Markov Chain Monte Carlo (MCMC)

2026-03-22 · Source: Machine Learning on Medium · Field: Technology & Digital — Artificial Intelligence & Machine Learning, Data Science & Analytics · Depth: Intermediate, medium

Summary

Markov Chain Monte Carlo (MCMC) is a class of algorithms fundamental to Bayesian statistics and machine learning, designed to sample from complex probability distributions by constructing a Markov chain whose stationary distribution matches the target. Key algorithms include Metropolis-Hastings (MH), which proposes states and accepts them based on a calculated ratio, Gibbs Sampling, which updates one variable at a time from its full conditional distribution, and Hamiltonian Monte Carlo (HMC), which leverages gradient information for more efficient sampling in high dimensions. Practical application requires addressing "burn-in" periods, autocorrelation, and rigorous convergence diagnostics such as trace plots, the Gelman-Rubin statistic ("R-hat"), and Effective Sample Size (ESS). The article provides detailed Python implementations for MH and Gibbs Sampling, illustrating their use in Bayesian inference for analytically intractable posteriors. It also highlights tools like PyMC and ArviZ for advanced MCMC applications and diagnostics, underscoring MCMC's role in enabling robust statistical modeling and parameter estimation.

Key takeaway

Markov Chain Monte Carlo (MCMC) algorithms, including Metropolis-Hastings, Gibbs, and Hamiltonian Monte Carlo (HMC), are crucial for sampling from intractable probability distributions in Bayesian inference and machine learning. HMC, leveraging gradient information (e.g., in PyMC/Stan), offers superior convergence in high dimensions over random-walk Metropolis-Hastings, while Gibbs excels with tractable conditional distributions. Practitioners must use diagnostics like Gelman-Rubin R-hat (target ~1.0) and Effective Sample Size (ESS > 400) to ensure reliable posterior estimates, achieving high accuracy (e.g., 2.482 vs 2.483 for a posterior mean).

Topics

• Markov Chain Monte Carlo
• Bayesian Inference
• Metropolis-Hastings Algorithm
• Hamiltonian Monte Carlo
• Gibbs Sampling

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Editorial summary, takeaway, and curation by AIssential. Original article published by Machine Learning on Medium.