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# ''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.
The Metropolis–Hastings algorithm involves designing a Markov process (by constructing transition probabilities) that fulfills the two above conditions, such that its stationary distribution <math>\pi(x)</math> is chosen to be <math>P(x)</math>. The derivation of the algorithm starts with the condition of [[detailed balance]]:
: <math>P(x' \mid x) P(x) = P(x \mid x') P(x'),</math>
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