Bayesian Analysis of X-ray Luminosity Functions

1
Abstract Often only a relatively small number of
sources of a given class are detected in X-ray
surveys, requiring careful handling of the
statistics. We previously addressed this issue in
the case of the luminosity function of
normal/starburst galaxies in the GOODS area by
fitting the luminosity functions using
Markov-Chain Monte Carlo simulations. We are
expanding on this technique to include
uncertainties in the redshifts (often
photometric) and uncertainties in the spectral
energy distributions of the sources. Future work
will also include performing a Bayesian analysis
of the correlations between X-ray emission and
fluxes from other bands (particularly radio, IR
and optical/NIR) and also between X-ray
luminosity and star-formation rate and stellar
mass estimates.
Bayesian Analysis of X-ray Luminosity
Functions A. Ptak (JHU)
Advantages of Bayesian Data Analysis Posterior
probability distribution contains full
information of the model parameters ? can be
visualized with histograms, density plots (i.e.,
2-D histograms) ? statistically-correct errors
(confidence intervals) MCMC draws can be
directly manipulated to calculate derived
quantities and then analyzed in the same way as
the model parameters, e.g., computing luminosity
density from model values of luminosity
functions
Markov-Chain Monte Carlo (MCMC) Simulation
Overview Motivation Need to integrate posterior
probability distribution, rarely easy MCMC is
similar to a random walk each MCMC chain
iteration is a set of model parameters drawn from
the posterior probability distribution. Metropoli
s-Hastings uses a proposal distribution to
select model parameters, with the proposal
distribution chosen to match final posterior
distribution shape, e.g., often a Gaussian is
used since in many cases the final posterior is
also close to Gaussian. Can be very inefficient
(and hence require a very large number of draws),
especially if some model parameters are
correlated or have posterior probability
distributions with wide wings, but only requires
evaluation of the posterior for a given set of
model parameters.
Luminosity Functions in the Poisson Limit
Marginalized posterior probabilities for
log-normal fit parameters
Binned approximation (Page and Carrera 2000)
C(L,z) completeness correction
Luminosity Function Posterior
Red dash lines show equivalent Gaussian
distribution with same mean and st. dev. as
posterior Change in L between z 0.25 and z
0.75 was found to have an Gaussian distribution
and independent of whether LFs were fit with
Schechter or log-normal functions. Main result
assuming pure luminosity evolution L?
(1z)p Early type galaxies p 1.6
(0.5-2.7) Late-type galaxies p 2.3 (1.5-3.1)
Prior
Likelihood
log f
log L
G Gaussian probability of observing model
parameter ?i with mean ? and st. dev. and ? Nj
Number of galaxies detected in luminosity
function bin j ?Vj Co-moving volume integrated
over bin j ?Lj Luminosity interval of bin j
s
a
Note a and s tightly constrained by (Gaussian)
prior, rather than being fixed

• Future Work
• Include the 1 Ms of additional CDF-S data
  (funded through the special CDF-S archival
  special call)
• Include Spitzer and VLA data into the galaxy
  selection, star-formation rate estimation, and
  photometric zs
• Incorporate uncertainties in z and source
  spectral shape into the analysis (by adding terms
  to likehood)
• Bayesian correlation studies for LX/SFR and
  LX/stellar mass (see Lehmer et al. 2008,
  astro-ph/0803.3620)

For more details, see Ptak et al. (2007, ApJ,
667, 826)