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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). Goal of Many Surveys: Understand the distribution and evolution of astronomical populations.

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Brandon c kelly cfa hubble fellow bckelly cfa harvard edu

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


Goal of many surveys understand the distribution and evolution of astronomical populations
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


Simple vs advanced approach
Simple vs. Advanced Approach evolution of astronomical populations

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


Example fitting a luminosity function via mcmc techniques
Example: Fitting a Luminosity Function via MCMC techniques evolution of astronomical populations

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


More complicated example the quasar black hole mass function
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


Summary and additional resources
Summary and Additional Resources Function

  • 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