Hypothesis testing for Poisson vs. geometric distributions using Stochastic complexity

We illustrate the concept of hypothesis testing using stochastic complexity, in the modern sense of normalized maximum likelihood codes, via the simple example of deciding whether a Poisson or a geometric model better matches the collected data. The Poisson model is generally found to have more power in describing data than the geometric model. Hence, the Poisson model is more harshly penalized by the stochastic complexity criterion. The integral of the square root of the Fisher information of both the Poisson and geometric models is found to be infinite. Hence, the allowed parameter range must be restricted somehow to make this integral finite. Some of the consequences of this are explored.