Distributions%20and%20Uncertainties

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Distributions and Uncertainties

• Tom Aldcroft, David van Dyk,
• Aneta Siemiginowska
• and the
• ChaMP collaboration

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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

Distribution
NH
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Flux dependence of NH distribution (binned
best-fit values)
Intrinsic absorption NH
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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
  (i1,..,Nagn)
• xi,b Observed probability distribution that ith
  AGN has NH in bin b
• pb gt Proportion of AGN having NH in bin b , the
  histogram with b1,..,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 chi2 df 2(1 of Xi in bin b)
• pb pb / sum(pb)

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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

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Results
1024
1023
1020
1022
1021
Intrinsic absorption NH
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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