#acl All:read

This page shows a simple example of MCMC analysis in XSPEC. We use for data file1.pha in spectral/session and a simple absorbed power-law for the model :
{{{
XSPEC12> data file1
XSPEC12> model phabs(pow)
}}}
start by doing a fit
{{{
XSPEC12> fit
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to give the result
{{{
========================================================================
Model phabs<1>*powerlaw<2> Source No.: 1   Active/On
Model Model Component  Parameter  Unit     Value
 par  comp
   1    1   phabs      nH         10^22    9.5831062      +/-  0.29171265     
   2    2   powerlaw   PhoIndex            1.3142755      +/-  3.3257374E-02  
   3    2   powerlaw   norm                0.12037190     +/-  8.2219031E-03  
________________________________________________________________________


 Chi-Squared =          78.09 using 64 PHA bins.
 Reduced chi-squared =          1.280 for     61 degrees of freedom 
 Null hypothesis probability =   6.917881e-02
}}}
We will start by estimating a good proposal distribution. Start by using a multivariate cauchy using the covariance matrix from the fit. Set up the chain length and burn-in for a short test chain and run it :
{{{
XSPEC12> chain proposal cauchy fit
XSPEC12> chain length 1000
XSPEC12> chain burn 0
XSPEC12> chain run test1.fits
}}}
To find out how well the chain worked we use chain stat on the first parameter :
{{{
 Means in chains :      9.6619207
 Mean over all chains :      9.6619207 and variance of chain means :   0.
 Variance over all chains :    0.075465715
 Rubin-Gelman convergence measure :     0.99900000
 Fraction of repeated values: 0.823000  
     (rule of thumb target is 0.75)
}}}
This looks pretty good: the fraction of repeated values is close to the target of 0.75. Let's do another test by halving all the covariance matrix values and see what effect that has :
{{{
XSPEC12> rescalecov 0.5
XSPEC12> chain run test2.fits

XSPEC12> chain stat 1
 Means in chains :      9.6619207     9.5525420
 Mean over all chains :      9.6072314 and variance of chain means :   0.0059818499
 Variance over all chains :    0.089958227
 Rubin-Gelman convergence measure :      1.0655624
 Fraction of repeated values: 0.823000  0.695000  
     (rule of thumb target is 0.75)
}}}
This looks OK so let's try a longer run :
{{{
XSPEC12> chain length 50000
XSPEC12> chain burn 1000
XSPEC12> chain run run1.fits
}}}