Understanding Markov Chain Monte Carlo (MCMC)

AI 'could solve Britain's productivity challenge'

March 19, 2025

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  • BT's CEO, Allison Kirkby, notes that nearly 50,000 BT employees utilize AI for operations, including fraud prevention and customer service improvements.

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🧠 Markov Chain Monte Carlo (MCMC): Sampling for Probabilistic Inference

Markov Chain Monte Carlo (MCMC) is a powerful method used for sampling from probability distributions where direct sampling is challenging. It is widely used in Bayesian inference, statistical physics, and machine learning. MCMC algorithms construct a Markov chain whose equilibrium distribution matches the desired target distribution, allowing for approximate probabilistic inference.

Markov Chains and Monte Carlo Methods

A Markov chain is a stochastic process where the next state depends only on the current state, not on previous states. Monte Carlo methods use random sampling to estimate numerical results. MCMC combines these concepts by generating a sequence of samples that approximate a given probability distribution over time.

Key MCMC Algorithms

Several MCMC algorithms exist, with the most common being:

  • Metropolis-Hastings Algorithm: This algorithm generates a new sample by proposing a candidate state based on a proposal distribution. The candidate is accepted with a probability that ensures convergence to the target distribution. If rejected, the chain remains in the current state.

  • Gibbs Sampling: Gibbs sampling is a special case of MCMC used when sampling from conditional distributions is easier than sampling from the full joint distribution. It updates one variable at a time while keeping the others fixed.

  • Convergence and Mixing: MCMC methods rely on the assumption that the Markov chain will eventually converge to the target distribution. The rate at which this happens depends on the mixing properties of the chain. Slow mixing results in correlated samples, requiring longer runs for accurate inference.

Applications

MCMC methods are widely applied in:

  • Bayesian statistics for posterior distribution estimation

  • Machine learning for probabilistic graphical models

  • Statistical physics for simulating complex systems

  • Computational biology for phylogenetic analysis

Conclusion

Markov Chain Monte Carlo is a crucial tool for probabilistic inference in complex models. Despite its computational cost, its ability to sample from intractable distributions makes it indispensable in various scientific fields. Careful implementation and convergence diagnostics are necessary for reliable results.

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