# Markov Chain Monte Carlo (MCMC)

## What is Markov Chain Monte Carlo (MCMC)?

Markov Chain Monte Carlo (MCMC) is a family of algorithms for sampling from a probability distribution. MCMC algorithms are primarily used in Bayesian statistics and statistical physics to estimate complex, high-dimensional probability distributions that are difficult to compute analytically. By generating a sequence of samples from a Markov chain, MCMC algorithms can approximate the desired probability distribution, allowing for the estimation of parameters, model selection, and other statistical inferences.

## How does MCMC work?

MCMC algorithms work by constructing a Markov chain, a stochastic process where the future state of the system depends only on the current state and not on the past states. The Markov chain is constructed in such a way that its stationary distribution (the distribution it converges to after a large number of steps) is the target distribution we want to sample from.

The most common MCMC algorithm is the Metropolis-Hastings algorithm, which proceeds as follows:

2. Propose a new state based on the current state and a proposal distribution.
3. Calculate the acceptance probability, which depends on the target distribution and the proposal distribution.
4. Accept or reject the proposed state based on the acceptance probability.
5. Repeat steps 2-4 for a large number of iterations.

By repeating this process, the algorithm generates a sequence of samples that, after a sufficient number of steps, approximates the target distribution.

## Applications of MCMC

MCMC algorithms have been widely used in various fields, such as:

• Bayesian statistics: MCMC is a fundamental tool for Bayesian inference, allowing for the estimation of posterior distributions and model parameters.
• Statistical physics: MCMC algorithms can be used to simulate complex physical systems and estimate properties such as phase transitions and equilibrium states.
• Machine learning: MCMC can be used in unsupervised learning tasks, such as clustering and dimensionality reduction, as well as in the training of probabilistic models, such as Bayesian networks and Gaussian processes.
• Computational biology: MCMC algorithms can be used to estimate evolutionary parameters, such as mutation rates and selection coefficients, from genetic data.