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Abstract
Bayesian Analysis of X-ray
Luminosity Functions
Often only a relatively small number of sources of a given class are detected in Xray 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.
A. Ptak (JHU)
Advantages of Bayesian Data Analysis:
Markov-Chain Monte Carlo (MCMC) Simulation
Overview
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)
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.
Metropolis-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.
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
(Ugly) Posterior Probabilities
Luminosity Functions in the Poisson Limit
z< 0.5 Normal Galaxy X-ray Luminosity Functions
dV
N   ( L, z )dL
dz
dz
Red crosses show 68% confidence intervals
Marginalized
posterior
probabilities for
log-normal fit
parameters
Early-type Galaxies
Late-type Galaxies
Binned approximation (Page and Carrera 2000):
N ~  ( L, z ) 
Lmin
a

z max
z min
dV
C ( L, z )dL
dz   ( L, z )VL
dz
C(L,z) = completeness correction
Luminosity Function Posterior:
Red dash lines show “equivalent” Gaussian distribution with same
mean and st. dev. as posterior
log φ*
Lmax
log L*

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* (1+z)p:
Early type galaxies: p = 1.6 (0.5-2.7)
Late-type galaxies: p = 2.3 (1.5-3.1)
Prior
Likelihood
p( | D)   G(i | i ,  i )   Pois N j |  (1,..., n , LX , j , z j )V j L j 
n
m
i 1
j 1
G = Gaussian probability of observing model parameter θi 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
Note: α and σ 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)