Transcript Presentation

Appetizer:
Statistical methods in XSPEC
Entrée:
A modest proposal on Bayesian methods
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.
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 Loredo’s
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.
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 ?
Likelihood
Prior
p(H|D,I) = p(D|H,I) x p(H|I)
p(D|I)
Posterior
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 don’t save the posterior
pdfs and make them publically available.
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.
Two possible ways of saving pdfs.
1. A multi-dimensional grid
2. 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.