  Over the past 10 years Bayesian methods have rapidly grown more popular as
several computationally intensive statistical algorithms have become feasible
with increased computer power. In this paper, we begin with a general
description of the Bayesian paradigm for statistical inference and the various
state-of-the-art model fitting techniques that we employ (e.g., Gibbs sampler
and Metropolis- Hastings). These algorithms are very flexible and can be used
to fit models that account for the highly hierarchical structure inherent in
the collection of high-quality spectra and thus can keep pace with the
accelerating progress of new space telescope designs. The methods we develop,
which will soon be available in the CIAO software package, explicitly model
photon arrivals as a Poisson process and, thus, have no difficulty with high
resolution low count X-ray and gamma-ray data. We expect these methods to be
useful not only for the recently launched Chandra X-ray observatory and XMM but
also new generation telescopes such as Constellation X, GLAST, etc. In the
context of two examples (Quasar S5 0014+813 and Hybrid-Chromosphere Supergiant
Star alpha TrA) we illustrate a new highly structured model and how Bayesian
posterior sampling can be used to compute estimates, error bars, and credible
intervals for the various model parameters.
