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
Derive conditional probability of each parameter given values of the other parameters
Pick parameter at random
Draw from conditional probability of that parameter given values of all other parameters from previous iteration
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 LFΦ
Fit Saunders et al (1990) LF
assuming Gaussian errors and
ignoring upper limits
Param. S MCMC
α ± ± 0.08
σ ± ± 0.02
Φ* ± ± 0.003
log L* ± ± 0.15
log L/L○

7. (Ugly) Posterior Probabilities
z< 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 φ*
log L* s a a s
Note: α and σ 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

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