Metropolis–Hastings algorithm: Difference between revisions

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A Markov process is uniquely defined by its transition probabilities <math>P(x' \mid x)</math>, the probability of transitioning from any given state <math>x</math> to any other given state <math>x'</math>. It has a unique stationary distribution <math>\pi(x)</math> when the following two conditions are met:<ref name=Roberts_Casella/>

# ''Existence of stationary distribution'': there must exist a stationary distribution <math>\pi(x)</math>. A sufficient but not necessary condition is [[Markov chain#Reversible Markov chain|detailed balance]], which requires that each transition <math>x \to x'</math> is reversible: for every pair of states <math>x, x'</math>, the probability of being in state <math>x</math> and transitioning to state <math>x'</math> must be equal to the probability of being in state <math>x'</math> and transitioning to state <math>x</math>, <math>\pi(x) P(x' \mid x) = \pi(x') P(x \mid x')</math>.

# ''Uniqueness of stationary distribution'': the stationary distribution <math>\pi(x)</math> must be unique. This is guaranteed by [[Markov Chain#Ergodicity|ergodicity]] of the Markov process, which requires that every state must (1) be aperiodic—the system does not return to the same state at fixed intervals; and (2) be positive recurrent—the expected number of steps for returning to the same state is finite.