadcast of a baseball game, a fan hears how baseball hitters perform in various situations, such as at home and on the road, on grass and on turf, in clutch situations, and ahead and behind in the count. From this discussion by the media, fans get the misleading impression that much of the variability in players' hitting performance can be explained by one or more of these situational variables. For example, an announcer may state that a particular player struck out because he was behind in the count and was facing a left-handed pitcher. In baseball one can now investigate the effect of various situations, as hitting data is recorded in very fine detail. This article looks at the hitting performance of major league regulars during the 1992 baseball season to see which situational variables are “real” in the sense that they explain a significant amount of the variation in hitting of the group of players. Bayesian hierarchical models are used in measuring the size of a particular situational effect and in identifying players whose hitting performance is very different in a particular situation. Important situational variables are identified together with outstanding players who make the most of a given situation.
The problem of interest is to estimate the home run ability of 12 great major league players. The... more The problem of interest is to estimate the home run ability of 12 great major league players. The usual ca-reer 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 player's ...
This paper considers the tting, criticism and comparison of three ordinal regression models { the... more This paper considers the tting, criticism and comparison of three ordinal regression models { the cumulative, sequential and two-step models. E cient algorithms based on Markov chain Monte Carlo methods are developed for each model. In the case of the cumulative model, a new Metropolis-Hastings procedure to sample the cut points is proposed. This procedure relies on a simple transformation of the cut-points that leaves the transformed cut-points unordered. For comparing these models, we develop a coherent approach based on marginal likelihoods and Bayes factors. To help in the assignment of prior distributions to regression parameters and the cut-points, di erent methods for forming and representing prior beliefs are provided. One set of methods is based on the idea of a training sample and a prior imaginary sample. Another method is based on the direct assessment of distributions on the multinomial response, followed by change of variable to a distribution on the parameters of the model. All of the ideas are illustrated in two data sets, one from
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