Brandon C. Kelly (CfA, Hubble Fellow, bckelly@cfa.harvard.edu)

1. Advanced Methods for Studying Astronomical Populations: Inferring Distributions and Evolution of Derived (not Measured!) Quantities Brandon C. Kelly (CfA, Hubble Fellow, bckelly@cfa.harvard.edu) AAS Jan 2010, bckelly@cfa.harvard.edu

2. Goal of Many Surveys: Understand the distribution and evolution of astronomical populations • Understand the growth and evolution of black holes, and its relation to galaxy evolution • E.g., infer the BH mass function, accretion rate distribution, and the spin distribution • Understand how the stellar mass of galaxies is assemble • E.g., infer the stellar mass function, star formation histories of galaxies (red sequence vs. blue cloud) But all we can observe (measure) is the light (flux density) and location of sources on the sky! AAS Jan 2010, bckelly@cfa.harvard.edu

3. Simple vs. Advanced Approach Simple but not Self-consistent Advanced and Self-Consistent Derive distribution and evolution of quantities of interest directly from observed distribution of measurable quantities Circumvents fitting of individual sources Self-consistently accounts for uncertainty in derived quantities and selection effects (e.g., flux limit) • Derive ‘best-fit’ estimates for quantities of interest (e.g., mass, age, BH spin) • Do this individually for each source • Infer distribution and evolution directly from the estimates • Provides a biased estimate of distribution and evolution AAS Jan 2010, bckelly@cfa.harvard.edu

4. Example: Fitting a Luminosity Function via MCMC techniques Intrinsic Distribution of Measurables Observed Distribution of Measurables Selection Effects Luminosity Luminosity Flux Limit Redshift Redshift Play luminosity function movie AAS Jan 2010, bckelly@cfa.harvard.edu

5. More Complicated Example: The Quasar Black Hole Mass Function Intrinsic Distribution Of Derived Quantities Intrinsic Distribution of Measurables Selection Effects Observed Distribution of Measurables Black Hole Mass Luminosity Luminosity Emission Line Width Emission Line Width Flux Limit Eddington Ratio Redshift Redshift Play BHMF Animation AAS Jan 2010, bckelly@cfa.harvard.edu

6. Summary and Additional Resources • Gelman et al. , Bayesian Data Analysis, 2004, (2nd Ed.; Chapman-Hall & Hall / CRC) • Gelman & Hill, Data Analysis Using Regression and Multilevel/Hierarchical Models, 2006 (Cambridge Univ. Press) • Kelly et al., A Flexible Method for Estimating Luminosity Functions, 2008, ApJ, 682, 874 • Kelly et al., Determining Quasar Black Hole Mass Functions from their Broad Emission Lines: Application to the Bright Quasar Survey, 2009, ApJ, 692, 1388 • Little & Rubin, Statistical Analysis with Missing Data, 2002 (2nd Ed.; Wiley) Bottom Line: When inferring distributions of derived quantities (e.g., mass, age, spin), one cannot simply calculate the distribution of the best-fit values. Instead, it is necessary to find the set of distributions for the derived quantity (e.g., mass) that are consistent with the observed distribution of the measurable quantity (e.g., flux). References and Further Reading AAS Jan 2010, bckelly@cfa.harvard.edu