Bayesian Statistics -- Test 1 Review Sheet, Spring 2014

I. Introduction to Bayesian Data Analysis
 A. Rationale behind Bayesian Methods vs. Classical Methods
 B. Motivation for Bayesian Modeling
   1. Specification of Joint Density and Prior Distributions
   2. Exchangeability (and relationship with iid)
 C. Interpretations of Probability
   1. Frequentist Definition of Probability
   2. Subjective Probability
   3. Conditional, marginal, and joint probability distributions
 D. Bayes' Law 
   1. Bayes' Law with one or more simple events
   2. Role of Bayes' Law in developing posterior distribution for inference
   3. Posterior mean and variance

II. Specifying Bayesian Models
 A. Likelihood Theory
   1. Definition of Likelihood Function
   2. Definition of MLE's
   3. Likelihood Principle
 B. Bayesian Framework
   1. Prior and Likelihood => Posterior
   2. Proportionality and role of Normalizing Constant
 C. Posterior Intervals
   1. Bayesian vs. Frequentist coverage
   2. Pre-experimental vs. Post-experimental coverage
   3. Frequentist coverage for Bayesian Intervals
   4. Formal Definition of Credible Interval
   5. Quantile-based Intervals
   6. HPD Intervals
 D. Conjugate Priors
   1. Definition of Conjugate prior for a sampling model
   2. The Beta-Binomial Bayesian Model
   3. The Poisson-Gamma Bayesian Model
 E. Possible Bayesian point estimators
   1. Posterior mean as a combination of the sample mean and prior mean
 F. Bayesian Learning/Updating

III. Bayesian Models for Normal Data
 A. Reasons for using a Normal Model for Data
 B. Conjugate Analysis for Normal Data (Mean unknown, Variance known)
   1. Prior precision, data precision, posterior precision
   2. Posterior mean as a combination of the sample mean and prior mean
 C. Conjugate Analysis for Normal Data (Mean known, Variance unknown)
   1. Inverse gamma prior distribution
 D. Conjugate Analysis for Normal Data (Mean unknown, Variance unknown)
   1. Prior for mu depends on sigma^2
   2. Role of n and s_0 in weighting of sample mean and prior mean
 E. Bayesian Analysis for Multivariate Normal Data
 F. Noninformative Bayesian Models for Normal Data
   1. Choices of Vague Priors for mu and sigma
   2. Resulting posterior distributions for mu and sigma^2
   3. How do these compare to the posteriors from the conjugate analysis?