Kolmogorov forward and backward equations

[sfischer]

For the Kolmogorov forward and backward equations, as I understand it, for the time-homogeneous state, we include paths where the transition rates go from state i to i (or j to j) whereas for the time-inhomogeneous case we only include paths where the transition rates transition from a different state. Does that sound correct. And if so, can someone explain the difference? Thanks.

 

[Mark Mitchell]

No, that's not correct.

In both the time-homogeneous and the time-inhomogeneous cases, the forward and backward differential equations are obtained by summing over ALL states. So we consider transitions from state i to state j (where i and j are different) and from state i to state i.

The difference between time-homogeneous and time-inhomogeneous is as follows:

- in a time-homogeneous model, the transition probabilities depend only on the length of the time interval considered. This type of model occurs when the transition rates between states are constant. So, the probability of moving from state A to state B between times 6 and 10 would equal the probability of moving from state A to state B between times 16 and 20, as both intervals are of length 4.

- in a time-inhomogeneous model, the transition probabilities depend on the actual times involved, not just the length of the interval. This type of model occurs when the transition rates between states vary over time. In this case, the probability of moving from state A to state B between times 6 and 10 would not be equal to the probability of moving from state A to state B between times 16 and 20 (other than by chance).

 

[sfischer]

Right. The reason I raised the point was that in section 9.1 (Marriage Example), we don't include the path of B to M then transition of M to M. Also in the Q&A 2.9 (vi) example, we don't include the transition from Sick to Sick followed by the probability of Sick to Dead. Whereas in the time-homogeneous examples, we do include these pathways. Am I missing something in these examples? Thanks,

 

[Mark Mitchell]

My original post explains the difference between time-homogeneous and time-inhomogeneous cases, and I said that for the DIFFERENTIAL equations, you sum over all states.

Section 9.1 of Chapter 6 is an example of the INTEGRATED form of the equations. For these integral equations, the summation is over all states excluding:
- the opening state (in the backward case, focussing on the first transition - see Section 7 of Chapter 6)
- the closing state (in the forward case, focussing on the last transition - see section 8 of Chapter 6).

Section 9.1 effectively relates to a forward case as the probability specifies a duration in the closing state forcing us to think about the last transition.