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Distributions and Uncertainties

Tom Aldcroft, David van Dyk, Aneta Siemiginowska and the ChaMP collaboration. Distribution and Uncertainty of Hydrogen (DUH). Distributions and Uncertainties. Distribution function (with confidence limits) from an ensemble of uncertain measurements.

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Distributions and Uncertainties

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

  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 Accurate measurement Poor measurement Probability NH NH NH Distribution NH

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

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