The Metropolis–Hastings algorithm proposes from at state and accepts it with probabilityThis acceptance rule enforces detailed balance with the target density .
Random-walk Metropolis proposes a symmetric increment from the current state and accepts a proposal from with probability .
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The Metropolis–Hastings algorithm is a Markov Chain Monte Carlo (MCMC) method used for sampling from probability distributions that are difficult to sample from directly. It is particularly useful in situations where the distribution is defined up to a normalization constant, making it challenging to derive samples analytically.