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25 May 2006
Dirichlet Spaces and Metropolis Traces
Drew Thomas
A problem I've had come up again and again is the ability to explore a space bound by a Dirichlet prior with a Metropolis-type algorithm. I've yet to find a satisfactory answer and I'm hoping someone else will have some insight.
The research question I have deals with allocating patients to hospitals, considering the effect of the number of beds - one example of the "supply-induced demand" question. (The analysis is being done under Prof. Erol Pekoz, who's visiting Harvard Stats this year.) Conjugate priors for this problem have eluded me, and so the quantity of interest, the probability that a patient will be sent to a particular hospital for inpatient care, is being inferred through a Metropolis algorithm.
Here's the thing: there are at most 64 different hospitals to which a patient can be assigned. Even after assuming that if a hospital has not yet received a patient from a particular area they won't ever, the number of hospitals is extreme.
One suggested proposal has been a Dirichlet distribution with parameters equal to the current values, times a constant. That way the expected value of the proposal will be the same as the last draw. However, when the number is too low, the smallest dimensions will have parameter value less than 1, which leads to trouble, as the value will tend to zero; when it's too high, the biggest parameters don't move at all, and the effect of moving some of its mass is lost.
I've considered implementing a parallel-tempering method but I'd like to keep it cleaner. Does anyone have a better method that's reasonably quick to run, rather than monkeying with each parameter individually?
Posted by Andrew C. Thomas at May 25, 2006 6:00 AM