April 2017 Question 9 v and vi

[Priyanka Malhotra]

Hi could someone please help me understand what does "Partial credit was given in parts (v) and (vi) for various alternative state spaces, notably {A, AB, AC, ABC} where the letters refer to the countries the foreign correspondent has already visited. This is not Markov, since the probability of moving from state AB to state ABC depends on whether he is in country A or in country B." mentioned in the examiner's report comment mean?
How is the diagram given in examiner's report is Markov and not the state space given in comments..? As per my understanding the diagram in the examiner report also has 2 different routes for coming to final state "all countries visited" and it depends whether it has been visited from "On A (A and B visited)" and "On B (A and B visited)". Thanks in advance!

 

[Andrew Martin]

Hi Priyanka

If we try to draw the transition diagram and add on probabilities of transition then we run into trouble when considering AB -> ABC. The state 'AB' only captures the fact that the correspondent has visited countries A and B, not where they are currently. If the correspondent is currently in country A, the probability of going A -> C is 0.2. If they are currently in country B, the probability of going B -> C is 0.15. So, what should we stick on the arrow showing the transition from AB to ABC?

That's why we split out this state into two separate states A'B and AB'. A'B represents the correspondent having visited A and B and currently being in A and AB' represents the correspondent having visited A and B and currently being in B. So, for example, the probability of going from A'B to ABC is 0.2 and the probability of going from AB' to ABC is 0.15.

When faced with these types of questions, one way to try and approach it is to think about whether there are any distinct 'groups' or 'types' within a particular state for which we have separate information. Here it is the two possibilities within the state AB, being in A or B.

This is similar to the perhaps more common example where we have no claims discount system (NCD) where say whether a driver moves down one or two levels of discount from a particular state when they have a claim depends on whether they had a claim in the prior year. In this case, knowing the current level of discount of the driver is not sufficient to determine transition probabilities as it depends on where they were previously. We have two 'groups' of drivers in this state, those that came up from not having a claim in the prior year and those that came down from having a claim the prior year. The transition probabilities out of this state are different for these two types of drivers so we need to split this into two.

Hope this helps!

Andy

 

[Priyanka Malhotra]

Hi Andy, thank you so much for explaining this in depth!