Markov-Chain Monte Carlo

1
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
• Sum histogram from best-fit value (often peak of
  histogram) in both directions
• Stop when x of values summed for an x
  confidence interval

2
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

3
Metropolis-Hastings

• 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)

4
M-H Issues

• Can be very slow to converge, especially when
  there are correlated variables
• Use multivariate proposal distributions (done in
  XSPEC approach)
• Transform correlated variables
• Convergence
• Run multiple chains, compute convergence
  statistics

5
MCMC Example

• 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)

6
Sanity Check Fitting local 60 mm LF
F
Fit Saunders et al (1990) LF assuming Gaussian
errors and ignoring upper limits Param. S1990
MCMC a 1.09 0.12
1.04 0.08 s 0.72 0.03 0.75
0.02 F 0.026 0.008 0.026 0.003 log
L 8.47 0.23 8.39 0.15
log L/L?
7
(Ugly) Posterior Probabilities
zlt 0.5 X-ray luminosity functions
Early-type Galaxies
Late-type Galaxies
Red crosses show 68 confidence interval
8
Marginalized Posterior Probabilities
Dashed curves show Gaussian with same mean st.
dev. as posterior
Dotted curves show prior
log L
log f
s
a
s
a
Note a and s tightly constrained by (Gaussian)
prior, rather than being fixed
9
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.
10
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)
11
Future

• Use physical priors have posterior from
  previous work be prior for current work
• 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