Over the past ten years Bayesian methods have rapidly grown more popular in many scientific disciplines 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. , the Gibbs sampler and the Metropolis-Hastings algorithm ) .
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 Chandra Interactive Analysis of Observations ( CIAO ) software , 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 .
Application of our method to the high-energy tail of the ASCA spectrum of \alpha TrA confirms that even at a quiescent state , the coronal plasma on this hybrid-chromosphere star is indeed at high temperatures ( > 10 MK ) that normally characterize flaring plasma on the Sun .
We are also able to constrain the coronal metallicity , and find that though it is subject to large uncertainties , it is consistent with the photospheric measurements .