Metropolis Method Condition Derivation

Metropolis-Hasting involves designing a Markov process by constructing Transition Probabilities, which has a unique stationary distribution π(x)\pi(x)π(x) if it fulfills two conditions:

Existitence of Stationary Distribution:
Reason----Detailed Balance:
π(x)P(x′∣x)=π(x′)P(x∣x′)\pi(x) P(x' | x) = \pi(x')P(x|x')π(x)P(x′∣x)=π(x′)P(x∣x′)
Uniqueness of Stationary Distribution:
Reason – Ergodicity: every states satisfies
1 Aperiodic: the system does not return to the same state at fixed intervals----
2 Be positive recurrent - the expected number of steps for returning to the same state is finite.

The derivation of the algorithm starts with the condition of detailed balance.
P(x′∣x)P(x)=P(x′,x)=P(x∣x′)P(x′)P(x'|x)P(x) = P(x', x)=P(x|x')P(x')P(x′∣x)P(x)=P(x′,x)=P(x∣x′)P(x′)
P(x′∣x)=g(x′∣x)A(x′,x)P(x'|x) = g(x'|x)A(x', x)P(x′∣x)=g(x′∣x)A(x′,x), where A(x’, x) is the acceptance ratio, the probability to accept the proposed state x’.

A(x′,x)A(x,x′)=P(x′)g(x∣x′)P(x)g(x′∣x)\frac{A(x', x)}{A(x, x')} = \frac{P(x')g(x|x')}{P(x)g(x'|x)}A(x,x′)A(x′,x)=P(x)g(x′∣x)P(x′)g(x∣x′)
The next step in the derivation is to choose an acceptance ratio that fulfils the condition above. One common choice is the Metropolis choice:
r=A(x′,x)=min(1,P(x′)P(x)g(x∣x′)g(x′∣x))r=A(x', x) = min(1, \frac{P(x')}{P(x)} \frac{g(x|x')}{g(x'|x)})r=A(x′,x)=min(1,P(x)P(x′)g(x′∣x)g(x∣x′))

reference
https://en.wikipedia.org/wiki/Metropolis–Hastings_algorithm

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