Metropolis Method Condition Derivation

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

1. Existitence of Stationary Distribution:
   Reason----Detailed Balance:
   $\pi(x) P(x' | x) = \pi(x')P(x|x')$
2. 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.
$P(x'|x)P(x) = P(x', x)=P(x|x')P(x')$
$P(x'|x) = g(x'|x)A(x', x)$, where A(x’, x) is the acceptance ratio, the probability to accept the proposed state x’.

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

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