Appetizer: Statistical methods in XSPEC

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Appetizer Statistical methods in
XSPEC Entrée A modest proposal on
Bayesian methods
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XSPEC compares data with parametrized theoretical
models modified by the instrumental response. It
has been traditional (since the mid-60s) to use
c2 to do this comparison and perform frequentist
calculation of confidence regions for model
parameters. XSPEC does allow different options
for the estimator for the variance used as the
denominator in c2. As is well known, c2 is not
appropriate when there are small numbers of
counts in a spectral bin. XSPEC includes the max.
likelihood statistic of Cash with an extension to
work in the presence of background.
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Although most X-ray spectroscopy follows the
frequentist path, most astronomers then interpret
the results in a Bayesian fashion. For those who
wish to be more consistent, XSPEC has
capabilities to perform Bayesian analysis. It is
based on Tom Loredos likelihood for a Poisson
source in the presence of background. The model
parameters can be assigned constant or
exponential priors. The posterior pdf can be
constructed using Markov Chain Monte Carlo
chains.
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What is required to support widespread use of
Bayesian methods in astronomy ? Is there
infrastructure we can put in place which will
make it easier to use Bayesian methods ?
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The problem is (almost) always what to choose as
the prior pdf. The appropriate part of the
statistics literature is full of arguments on how
to choose your prior. However, in astronomy the
prior pdf is usually the posterior pdf of the
previous observation. But in general we dont
save the posterior pdfs and make them publically
available.
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We need a standard file format for probability
density functions. This will provide a method
(and incentive) for those performing Bayesian
analysis to save their posteriors. These can then
be used as input by themselves or others as
priors when new data becomes available.
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• Two possible ways of saving pdfs.
• A multi-dimensional grid
• MCMC chain
• Perhaps need standard file formats for both. Also
  need considerable thought on what other
  information should be included to ensure that
  these pdfs will not be misused.