Learning Objectives
5 objectives- Understand the fundamental concepts and principles of Bayesian statistics.
- Apply Bayesian inference including prior, likelihood, and posterior distributions.
- Utilize Markov Chain Monte Carlo methods for sampling from complex distributions.
- Implement Bayesian models such as linear regression and hierarchical models.
- Evaluate and compare Bayesian models using appropriate statistical criteria and tools.
Content Outline
PreviewUnit 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
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