Advanced Methods for Studying Astronomical Populations: Inferring Distributions and Evolution of Derived (not Measured!) Quantities Brandon C. Kelly (CfA, Hubble Fellow, 1/7/10AAS Jan 2010,

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) 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) 1/7/10AAS Jan 2010, But all we can observe (measure) is the light (flux density) and location of sources on the sky!

Simple vs. Advanced Approach Simple but not Self-consistent 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 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 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 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) 1/7/10AAS Jan 2010,

Example: Fitting a Luminosity Function via MCMC techniques 1/7/10AAS Jan 2010, Play luminosity function movie Luminosity Redshift Flux Limit Luminosity Redshift Intrinsic Distribution of Measurables Intrinsic Distribution of Measurables Selection Effects Selection Effects Observed Distribution of Measurables Observed Distribution of Measurables

More Complicated Example: The Quasar Black Hole Mass Function 1/7/10AAS Jan 2010, Luminosity Redshift Flux Limit Luminosity Redshift Intrinsic Distribution of Measurables Intrinsic Distribution of Measurables Selection Effects Selection Effects Observed Distribution of Measurables Observed Distribution of Measurables Intrinsic Distribution Of Derived Quantities Intrinsic Distribution Of Derived Quantities Black Hole Mass Eddington Ratio Emission Line Width Emission Line Width Emission Line Width Emission Line Width Play BHMF Animation

Summary and Additional Resources Gelman et al., Bayesian Data Analysis, 2004, (2 nd 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 (2 nd Ed.; Wiley) Gelman et al., Bayesian Data Analysis, 2004, (2 nd 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 (2 nd Ed.; Wiley) 1/7/10AAS Jan 2010, 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