In a non-homogenous, continuous time Markov model, 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 adjacent states, say s1->s2->s3?
In other words: only such transitions are of interested which do not deviate from s1->s2->s3, e.g. s1->s4->s2->s3 is not allowed.
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!
