Markov chain Monte Carlo

MCMC samples from complex posterior distributions

Markov chain Monte Carlo (MCMC) is a powerful statistical tool used to sample from complex probability distributions that are difficult to analyze analytically. By constructing a Markov chain with an equilibrium distribution matching the target distribution, MCMC methods enable researchers to approximate these distributions through iterative sampling. This approach is particularly useful for high-dimensional problems where traditional methods fall short.

Example

In Bayesian statistics, MCMC can be used to estimate the posterior distribution of model parameters given observed data. For instance, in a Bayesian linear regression model, MCMC can help sample from the posterior distribution of the regression coefficients, allowing for uncertainty quantification and prediction intervals.

MCMC is crucial for statistical inference in complex models, providing a practical way to approximate distributions that are otherwise intractable.

Related concepts

Metropolis–Hastings algorithm

Metropolis-Hastings algorithm samples from difficult distributions

Langevin dynamics does: adds noise to gradient descent to sample from a distribution

Langevin dynamics adds noise to gradient descent to sample from a distribution

the Dirichlet distribution does: distribution over probability simplices

The Dirichlet distribution generates random probability vectors over a simplex

t-SNE preserves local structure

t-SNE preserves local structure by converting distances to probabilities and minimizing Kullback-Leibler divergence

Fisher information

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Top-k vs top-p sampling: top-k fixes candidate count, top-p fixes cumulative probability mass

Top-k sampling fixes candidate count; top-p sampling fixes cumulative probability mass

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