The Nelson-Aalen estimator of the transition matrix P(s,t) estimates the transition probability in the time interval [s,t] from any state s1 to any other state s2.
Is it possible to estimate the transition probability within the time interval [s,t] along a given fixed path of states, say s1->s2->s3?
It is clear that this is not given by the P(s,t)(1,3), i.e. the matrix element in 1st row and 3 column, because this estimates the transition probability of ALL paths from 1->3.
Is there any idea how to estimate the transition probability along a given path?
Any idea would be very appreciated.
Thanks!
Am I right that you are looking for probability of a certain trajectory? And on top of this the chain itself is non-homogenous?
If the latter is true, I guess, the only option is to discretize your process and do brute force calculations.
Dear Dmitri,
thank you for your comment, yes you'are right.
I will consider your suggestion and I wonder a bit why it is difficult to find an answer for this question.
Dear Wolfgang,
the major difficulty here stems from non-homogenous nature of the process at the rate matrix (infinitesimal generator) changes in time. This type of MC is not well studied in general. On top, the metric itself is very difficult to estimate even under simpler assumptions, e.g., homogeneity. In short it is related to the first passage time to a state which is very difficult for general MCs but here the process should also follow a predefined trajectory. One may expect some results available when MC has a certain structure such as birth-death process of even simpler.
Still working in discrete time you can easily (well relatively) get you result even for non-homogenous case. Assuming that you have side information about evolution of the rate matrix you can define its discrete-time version - transition probability matrix and get your result straightforwardly. There might also be chances for computationally friendly result via fundamental matrix, you may want to take a look at book by Kemeny and Snell [1]. Here, you can make the final state absorbing and browse the book for results telling probability of trajectory for absorbing Markov chains. I remember they've has some expositions there.
Sorry Wolfgang for not being very specific, I just remember I was run into a similar problem and spent a lot of time trying to solve it or find a solution before I went with discretization. I would be more than happy if something might help you.
[1] Kemeny J. G., Snell J. L. Finite Markov chains: with a new appendix" Generalization of a fundamental matrix". – Springer, 1983.
Dear Wolfgang Hitzl , thank you for your nice question. If I understand your question correctly I think you can resolve your problem by applying the Chapman–Kolmogorov equation. The Chapman–Kolmogorov equations can be written in terms of the n -step transition probability matrices as P(n)=P(m)P(n-m). See for example, Scott L. Miller and Donald Childers, Probability and Random Processes , With Applications to Signal Processing and Communications, Second Edition • 2012. Kind Regards. Rob
Rob Christiaanse : Thank you for your reply. Well, I think the Chapman-Kolmogorov equations are not the solution of the above problem. The transition probability matrix P(s,t) gives the probability of any transition from say a state s1->s2 within the time interval [s,t]. This means a "particle" can travel along ANY path from s1->s2. The above question - however - refers to the transition probability of one GIVEN path.
Thanks for the reference. I will try to find a solution.
- Badges
- Science topic
- Similar topics
- Engineering
- Computer Engineering
- Computer