Marginalization with Moment Generating Functions with Applications in Astrostatistics

We present a new analytical method to derive a likelihood function that is marginalised over a population. This method can be used for computational advantage in the context of Bayesian hierarchical models and marginal likelihood calculations in Bayesian models. The key innovation is the specification of the necessary integrals in terms of high-order (sometimes fractional) derivatives of the population prior moment-generating function, if particular existence and differentiability conditions hold.
We confine our attention to Poisson and gamma likelihood functions. Under Poisson likelihoods, the observed Poisson count determines the order of the derivative. Under gamma likelihoods, the shape parameter, which is assumed to be known, determines the order of the fractional derivative.
We also present examples validating this new analytical method. In some of the examples, the new method is the only known analytical method to calculate the integral, giving instantaneous and accurate calculations.

model evidences

fractional derivatives

moment-generating function

integration

Bayesian modeling 

International Society for Bayesian Analysis (ISBA)