Markov chain: inferring transition rates from equilibrium

Question

I feel like I have been having a very dumb week trying to solve/research this problem and that I am missing an easy solution or that it is not possible. Given an equilibrium distribution (say from a Continuous time Markov chain) and the number of states is known, is it possible to find the transition rates/infinitesimal matrix that the equilibrium distribution came from?

All papers I have read only infer the rates when they have equilibrium distribution and sequence of data, thus they have the counts of each state visit during the process and can solve through MLE/Bayesian methods. But in this case the only information known is the equilibrium distribution and the structure of the chain (number of states and that there is detailed balance). Is it possible to infer the rates and if so some hint/nudge/paper in the direction I should be looking?

Answer 1

If you have a Markov chain with transition probability matrix $\mathbf{P}$ then any stationary distribution $\pi$ for that chain is a unit eigenvector of $\mathbf{P}$ with corresponding eigenvalue of one. This means that knowledge of a single stationary distribution for the chain is equivalent to knowledge of a single eigenvector/eigenvalue pair. It is well-known that this is insufficient to determine the entire transition probability matrix (for any chain with more than two states).