|How To Win with Non-Gaussian Data: Poisson Goodness-of-Fit
|371, Statistical Challenges in Modern Astronomy IV
|Connors, A.; van Dyk, D.A.
|We propose a new method to test for goodness-of-fit of a model for low-count Poisson data. Our approach does not resemble the usual methods of approximations to χ2, but instead explicitly uses the full Poisson distribution. First, we propose to use a simple (Poisson-specific) multiscale model to characterize the “mismatch” between a best-fit physical model and the data. Next, we embed this multiscale model into a probabilistic/ likelihood framework (via hierarchical Bayes), allowing us to handle statistical uncertainties. We then use MCMC to map out the shape of the joint posterior probability of all of the unknown parameters. Finally, we note that this is a generalization of a problem with a known solution: whether an additional component of known shape is justified by the data. Hence, when the multiscale structure that we use to model the “mismatch” between the data and the physical model accounts for a significant number of counts, and/or when the scale-factor for the best-fit physics model is significantly different than one, then the original fit was not “good-enough”. That is, the fitted multiscale “mismatch” isolates the discrepency between model and data. We use models of the Gamma-Ray sky as viewed by CGRO/EGRET as our example. We demonstrate that our method works nicely on several examples, but further work is needed to investigate the method’s power.
The authors and CHASC acknowledge support from the Chandra X-Ray Center, NSF(DMS 04-06085), and AISRP NAG5-12082. It is a particular pleasure to acknowledge the SAMSI06 Special Program in Astrostatistics.