0
$\begingroup$

I have an irreducible continuous-time Markov chain (CTMC) with a finite state space. The CTMC also does not have any one-step transitions from any state to itself. I have the transition rate matrix $Q$, so I can solve for the limiting probabilities (null space of $Q^T$ - computed via the SVD) and stationary transition probabilities (matrix exponential $e^{Qt}$).

Given some time interval $\Delta t$, how do I determine the probability density function (PDF) for the number of times state $i$ has a one-step transition to state $j$ in such an interval? I'm looking for something like "state $i$ goes to state $j$ 3 times with probability 0.6 and 4 times with probability 0.4 in time interval $\Delta t$." Do I need any other information?


note: My matrix $Q$ is very large (over three million states), so I would like to avoid computing a matrix exponential. The matrix is also very sparse - each row only have five non-zeros elements, but I don't think that would help me too much in computing a matrix exponential.

$\endgroup$
0
$\begingroup$

This is actually a fairly complex problem. In the Journal of Mathematical Biology, there is an article called "Counting labeled transitions in continuous-time Markov models of evolution" that attempts to do exactly what I am trying to do with respect to CTMCs. They were able to write down a solution for the case of only two states, but it quickly becomes more complicated with more states. This may not really be an answer, but no one else has tried to answer the question, so this will be the accepted solution.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.