Probabilistic ML - 20 - Markov Chain Monte Carlo

2025-07-11 · Source: Tübingen Machine Learning - YouTube · Field: Technology & Digital — Artificial Intelligence & Machine Learning, Mathematics & Computational Sciences · Depth: Advanced, extended

Summary

This lecture on "Probabilistic ML - 20 - Markov Chain Monte Carlo" explores various sampling methods for drawing from complex probability distributions, especially when cumulative density functions are intractable. It first outlines direct Monte Carlo techniques like rejection sampling, which discards samples based on a proposal distribution, and important sampling, which reweighs all samples. Both face challenges in high dimensions, with rejection sampling suffering from exponentially increasing rejection rates and important sampling from high variance. The lecture then delves into Markov Chain Monte Carlo (MCMC) methods, detailing the Metropolis-Hastings algorithm, which iteratively proposes and accepts samples to explore the distribution, though it can exhibit slow random walk behavior. Specialized MCMC variants include Gibbs sampling, which achieves an acceptance rate of one by leveraging analytically tractable conditional distributions, and Hamiltonian Monte Carlo (HMC). HMC introduces momentum to guide sampling along deterministic paths, significantly improving exploration and also achieving an acceptance rate of one, with the No-U-Turn Sampler (NUTS) further enhancing its efficiency. These methods are foundational to probabilistic programming frameworks.

Key takeaway

For Machine Learning Engineers building high-dimensional probabilistic models, understand that basic Monte Carlo methods like rejection or important sampling are often inefficient due to the curse of dimensionality. You should prioritize advanced Markov Chain Monte Carlo (MCMC) techniques such as Hamiltonian Monte Carlo (HMC) or Gibbs sampling when applicable. These methods offer superior exploration and acceptance rates, providing more reliable samples for Bayesian inference, especially within probabilistic programming frameworks.

Key insights

MCMC methods address high-dimensional sampling challenges by generating dependent sequences that asymptotically represent the target distribution.

Principles

• Probability measures are deterministic objects, not inherently random.
• Monte Carlo estimates converge at a rate proportional to the square root of samples.
• High-dimensional sampling requires methods that overcome random walk behavior.

Method

Metropolis-Hastings proposes samples from a local distribution and accepts them based on target and proposal ratios. Hamiltonian Monte Carlo (HMC) simulates physical dynamics in an enlarged state space for efficient, high-acceptance samples.

In practice

• Explore Laplace approximation or variational bounds for fast posterior approximations.
• Apply Gibbs sampling when conditional distributions are analytically tractable.
• Utilize probabilistic programming frameworks like Pyro for Bayesian inference.

Topics

• Markov Chain Monte Carlo
• Hamiltonian Monte Carlo
• Gibbs Sampling
• Rejection Sampling
• Probabilistic Programming
• Bayesian Inference

Best for: Research Scientist, AI Scientist, Machine Learning Engineer, AI Student

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Editorial summary, takeaway, and curation by AIssential. Original article published by Tübingen Machine Learning - YouTube.