Probabilistic Cross-Identification of Astronomical Sources

Tamás BudaváriAlexander S. Szalay

María Nieto-Santisteban

The Johns Hopkins University

10/26/2007 Tamás Budavári 2

MotivationThe problem

Cross-identification of sources in N number of catalogs

Current practice2-way matching by some radius cut based on σ, etc.N-way matching via some chaining rules

We needReliable measure of quality, e.g., to make sensible cutsUnification w/ physical measurements, modelling & priorsMethodology symmetric in the catalogs

10/26/2007 Tamás Budavári 3

Cross-Identification

What is the right question?How good…What is the probability…What is the observational evidence… ??

10/26/2007 Tamás Budavári 4

Cross-Identification

What is the right question?How good…What is the probability…What is the observational evidence…

Bayesian hypothesis testingIntroducing the Bayes factor

??

10/26/2007 Tamás Budavári 5

Bayesian View of Astrometry

Astrometric precision

10/26/2007 Tamás Budavári 6

Bayesian View of Astrometry

Astrometric precision

Where is the object?

10/26/2007 Tamás Budavári 7

Hypothesis Testing

The Bayes factor

H: the sources are from the same object

K: sources might be from separate objects

10/26/2007 Tamás Budavári 8

Hypothesis Testing

The Bayes factor

H: the sources are from the same object

K: sources might be from separate objects

10/26/2007 Tamás Budavári 9

Hypothesis Testing

The Bayes factor

H: the sources are from the same object

K: sources might be from separate objects

10/26/2007 Tamás Budavári 10

Astrometry:

Analytic results:

Normal Distribution

10/26/2007 Tamás Budavári 11

Astrometry:

Analytic results:

For the typical large weights and small separations

Normal Distribution

10/26/2007 Tamás Budavári 12

Tw

o-W

ay M

atch

ing

1-1 1-2 2-2

10/26/2007 Tamás Budavári 13

From Priors to Posteriors

Bayes factor provides the linkWhen H and K are complement

Simple picture for prior2-way: 1/Nn-way: 1/Nn-1

10/26/2007 Tamás Budavári 14

Uniform Prior

Partial overlap on sky

Footprint intersection

Radial selection fnSubset of sources

11

XX

22

21

)(NN

NHP X

10/26/2007 Tamás Budavári 15

Sky Coverage

Refines the prior PDF on the locationSimple scaling inside footprint: BA= B×(A/4)n-1

Edge correction affects small fraction

Changes the prior probability of HSmaller footprint, larger prior: P(H) ~ (A/4)1-n

Cancellation in posterior probability

10/26/2007 Tamás Budavári 16

Other Physical Input

Multi-color photometry commonModel for SEDs and filter transmissionsModel for photometric accuracy

Can fold in other measurementsStraightforward and completely separated

10/26/2007 Tamás Budavári 17

Efficient Incremental Evaluation

Recycle fast two-way matching tools

Recursive computation

10/26/2007 Tamás Budavári 18

Summary

Theoretically any astrometric modelBayesian hypothesis testing w/ generic PDFsProbabilistic interpretation of results

Spherical normal distribution is easyAnalytical formula for the observational evidence

Straightforward to fold in the physicsFor example, SED modelling and photometric errors

Efficient evaluation via fast 2-way toolsRecursive algorithm for high performance apps

10/26/2007 Tamás Budavári 19