Integration of product of functions on a probability simplex
I saw this trick in Wolpert and Wolf 1995, and I thought I would share it.
Given a product of functions over a probability mass function where K is the cardinality of alphabets, and real valued function , we want to compute the integral of this over the probability simplex ( are non-negative and sum up to 1). The trick is to realize that the integral can be written in convolutions, and use the convolution theorem for Laplace transform.
Note that is not integrated, since it is determined by the rest of the variables. Define . Note that . Rewrite the integral as,
Recognizing the convolution, it can be simplified as,
Now, just apply the Laplace transform, and you are good to go!
This trick is useful for Bayesian integrals on Dirichlet prior.
- David Wolpert, David Wolf. Estimating functions of probability distributions from a finite set of samples. Physical Review E, Vol. 52, No. 6. (December 1995), pp. 6841-6854.