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David A. van Dyk, Rostislav Protassov (Harvard University), Vinay L. Kashyap, Aneta Siemiginowska (Harvard-Smithsonian Center for Astrophysics), Alanna Connors (Wellesley College)
Recently Bayesian methods have grown rapidly in popularity in many scientific disciplines as several computationally intensive statistical algorithms have become feasible with modern computer power. In this paper, we demonstrate how we have employed these state-of-the-art techniques (e.g., Gibbs sampler and Metropolis-Hastings) to fit today's high-quality, high resolution astrophysical spectral data. These algorithms are very flexible and can be used to fit models that account for the highly hierarchical structure in the collection of high-quality spectra and thus can keep pace with the accelerating progress of new telescope designs. We explicitly model photon arrivals as a Poisson process and, thus, have no difficulty with high resolution low count X-ray and gamma-ray data. These methods will be useful not only for the soon-to-be-launched Chandra X-ray observatory but also such new generation telescopes as XMM, Constellation X, and GLAST.
We also explicitly incorporate the instrument response (e.g. via a response matrix and effective area vector), plus background contamination of the data. In particular, we appropriately model the background as the realization of a second Poisson process, thereby eliminating the need to directly subtract off the background counts and the rather embarrassing problem of negative photon counts. The source energy spectrum is modeled as a mixture of a Generalized Linear Model which accounts for the continuum plus absorption and several (Gaussian) line profiles. Generalized Linear Models are the standard method for incorporating covariate information (as in regression) into non-Gaussian models and are thus an obvious but innovative choice in this setting. Using several examples, we illustrate how Bayesian posterior sampling can be used to compute point (i.e., ``best'') estimates of the various model parameters as well as compute error bars on these estimates and construct statistical tests.
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