Bayesian Statistical Methods

# Bayesian Statistical Methods

Bayesian statistics offers a powerful alternative to frequentist approaches, incorporating prior knowledge and providing intuitive probability interpretations.

## Learning Objectives

In this lecture, you will learn:
- Bayesian vs. frequentist approaches
- Prior and posterior distributions
- Bayes' theorem applications
- MCMC sampling methods
- Practical implementation in R

## Bayesian Framework

**Bayes' Theorem**: P(θ|data) = P(data|θ) × P(θ) / P(data)

Where:
- P(θ|data) is the posterior distribution
- P(data|θ) is the likelihood
- P(θ) is the prior distribution
- P(data) is the marginal likelihood

## Key Advantages

- Incorporates prior knowledge
- Provides probability statements about parameters
- Handles complex models naturally
- Quantifies uncertainty in predictions

## MCMC Methods

- **Gibbs Sampling**: For conjugate priors
- **Metropolis-Hastings**: General purpose
- **Hamiltonian Monte Carlo**: Efficient for high dimensions

## Practical Applications

- Clinical trials and medical research
- Machine learning and AI
- Environmental modeling
- Financial risk assessment