Unit 3094: Comprehensive Bayesian Statistics

1. Introduction to Bayesian Statistics

• Overview of Bayesian vs Frequentist approaches
• Bayes' Theorem: formula and intuition
• Key components: Prior, Likelihood, Posterior
• Applications and significance in statistical inference

2. Prior Distribution

• Definition and role of prior distributions
• Types of priors: informative, non-informative, conjugate priors
• Criteria for choosing appropriate priors
• Effect of different priors on posterior results

3. Likelihood Function

• Definition and interpretation of likelihood
• Likelihood as probability of observed data given parameters
• Calculating likelihood for common statistical models (e.g., binomial, normal)
• Relationship between likelihood and data

4. Posterior Distribution

• Deriving posterior distribution using Bayes' theorem
• Mathematical relationship: Prior × Likelihood → Posterior
• Interpretation and practical implications of posterior distributions
• Examples of posterior updates with data

5. Markov Chain Monte Carlo (MCMC) Methods

• Introduction to MCMC and motivation
• Metropolis-Hastings algorithm: concept and steps
• Gibbs sampling: mechanism and applications
• Advantages of MCMC in sampling from complex posteriors
• Practical considerations and convergence diagnostics

6. Bayesian Hypothesis Testing

• Concept of Bayesian hypothesis testing
• Bayes factor: definition and interpretation
• Posterior predictive p-values and their use
• Comparison with classical hypothesis testing methods

7. Bayesian Linear Regression

• Applying Bayesian methods to linear regression
• Specifying priors for regression coefficients and variance
• Estimating regression parameters and uncertainty
• Comparison with frequentist linear regression results

8. Hierarchical Bayesian Models

• Concept of hierarchical or multi-level models
• Nesting parameters and sharing information across groups
• Advantages of hierarchical modeling (e.g., partial pooling)
• Specifying priors at different hierarchy levels

9. Bayesian Model Comparison

• Techniques for model comparison: Bayes factors, Deviance Information Criterion (DIC)
• Assessing model fit and complexity
• Practical examples of model selection in Bayesian framework

10. Bayesian Computational Tools

• Overview of popular Bayesian software: Stan, JAGS, PyMC3
• Setting up and running Bayesian analyses using these tools
• Interpreting outputs and diagnostics
• Best practices and resources for computational Bayesian analysis