A Course in Bayesian Statistics

For the past four years we have been developing a new course in Bayesian statistics within the College of Education of the University of Iowa. This course runs in parallel with a divisional statistics sequence that focuses largely on classicial methods. My aim here is to describe some unusual features of our new course that may be of interest to others and that could be easily adapted and integrated into approaches differing somewhat from our own. Students for this course have come largely from our College of Education. A majority are students of Educational Measurement and Statistics, though others are students of Educational Psychology, Counseling and Guidance, Student Personnel, Mathematics Education, Physical Education, Educational Administration or Mathematical Statistics. The course is listed as a second semester statistics course with a sole prerequisite of a one-semester course in descriptive and classical inferential statistics. Usually students have had this course from the Blommers-Lindquist (1960) or Hayes (1973) text. The course does not have a calculus prerequisite; however, students typically understand the interpretation of an integral as a symbolic notation for an area under a curve. Students are carefully selected through pretest and many are advised to obtain a second semester of preparation. Our purpose in this course is to provide coverage of Bayesian methods through the Behrens-Fisher comparison of means, simple linear regression, multinomial analysis, and some simple work with several arc-sine transformed proportions. We seldom get all of this done because we spend part of the semester motivating Bayesian methods by looking at educational data and formulating some basic problems in simple regression and contingency-table terms. Our language is that of elementary decision theory, with least squares being taught as the minimization of squared-error loss, and generally the choice of estimator being seen as dependent on the loss structure of the problem. The unifying concept through all of this data analysis is that of conditional probability, which is thought of as probability in a specified subpopulation. Regression is taught as conditional expectation and variance in the subpopulation defined by the value of the conditioning variable, which may or may not be a random variable. Similarly, contingency tables are discussed and referenced to joint, marginal and conditional probabilities. Thus the student receives a coverage of probability theory that provides a clear motivation for Bayesian, that is, conditional probability inference. As we work through our analysis of data structured in the context of very real educational problems, we do, in fact, cover the derivation of the F, t, x2, x-2, and x-l distributions as they naturally appear in Bayesian analysis. But this work is almost always done only after the need for such distributions and derivations have been motivated by real problems. This contrasts with treatments which proceed in a mathematical way to state and prove a theorem and then give an example of how the theorem is used in application (e.g., Lindley, 1965). The main reason for motivating our work through data analysis (teaching this first and only then teaching some structure and mathematics) is that our primary goal is to develop in our students an ability to analyze data pertinent to educational problems. We believe that this approach develops a sensitivity to data that is hard to foster in a more mathematics-centered presentation. On the other hand, some of our students develop mathematical skills quite comparable to those of students coming out of a mathematical statistics department. Others, of course, do not. And in fact, some of our students really learn very little about the mathematical structure of Bayesian inference other than to know that under certain circumstances prior and posterior distributions are "t", x-2, or F, and that these distributions are used in certain ways for inference and decision making. But they do know precisely why and how to use these distributions to answer specific scientific questions. The next perhaps unique feature is that throughout this course we insist on the use of proper prior distributions. A major part of any analysis is the fitting, evaluation, re-evaluation, and defense of a proper prior distribution by the data analyst. We view statistical analysis and scientific report writing as principally an adversary proceeding in which a scientific investigator is attempting to convince * College of Education, Div. of Educational Psychology, Measurement & Statistics, Univ. of Iowa, Iowa City, IA 52242. I am grateful to my colleagues Paul Blommers, William Coffman, Leonard Feldt, and Robert Forsyth of the Division of Educational Psychology, Measurement and Statistics and Robert Hogg and George Woodworth of the Division of Mathematical Sciences for their comments on an earlier draft of this paper. I should also like to express my gratitude to the faculties of each of these Divisions for their toleratiQn of a vigorous presentation of ideas that conflict with their established practice. In this course we propound the thesis that, when available, Bayesian methods provide deeper analyses than do classical methods. At the same time we point out that Bayesian methods are not available for many problems and that the usefulness of a data analysis depends primarily on the care and skill exhibited in the analysis. We believe that our students benefit from the directness of these parallel presentations and we leave to them the choice of what methods they will use in their work. Instructional materials discussed in this paper are available from the Psychometric Research Department of the Iowa Testing Programs, Lindquist Center for Measurement, The University of Iowa, Iowa City, Iowa 52242.