Bayesian Statistics -- Post-Test 2 Review Sheet, Spring 2011

I. Assessing Model Quality
 A. Sensitivity Analysis
   1. Checking sensitivity to likelihood specification
   2. Checking sensitivity to prior specification
     a.  Altering the functional form of the prior
     b.  Altering the prior parameter values
     c.  How much does the posterior change for the prior choices?
     d.  Remedial action if model is sensitive to prior choice
 B. Posterior predictive distribution
   1. Definition of prior predictive distribution
   2. Definition of posterior predictive distribution
   3. Deriving posterior predictive distributions (when possible)
   4. Using Monte Carlo to approximate posterior predictive distributions
   5. Predictions and prediction intervals in regression based on posterior predictive distribution

II. Bayesian Hypothesis testing
 A. Problems with P-values and classical hypothesis testing
 B. One-sided Hypothesis Tests
   1. Finding posterior probability that H0 is true
 C. Two-sided Hypothesis Tests
   1. Issues with using a continuous prior with a "point null"
   2. Possible (imperfect) solutions
 D. Bayes Factor
   1. Testing two competing models
   2. Relationship to classical likelihood ratio tests
   3. Bayes factors as a ratio of posterior odds to prior odds
   4. Rules of thumb for interpreting Bayes factors
   5. Using Bayes factor to find posterior probability of one of two possible models
 E. Comparing Two Normal Means
   1. Bayes factor approach for the two-sided test
   2. Choosing between a central-t and noncentral-t model for the test statistic T
   3. Gibbs sampling approach for the one-sided test
   4. Finding posterior probability that H0 true
   5. Finding relevant posterior predictive probabilities
 F. Other Bayes Factor Issues
   1. Problems with Bayes factors with improper priors
   2. BIC as an approximation to the Bayes factor
   3. Definition of BIC
   4. Lack of prior information in BIC

III. Bayesian Hierarchical Models
 A. Hierarchical (Multilevel) Data Structures
 B. Hierarchical Bayes Estimation
   1. Hierarchy of Priors
   2. Hyperpriors on the prior parameters
   3. Need for MCMC methods
 C. Exchangeability and its role in Hierarchical Bayes
 D. Hierarchical Bayes Model for Comparing Several Normal Means
   1. Structure of likelihood, priors and hyperpriors
   2. Functional forms of the full conditionals
   3. Posterior approximation using the Gibbs sampler
 E. Bayesian Estimation and Shrinkage
   1. What is shrinkage?
   2. Relationship of shrinkage to (group) sample size
 F. Empirical Bayes Estimation
   1. Using the "marginal likelihood" to estimate prior parameters
   2. Why is it not "purely" Bayesian?
   3. Comparison of Hierarchical and Empirical Bayes
 
IV. Binary and Ordinal Probit Regression
 A. Regression with Ordinal Responses
   1. Role of the latent variable Z
   2. Function g(.) that relates categorical Y to continuous Z
   3. Definition of "category thresholds"
   4. Functional forms of the necessary full conditionals
   5. Posterior inference on the regression parameters
 B. Regression with Binary Responses
   1. Analogous model with K=2 categories
   2. Posterior predictive probability for each category for an individual