1. Distributions and Uncertainties
Distribution and Uncertainty of Hydrogen (DUH)
Tom Aldcroft, David van Dyk,
Aneta Siemiginowska
and the
ChaMP collaboration

2. 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
Typical
Measurement
Probability
Accurate
measurement
Poor NH
Distribution NH

3. (binned best-fit values)
Unobscured
Obscured
Flux dependence of
NH distribution
(binned best-fit values)
Flux
Faint
Bright
1020
1024
1023
1022
1021
Intrinsic absorption NH

4. 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)

5. 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

6. Results Intrinsic absorption NH Flux Unobscured Obscured Faint Bright
1020
1021
1022
1023
1024
Intrinsic absorption NH

7. 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