Estimating Expectations is Difficult: Why do we need MCMC?

Markov Chain Monte Carlo is a powerful framework for generating samples from probability distributions. In this video, you'll discover how MCMC unlocks practical solutions for sampling from high-dimensional probability distributions, where simpler methods like rejection sampling and importance sampling struggle. Curious what you'll uncover? In this video, you'll explore: 1. The core challenges of making inferences from real-world probability distributions 2. The strengths and weaknesses of Rejection Sampling and Importance Sampling 3. The building blocks of MCMC and an overview of the mathematics behind it. This is the first episode in a multi-part series leading up to Hamiltonian Monte Carlo (HMC). Subscribe and join the journey as we lay the groundwork to master advanced MCMC techniques. Timestamps 0:00 Why Sampling is Necessary 1:55 Grid Approximation 3:27 The "Typical Set" 4:29 Monte Carlo Estimation 5:32 Rejection Sampling 7:16 Rejection Sampling in High Dimensions 8:21 Importance Sampling 10:45 Importance Weight Problems 12:02 Markov Chain Monte Carlo 15:06 Why Choose MCMC? 16:35 What Comes Next? References/Further Reading 1. MacKay, D. J. (2003). Information theory, inference and learning algorithms. Cambridge university press. Chapter 29 2. Gelman, A., Carlin, J. B., Stern, H. S., & Rubin, D. B. (1995). Bayesian data analysis. Chapman and Hall/CRC. Chapter 10 & 11 (third edition) 3. Bishop, C. M., & Nasrabadi, N. M. (2006). Pattern recognition and machine learning (Vol. 4, No. 4, p. 738). New York: springer. Chapter 11 #statistics #MCMC #drawingdistributions