Metropolis–Hastings algorithm

The Metropolis-Hastings algorithm is a Markov chain Monte Carlo (MCMC) method used for obtaining random samples from complex probability distributions. It works by proposing new samples based on previous ones and then deciding whether to accept or reject them based on the probability distribution's value at that point.

The algorithm generates a sequence of samples that can be used to approximate the target distribution or compute integrals like expected values. This makes it particularly useful for high-dimensional distributions where direct sampling is challenging.

While Metropolis-Hastings is powerful for multi-dimensional distributions, single-dimensional distributions often benefit from simpler methods like adaptive rejection sampling, which avoids autocorrelation issues inherent in MCMC methods.