Like I undestand MCMC sampling, the fulfillment of the detailed balance equation guarantees that our MC has reached its stationary distribution (given we ensure ergodicity).
Detailed Balance is:
$\pi(x)q(x\rightarrow x')=\pi(x')q(x'\rightarrow x)$
with $\pi(x)$ being the probability to be in state $x$ and $q(x\rightarrow x')$ the transition probability from state $x$ to $x'$ at time $T$. (According to Russell, Stuart, and Peter Norvig. "AI a modern approach")
A problem I came across in MCMC is to find the right burn-in time $T$, the amount of samples needed to reach the stationary distribution. Why can we not use DBE as a stopping criterion? Why can we not compute whether DBE is fulfilled after each sample and then stop sampling as soon as it is fulfilled? Naively, it looks like $\pi(x)$ and $q(x\rightarrow x')$ could be computed emperically based on the samples obtained so far.