Markov Chain Monte Carlo (MCMC) refers to a class of algorithms used to sample from probability distributions, particularly when direct sampling is challenging. These methods construct a Markov chain whose equilibrium distribution matches the target distribution, allowing for the estimation of properties of the distribution through the chain’s states.
Key Concepts:
- Markov Chain: A sequence of random variables where each variable depends only on the previous one, exhibiting the Markov property.
- Monte Carlo Method: A computational technique that relies on repeated random sampling to obtain numerical results, often used for estimating integrals and expectations.
Applications of MCMC:
- Bayesian Inference: MCMC is widely used to estimate posterior distributions in Bayesian statistics, enabling the computation of complex integrals that are otherwise analytically intractable.
- Statistical Physics: In physics, MCMC methods are employed to simulate systems with numerous degrees of freedom, such as molecular dynamics and lattice field theories.
- Machine Learning: MCMC techniques are utilized in various machine learning algorithms, including those for parameter estimation and model selection.