POST-TEST 1 REVIEW SHEET
(Also review the Test 1 Review Sheet posted on the course website)

I.  Markov Chain Monte Carlo Techniques
 A. The Monte Carlo Method
   1. Using the law of large numbers to approximate population quantities of interest
   2. Sampling from common distributions in R
 B. MCMC Methodology
   1. When is MCMC useful?
   2. What is a Markov Chain and the Markovian Property?
###
###  NOTE:  For Test 3, you should be prepared to answer conceptual questions about 
###  the Gibbs Sampler and M-H Algorithm (e.g., short-answer, multiple-choice, or
###  True-False type questions), but you will not have to *implement* these algorithms
###  on a data set on Test 3.
###
 C. Gibbs Sampling
   1. When can we use the Gibbs Sampler?
   2. The formal Gibbs Algorithm
   3. What is the result of the Gibbs Sampling process?
   4. Implementations of Gibbs Sampling in R (Don't worry about coding details)
   5. Burn-in and Convergence Diagnostics like trace plots
 D. Metropolis-Hastings Method
   1. When should we use the Metropolis-Hastings Method?
   2. The formal Metropolis-Hastings Algorithm
   3. Role of the acceptance ratio in the algorithm
   4. Acceptance rate
   5. Autocorrelation and the role of "thinning"


II. 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 
   (You won't have to do the details of deriving/simulating posterior predictive distributions on the test.
    Just understand what they are used for and be prepared for conceptual questions about their use.)
   1. Definition of prior predictive distribution
   2. Definition of posterior predictive distribution
   3. Deriving posterior predictive distributions
   4. Using Monte Carlo to approximate posterior predictive distributions
   5. Predictions and prediction intervals in regression based on posterior predictive distribution

III. 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

IV. Bayesian Linear Regression
 A. Regression Setup (with Matrix-Vector Notation)
 B. Noninformative Analysis for the Regression Situation
   1. Choices of Vague Priors for Beta and sigma^2
   2. Resulting posterior distributions for mu and sigma^2
 C. Conjugate Analysis for the Regression Situation
   1. Specification of Conjugate Prior Information
   2. Role of the "Hypothetical" Prior Observations
   3. Role of delta, a and b in prior specifications
   4. Rules of Thumb for weighting the worth of the prior information
   5. Form of Posterior distributions for precision tau and for beta|tau
 D. Bayesian Model Selection
   1. Partitioning beta_j into z_j b_j
   2. Role of the indicator vector, z
   3. Finding posterior probabilities for each possible value of the vector z
   4. Implementing the approach via Gibbs Sampling code (don't worry about coding details)
   5. Restricting possible choices of model to a particular subset of models
      by only specifying certain z vectors
 E. Posterior Predictive Distribution for the Data
   1. What is the definition of the posterior predictive distribution in the regression setting?
   2. Form of the posterior predictive distribution for the normal-error regression model
   3. Using the posterior predictive distribution for check model fit
 F. Model Selection Criteria
   1. Basic interpretation of what is a "better" MAE, BIC, ELPD (in a RELATIVE sense)

V. Bayesian Count Regression
 A. Poisson Regression Model
   1. Regression Equation for Poisson model
   2. Interpretation of estimated regression coefficients
 B. Negative Binomial Regression Model
   1. Difference between Poisson and Negative Binomial distributions
   2. Concept of Overdispersion
   3. Parameters in Negative Binomial distribution
 C. Use of Posterior Predictive Distribution to assess model fit in count regression models
 D. Interpreting Interactions in any Regression Model
   1. See link with R example about interpreting interactions on the course website

VI. Bayesian Logistic Regression and Naive Bayes Classification
 A. Regression with Binary Responses
   1. Probability and Odds connection
   2. Form(s) of Regression Equation for Logistic model
   3. Interpretation of estimated regression coefficients
   4. Using logistic regression for prediction/classification
   5. Classification Accuracy and Confusion matrix
   6. Sensitivity, specificity, and cutoff value
 B. Naive Bayes Classification
   1. Ability to do multicategory classification
   2. Ways of specififying prior probabilities
   3. Connection to Bayes Rule
   4. "Naive" assumption of independence of predictors
   5. Comparisons of Naive Bayes Classification and Logistic Regression

III. Bayesian Models for Hierarchical/Grouped Data
 A. Hierarchical (Multilevel) Data Structures
   1. What characterizes grouped data?
   2. Examples of grouped data
 B. Hierarchical Models
   1. Complete pooling and No-pooling models (pros and cons)
   2. Hierarchical Models that use partial pooling
   3. Within-group and between-group variation
   4. Overall model parameters and Group-specific parameters
   5. Predictions and Estimation
 C. Bayesian Estimation and Shrinkage
   1. What is shrinkage?
   2. Relationship of shrinkage to (group) sample size
 D.  Normal Hierarchical Models with Predictors
   1. Varying Intercepts Model (big-picture concept)
   2. Varying Intercepts and Slopes Model (big-picture concept)
   3. Posterior Prediction of a response for a subject in the study, and for a new subject