Metropolis–Hastings Algorithms

# Metropolis-Hastings Algorithms in R: MCMC, Langevin Dynamics, and Model Selection Unlock the power of Markov Chain Monte Carlo (MCMC) methods! In this lecture, we dive deep into the **Metropolis-Hastings algorithm**, exploring its implementation, variations, and practical applications in R. This lecture is based on Chapter 6 of *"Introducing Monte Carlo Methods with R"* by Christian P. Robert and George Casella. We use the `mcsm` package to demonstrate how these algorithms work in real-world statistical modeling. ### What You’ll Learn: * **The Fundamentals:** Understanding the Metropolis-Hastings transition kernel and acceptance probability. * **Langevin Metropolis-Hastings:** Implementing more efficient sampling using gradient information, with examples from: * **Probit Posteriors:** Sampling from a probit model using the `Pima.tr` dataset. * **Mixture Posteriors:** Handling complex, multi-modal distributions. * **MCMC Model Selection:** Using Gibbs sampling and indicator variables to navigate model spaces (demonstrated with the `swiss` fertility dataset). * **Acceptance Rates:** How to compare different proposal distributions (e.g., double-exponential vs. normal) to optimize your algorithm. ### Code & Resources: * **R Packages Used:** `mcsm`, `magrittr`, `ggplot2`. * **Key Functions:** `hastings()`, `pimamh()`, `mhmix()`, and `mochoice()`. * **Textbook Reference:** Robert, C. P., & Casella, G. (2010). *Introducing Monte Carlo Methods with R*. Springer-Verlag. * **UCF Students:** You can download the textbook for free through the UCF Library: [https://library.ucf.edu/](https://library.ucf.edu/) **About the Instructor:** Lecture by Aaron Smith. #RStats #DataScience #MCMC #BayesianStatistics #MetropolisHastings #MachineLearning