Page "Metropolis–Hastings algorithm" Paragraph 28

which defines the transition probability from the current state x to a new state x '.

It can be shown that two conditions are necessary and sufficient for generating states according to.

One is the ergodicity of the process which ensures that, at most, one asymptotic distribution exists.

The other is the condition of balance of the matrix which is generally satisfied if the matrix fulfills detailed balance.

Detailed balance ensures that there exists at least one asymptotic distribution and that this asymptotic distribution is.

Formally, a detailed balanced process is the one on which the probability to be in the state x and transit to x ' must be the same as to be in the state x ' and transit to x.

This can be written as