This is a follow up to an earlier question that I asked here:
In that question I asked how I could estimate a finite distribution of probabilities for a six sided die, and update them as I learned the outcome of rolls. I was directed towards Dirichlet distribution and it is ideal for this. My prior distribution is Dir(1,1,1,1,1,1), then after each roll outcome I update by adding 1 to the relevant parameter. This is ideal since I can calculate credible intervals, the expected value of various functions over the whole simplex, etc.
However I have realised that the real situation is somewhat more complicated.
Suppose that each outcome is not the roll of the die, but a roll of a modified die (modified in a known way). For instance, on certain rolls a 1 could be prohibited (In other words a copy of the die is rolled which has p1 set to 0, the other probabilities scaled to add up to 1). On certain other rolls the probability of 3 is doubled (relative to the other probabilities). To avoid problems let's suppose all probabilities of the original die are strictly between 0 and 1.
For example, say that on the first roll the die is modified so that an odd roll is not permitted. If the actual die has probabilities p1,p2,p3,p4,p5,p6 then this modified die will only have non-zero probabilities for landing on 2, 4 or 6, and these will be p2/(p2+p4+p6), p4/(p2+p4+p6), p6/(p2+p4+p6).
Is there still a way to update my Dirichlet Distribution with the outcomes of these 'modified rolls'? What is the most feasible approach?