Transcript MCMC in XSPEC

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:
1. Sum histogram from best-fit value (often peak of histogram)
in both directions
2. Stop when x% of values summed for an x% confidence
interval
Gibbs Sampling
1. Derive conditional probability of each
parameter given values of the other parameters
2. Pick parameter at random
3. Draw from conditional probability of that
parameter given values of all other parameters
from previous iteration
4. Repeat until chain converges
Metropolis-Hastings
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Can be visualized as similar to the rejection
method of random number generation
Use a “proposal” distribution that is similar in
shape to the expected posterior distribution to
generate new parameter values
Accept new step when probability of new values
is higher, occasionally accept new step otherwise
(to go “up hill”, avoiding relative minima)
M-H Issues
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Can be very slow to converge, especially when
there are correlated variables
Use multivariate proposal distributions (done in
XSPEC approach)
 Transform correlated variables
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Convergence
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Run multiple chains, compute convergence statistics
MCMC Example
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In Ptak et al. (2007) we used MCMC to fit the
X-ray luminosity functions of normal galaxies in
the GOODS area (see poster)
Tested code first by fitting published FIR
luminosity function
Key advantages:
visualizing full probability space of parameters
 ability to derive quantities from MCMC chain value
(e.g., luminosity density)
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Sanity Check: Fitting local 60 mm LF
Φ
Fit Saunders et al (1990) LF
assuming Gaussian errors and
ignoring upper limits
Param. S1990
α
1.09 ± 0.12
σ
0.72 ± 0.03
Φ*
0.026 ± 0.008
log L* 8.47 ± 0.23
MCMC
1.04 ± 0.08
0.75 ± 0.02
0.026 ± 0.003
8.39 ± 0.15
log L/L○
(Ugly) Posterior Probabilities
z< 0.5 X-ray luminosity functions
Early-type Galaxies
Late-type Galaxies
Red crosses show 68% confidence interval
Marginalized Posterior Probabilities
Dashed curves show Gaussian with same mean & st. dev. as posterior
Dotted curves show prior
log φ*
a
log L*
s
a
s
Note: α and σ tightly constrained by (Gaussian) prior, rather than being “fixed”
MCMC in XSPEC
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.
XSPEC MCMC Output
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)
Future
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Use “physical” priors… have posterior from previous
work be prior for current work
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Use observed distribution of photon indices of nearby AGN
when fitting for NH in deep surveys
Incorporate calibration uncertainty into fitting (Kashyap
AISR project)
XSPEC has a plug-in mechanism for user-defined
proposal distributions… would be good to also allow
user-defined priors
Code repository/WIKI for MCMC analysis in
astronomy