when to split Markov states. April 2011 Q12iia

[vegan]

My understanding is some states of a model need to be split to satisfy the markov property if the model incorporates some "previous" claims history, whereby "previous" means the year prior to the last year. (i.e. if the next movement is defined as at time n; then "last year's" movement is n-1 and the "previous year's" movement is n-2).

2. How can we examine which states exactly need to be split and do not need to be split in any model to satisfy the markov property?

In the question in the subject above, my understanding is that the first two bullet points in the question only refer to order changes from n-1. The last two bullet points include the history or order changes from year n-2, which has prompted for some states in the model to be split.

1. Why is state L split into two (specifically, L+ and L-) - would it not require a division into a third state as well?
3. Why has state S not been split?
4. Why has state B not been split, like L?
5. Why has state G not* been split, like L?

Thanks,

 

[John Potter]

How do we know when to split states?

Let's use April 2011 Q12 as an example.

Draw a diagram with B, G, L and S.
Draw the arrows from one state to another.
Can increase order so arrow from B to G, G to L, L to S and S to S
Can decrease order so arrow from S to L, L to G, G to B and B to B
Can maintain order so arrow from B to B (already there), G to G, L to L, S to S (already there)

Then we would just start writing the probabilities on the arrows. The problem comes when we get to State L because the probability we write on the arrow from L to G depends on how we got to L. This is NOT Markov. When it is Markov, you never get this problem.

If last year we came from S to L and decrease order again, we will go from L to B.
If last year we came from G or L to L and decrease order, we will go from L to G.

This is why we split L. So, split L into 2 states L+ and L- say, and draw the new arrows. Then you can write the probabilities on the arrows without having to worry about what happened the previous year

We don't need to split the other states because we don't get this problem with any of the other states. We can write the probabilities on the arrows without having to worry about what happened the previous year.

Good luck!
John