The general Bayes' theorem is well known as:
$$P(X|Y) = \frac{P(Y|X) \cdot P(X)}{P(Y)} $$
where $P(X)$ a prior probability distribution (belief), and $P(Y)$ a data probability distribution (observation).
If we would write:
$$ P(X|Y)\cdot P(Y) = P(Y|X) \cdot P(X) $$
and interpret the conditional probabilities as transition probabilities in the context of master equations (first order Markov chains assumed) then:
$$ 0= w(Y|X) \cdot P(X) - w(X|Y)\cdot P(Y)$$
Looks like a typical master equation under the very condition of detailed balance, derived from:
$$\partial_tP(X) = w(Y|X) \cdot P(X) - w(X|Y)\cdot P(Y)$$
Question: Is this interpretation mathematically correct, namely to interpret the Bayes theorem as a stationary solution of a master equation in detailed balance? If its is correct, then please describe the corresponding Markov process; if it is not correct, then please explain why?
I could meanwhile answer this question and like to reference the following article with a complete solution presented to this question; https://medium.com/@vaseghisam/1763-meets-1968-how-bayes-illumes-the-detailed-balance-in-chapman-and-kolmogorovs-equation-7fc48e68140a