Distribution and Uncertainty of
Distributions and Uncertainties
Hydrogen (DUH)
Tom Aldcroft, David van Dyk,
Aneta Siemiginowska
and the
ChaMP collaboration
Distribution function (with confidence limits) from an
ensemble of uncertain measurements
Based on Chandra spectra of 900 AGN, estimate how distribution of
intrinsic absorbing column (NH) depends on source flux
Some AGN have well constrained values of NH while others (with very
few counts) give poor constraints
Accurate
measurement
NH
NH
Distribution
Probability
Typical
Measurement
NH
Poor
measurement
NH
Unobscured
Obscured
Flux dependence of
NH distribution
(binned best-fit values)
Faint
Flux
Bright
1020
1021
1022
1023
1024
Intrinsic absorption NH
Need uncertainties!
Use Gibbs sampler to construct a Markov chain with stationary
distribution equal to the posterior distribution
Xi = Ideal measurement of NH for ith AGN (i=1,..,Nagn)
xi,b = Observed probability distribution that ith AGN has NH in bin b
pb => Proportion of AGN having NH in bin b , the histogram with
b=1,..,B
Initialize pb = 1/B
Sample Xi for each AGN given xi,b and pb
Xi ~ Multinomial ( ( pb * xi,b ) / sum( pb * xi,b ) ) ( * is convolution)
Sample pb from its distribution
pb ~ chi^2 [ df = 2*(1 + # of Xi in bin b) ]
pb = pb / sum(pb)
Need uncertainties!
Use Gibbs sampler to construct a Markov chain with stationary
distribution equal to the posterior distribution
Randomly sample probability distribution for each AGN, weighting by
current estimate of NH histogram
Build up NH histogram
Resample each bin of NH histogram with c2 distribution
Iterate to convergence
Confidence limits for each bin from spread of points in Markov chain
Results
Unobscured
Obscured
Faint
Flux
Bright
1020
1021
1022
1023
1024
Intrinsic absorption NH
Summary
We have developed a method to calculate a distribution and
uncertainties for each bin when a probability distribution is available
for each input datum