Bayesian Statistics -- Test 1 Review Sheet, Spring 2024

I. Introduction to Bayesian Data Analysis
 A. Rationale behind Bayesian Methods vs. Classical Methods
 B. Motivation for Bayesian Modeling
 C. Interpretations of Probability
   1. Frequentist Definition of Probability
   2. Subjective Probability
 D. Applying Bayes' Rule 
   1. Bayes' Rule with two events (say, A and B)
   2. Bayes' Rule with multiple events
   2. Role of Bayes' Rule in developing posterior distribution for inference
   3. Posterior mean and variance

II. Specifying Bayesian Models
 A. Bayesian Framework
   1. Definition of Likelihood Function
   2. Definition of MLE's
   3. Likelihood Principle
   4. Prior and Likelihood => Posterior
   5. Proportionality and role of Normalizing Constant
 B. Conjugate Priors
   1. Definition of Conjugate prior for a sampling model
   2. The Beta-Binomial Bayesian Model
      a. Know the Formula for posterior parameters in the beta-binomial model
      b. Know the Formula for mean of a beta distribution 
   3. The Poisson-Gamma Bayesian Model 
      a. Know the Formula for posterior parameters in the gamma-Poisson model
      b. Know the Formula for mean of a gamma distribution (in terms of the shape and rate parameters)
      c. Know the Formula for variance/std.dev of a gamma distribution (in terms of the shape and rate parameters)
 C. Possible Bayesian point estimators
   1. Posterior mean as a combination of the sample mean and prior mean
 D. Bayesian Learning/Using Posterior as an Updated Prior
 E. Posterior Credible Intervals
   1. Formal Definition of Credible Interval
   2. Quantile-based Intervals
   3. HPD Intervals

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. Understanding the Normal-Normal model
   1. You don't need to memorize the Normal-Normal posterior formulas, but understand them if shown them
   2. You don't need to write the Normal-Normal R code for the test, but understand the code and output if shown it