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Metropolis Sampling

Learn about Metropolis and Metropolis-Hastings MCMC sampling methods.

Metropolis and Metropolis-Hastings Methods

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Overview

Markov Chain Monte Carlo (MCMC) methods are powerful computational techniques for sampling from complex probability distributions. The Metropolis and Metropolis-Hastings algorithms are foundational MCMC methods.

Key Concepts

  • Target Distribution: The distribution we want to sample from
  • Proposal Distribution: A distribution used to suggest new states
  • Markov Chain: A sequence of random variables where each depends only on the previous one
  • Acceptance Ratio: Determines whether to accept or reject proposed moves

The Metropolis Algorithm

The Metropolis algorithm uses a symmetric proposal distribution to generate samples from a target distribution.

Algorithm Steps

  1. Start with an initial state
  2. For each iteration:
    • Propose a new state from the proposal distribution
    • Calculate the acceptance ratio
    • Accept or reject the proposed state based on this ratio

The Metropolis-Hastings Algorithm

Metropolis-Hastings extends Metropolis by allowing asymmetric proposal distributions, providing greater flexibility for complex sampling problems.

Applications

  • Bayesian inference
  • Parameter estimation
  • Statistical physics
  • Machine learning

Further Reading

  • Hamiltonian Monte Carlo (HMC)
  • Gibbs sampling
  • PyMC3, Stan, or JAGS for practical implementations