## Importance sampling reweighting from an infinite mixture model to a finite mixture model …

I finally derived an expression for the Radon-Nikodym derivative of a k component finite Normal mixture model with respect to an equivalent infinite dimensional (Dirichlet process prior) mixture model; this was required for the revisions to my recursive marginal likelihood estimation paper.  One half of the formula is this horrible beast of combinatorial nastiness:

Surprisingly, it actually works (i.e. gives the correct solution for a couple of well-known benchmarks) and allows me to demonstrate practically the potential for importance sample reweighting over stochastic process priors.  A successful week!

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### One Response to Importance sampling reweighting from an infinite mixture model to a finite mixture model …

1. Congrats. I must say, that -is- warty-lookin’. I wonder if there is some kind of geometrical summary of it possible.

Successful week here, too: I am given a tabulation of counts for M categories, one tabulation per hour, the counts all coming from the same rich source. I am assuming the categories are independent (at least at first, intending to deal with depending using hyperpriors later). I use a Bayesian bootstrap to estimate the probabilities of choosing each of the Categories, right now taking a mean of the bootstrap replica probabilities and then renormalizing, even if that is barely necessary. I then repeat the process for the next hour.

I have done this from a huge, comparable parent population similar to but not including the specific subpopulations and wrangled that into a kind of empirical prior. I then use that prior in conjunction with the recipe above. I have also learned, from the parent, an ordering of the Categories from most common to least, and use that in presentations.