Time homogenous MJP

[James123]

Hi,

I know time inhomogenous markov jump chains account for duration and the actual times themselves, so for example if looking between time s and t, we would know the value of an and t eg. S=2 and t=5 so duration =3.

whereas for a time homogenous MJP we wouldn’t know the actual times, ie . No values for s and t but we would know the duration is 3, ie, t-s=3 so it may be that s =8 and t= 9 or any other possibility.

but I’m confused by exam paper 2023 April question 4(I) as the solution suggests the time inhomogenious model is more suitable here as it gives us length of time ie.duration details which we wouldn’t have with the time homogenous case.

but I thought we get duration details in both cases, in a time homogenous model we would still know the length of time the individual has been sick, we just wouldn’t know the time itself?

thanks in advance

 

[Andrew Martin]

Hi James

For a time-homogeneous MJP, it isn't that we can't / don't know the actual times but rather that the actual start and end times don't matter for the purposes of calculating probabilities, only the duration between them. So, for example:

P(X_10 = 2 | X_7 = 1) = p12(7,10) = P(X_435 = 2 | X_432 = 1) = p12(432,435) = p12(3).

For a time-inhomogeneous MJP, the start and end times matter. So P(X_10 = 2 | X_7 = 1) = p12(7,10) may not be the same as P(X_435 = 2 | X_432 = 1) = p12(432,435) even though both probabilities are over a duration of 3 time units.

In question 4, the transition probabilities depend not only on the duration (difference between start and end time) but also on the number of days after being bitten. So, the start and end times matter. For example, P(X_5 = Fully recovered | X_2 = Sick) is not the same as P(X_9 = Fully recovered | X_6 = Sick), as the transition rate is changing over time.

Hope this helps!

Andy