Markov-Chain Monte Carlo. Instead of integrating, sample from the posterior. The histogram of chain values for a parameter is a visual representation of the (marginalized) probability distribution for that parameter. Can then easily compute confidence intervals:
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Instead of integrating, sample from the posterior
The histogram of chain values for a parameter is a visual representation of the (marginalized) probability distribution for that parameter
Fit Saunders et al (1990) LF
assuming Gaussian errors and
ignoring upper limits
Param. S1990 MCMC
α 1.09 ± 0.12 1.04 ± 0.08
σ 0.72 ± 0.03 0.75 ± 0.02
Φ* 0.026 ± 0.008 0.026 ± 0.003
log L* 8.47 ± 0.23 8.39 ± 0.15
z< 0.5 X-ray luminosity functions
Red crosses show 68% confidence interval
Dashed curves show Gaussian with same mean & st. dev. as posterior
Dotted curves show prior
Note: α and σ tightly constrained by (Gaussian) prior, rather than being “fixed”
XSPEC MCMC is based on the Metropolis-Hastings algorithm. The chain proposal command is used to set the proposal distribution.
MCMC is integrated into other XSPEC commands (e.g., error). If chains are loaded then these are used to generate confidence regions on parameters, fluxes and luminosities. This is more accurate than the current method for estimating errors on fluxes and luminosities.
Histogram and probability density plot (2-d histogram) of spectral fit parameters from an XSPEC MCMC run produced by fv (see https://astrophysics.gsfc.nasa.gov/XSPECwiki)