Albert Jim - Probability and Bayesian Modeling

Probability and Bayesian Modeling

CHAPMAN & HALL/CRC

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Probability and Bayesian Modeling

Jim Albert
Jingchen Hu

CRC Press

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Contents

The Traditional Introduction to Statistics

A traditional introduction to statistical thinking and methods is the two-semester probability and statistics course offered in mathematics and statistics departments. This traditional course provides an introduction to calculus-based probability and statistical inference. The first half of the course is an introduction to probability including discrete, continuous, and multivariate distributions. The chapters on functions of random variables and sampling distributions naturally lead into statistical inference including point estimates and hypothesis testing, regression models, design of experiments, and ANOVA models.

Although this traditional course remains popular, there seems to be little discussion in this course on the application of the inferential material in modern statistical practice. Although there are benefits in discussing methods of estimation such as maximum likelihood, and optimal inference such as a best hypothesis test, the students learn little about statistical computation and simulation-based inferential methods. As stated in Cobb (2015), there appears to be a disconnect between the statistical content we teach and statistical practice.

Developing a New Course

The development of any new statistics course should be consistent with current thinking of faculty dedicated to teaching statistics at the undergraduate level. Cobb (2015) argues that we need to deeply rethink our undergraduate statistics curriculum from the ground up. Towards this general goal, Cobb (2015) proposes five imperatives that can help the process of creating this new curriculum. These imperatives are to: (1) flatten prerequisites, (2) seek depth in understanding fundamental concepts, (3) embrace computation in statistics, (4) exploit the use of context to motivate statistical concepts, and (5) implement research-based learning.

Why Bayes?

There are good reasons for introducing the Bayesian perspective at the calculus-based undergraduate level. First, many people believe that the Bayesian approach provides a more intuitive and straightforward introduction than the frequentist approach to statistical inference. Given that the students are learning probability, Bayes provides a useful way of using probability to update beliefs from data. Second, given the large growth of Bayesian applied work in recent years, it is desirable to introduce the undergraduate students to some modern Bayesian applications of statistical methodology. The timing of a Bayesian course is right given the ready availability of Bayesian instructional material and increasing amounts of Bayesian computational resources.

We propose that Cobbs five imperatives can be implemented through a Bayesian statistics course. Simulation provides an attractive flattened prerequisites strategy in performing inference. In a Bayesian inferential calculation, one avoids the integration issue by simulating a large number of values from the posterior distribution and summarizing this simulated sample. Moreover, by teaching fundamentals of Bayesian inference of conjugate models together with simulation-based inference, students gain a deeper understanding of Bayesian thinking. Familiarity with simulation methods in the conjugate case prepares students for the use of simulation algorithms later for more advanced Bayesian models.

One advantage of a Bayes perspective is the opportunity to input expert opinion by the prior distribution which allows students to exploit context beyond a traditional statistical analysis. This text introduces strategies for constructing priors when one has substantial prior information and when one has little prior knowledge.