• Bayesian model specification: Suppose You (a generic person wishing to reason sensibly in the presence of uncertainty) are uncertain about something: let's call it θ (this could be almost anything, but for concreteness think of a vector of real numbers of length k). You have an information source (data set) D that You judge to be relevant to decreasing Your uncertainty about θ (D could again be almost anything, but for concreteness think of a vector of real numbers of length n), and You also have background information B summarizing problem context and how D was gathered.

  A foundational theorem, proved independently by the statistician and actuary Bruno de Finetti (1937) and the physicist Richard Cox (1946), says that if You wish to summarize Your information about θ, in light of D and B, in a logically-consistent manner, then You need to specify two ingredients, for inference about θ and prediction of future data D^* -- namely, p ( θ | B ) (Your prior distribution for θ) and p ( D | θ B ) (Your sampling distribution for D) -- and two more ingredients for decision-making: a set A of possible actions and a utility function U ( a, θ ) quantifying Your judgments about the rewards (monetary or otherwise) that would ensue if You chose action a and the unknown actually took the value θ. This theorem then goes on, through a series of corollaries, to tell You how these ingredients should be combined for logically-consistent inference, prediction and decision-making, using the basic rules we examined in the first short course in Oct 2010 (for example, Bayes's Theorem for inference).

  The de Finetti-Cox theorem is great as far as it goes -- it tells You which four ingredients You have to specify -- but it's almost entirely silent on how to specify those ingredients. The first half of this short course will be about how to perform Bayesian model specification: how to go from {the context of the problem and the design of the data-gathering process} to good choices for the ingredients mentioned above.

  Let's agree to call { p ( θ | B ), p ( D | θ B ) } Your model M for Your uncertainty about θ -- such a model is needed for inference and prediction. In this short course I'll concentrate mainly on specifying M, not because decision-theory is unimportant but for lack of time to do justice to both inference/prediction and decision-making. Topics to be covered on model specification will include the following:

  • Four principles for good Bayesian model-building: the Calibration Principle, the Modeling-As-Decision Principle, the Prediction Principle and the Decision-Versus-Inference Principle.
  • How to specify diffuse (low-information-content) and non-diffuse (high-information-content) prior distributions.
  • How to deal with model uncertainty: Bayesian model averaging, Bayesian non-parametric methods, and {Bayesian cross-validation as a way to use Your data to find good models while paying the right price for Your data-driven search}.
  • How to provide good Bayesian answers to the questions "Is model M₁ better than model M₂?" and "Is model M₁ good enough to stop searching?". Answers to the first type of question involve Bayes factors and log scores; answers to the second type involve a properly-calibrated version of posterior predictive P-values.

• Bayesian hierarchical modeling: Many data sets have a nested, or multilevel, or hierarchical character. Two examples of fields in which this type of data arises frequently include education (when measuring educational quality, Your data set may have students nested in classrooms nested in schools nested in school districts) and medicine (when measuring quality of health care, Your data set may involve patients nested within hospitals nested within geographical regions).

  Another example involves the discipline of meta-analysis, in which there are a number of studies of the same basic phenomenon and Your goal is to combine information across studies to produce a more accurate answer than that obtainable from any single study.

  Even when Your data set is not hierarchical, there are many situations in which You'll find it useful to employ Bayesian hierarchical modeling, especially when You want to build a mixture model. Mixture models are helpful in settings (which occur frequently) in which You discover that Your data values exhibit unexplained heterogeneity and You want to realistically describe this heterogeneity to produce well-calibrated inferences and predictions.

As usual I'll motivate each unit of methodology in the context of a case study. The case studies in this short course will be drawn from health policy, medicine and the physical sciences, but the methods illustrated will apply to a broad range of subject areas in the natural and social sciences, business (including topics of direct relevance to pharmaceutical companies), and public policy.

The course will liberally illustrate user-friendly implementations of the methods covered via the freeware Bayesian analysis program WinBUGS and the freeware statistical computing package R.

The course is intended mainly for people who often use statistics in their research; some graduate coursework in statistics will provide sufficient mathematical background for participants, and -- since this is a follow-on course from the one a year ago -- I'll assume some familiarity with Bayesian reasoning and methods. To get the most out of the course, participants should be comfortable with hearing the course presenter mention (at least briefly) (a) differentiation and integration of functions of several variables and (b) discrete and continuous probability distributions (joint, marginal, and conditional) for several variables at a time, but all necessary concepts will be approached in a sufficiently intuitive manner that rustiness on these topics will not prevent understanding of the key ideas.