Say I was given some directed graph that satisfies the Markov property, has a stationary distribution, $\pi$, and I know the transition probabilities are functions of some unknown parameters $P_{i\to j}=p(\pmb{\alpha})$, where $\pmb\alpha$ is a vector of unknown parameters (pardon my poor probability notation here). I know that I can use Markov Chain Monte Carlo to simulate the stationary distribution, $\pi$, to calculate the posterior for $\pmb\alpha$. However, I am curious if MCMC can still be used to get the posterior if you were given some non-stationary distribution instead? By non-stationary, I mean non-equilibrium distribution here.
In example, let $t>0$ be time and $T$ be the time it takes for the graph to reach the equilibrium (stationary) state. Now say your experiment/data takes too long to reach equilibrium and you can only sample the data at some time $t_0<T$. Can you still use that distribution to get the posterior and thereby infer $\pi$?