1. Probabilistic Cross-Identification of Astronomical Sources Tamás Budavári Alexander S. Szalay María Nieto-Santisteban The Johns Hopkins University

2. 10/26/2007Tamás Budavári2 Motivation The problem Cross-identification of sources in N number of catalogs Current practice 2-way matching by some radius cut based on σ, etc. N-way matching via some chaining rules We need Reliable measure of quality, e.g., to make sensible cuts Unification w/ physical measurements, modelling & priors Methodology symmetric in the catalogs

3. 10/26/2007Tamás Budavári3 Cross-Identification What is the right question? How good … What is the probability … What is the observational evidence … ?

4. 10/26/2007Tamás Budavári4 Cross-Identification What is the right question? How good … What is the probability … What is the observational evidence … Bayesian hypothesis testing Introducing the Bayes factor ?

5. 10/26/2007Tamás Budavári5 Bayesian View of Astrometry Astrometric precision

6. 10/26/2007Tamás Budavári6 Bayesian View of Astrometry Astrometric precision Where is the object?

7. 10/26/2007Tamás Budavári7 Hypothesis Testing The Bayes factor H: the sources are from the same object K: sources might be from separate objects

8. 10/26/2007Tamás Budavári8 Hypothesis Testing The Bayes factor H: the sources are from the same object K: sources might be from separate objects

9. 10/26/2007Tamás Budavári9 Hypothesis Testing The Bayes factor H: the sources are from the same object K: sources might be from separate objects

10. 10/26/2007Tamás Budavári10 Astrometry: Analytic results: Normal Distribution

11. 10/26/2007Tamás Budavári11 Astrometry: Analytic results: For the typical large weights and small separations Normal Distribution

12. 10/26/2007Tamás Budavári12 Two-Way Matching 1-1 1-22-2

13. 10/26/2007Tamás Budavári13 From Priors to Posteriors Bayes factor provides the link When H and K are complement Simple picture for prior 2-way: 1/N n-way: 1/N n-1

14. 10/26/2007Tamás Budavári14 Uniform Prior Partial overlap on sky Footprint intersection Radial selection fn Subset of sources 1 X 2

15. 10/26/2007Tamás Budavári15 Sky Coverage Refines the prior PDF on the location Simple scaling inside footprint: B A = B×(A/4  ) n-1 Edge correction affects small fraction Changes the prior probability of H Smaller footprint, larger prior: P(H) ~ (A/4  ) 1-n Cancellation in posterior probability

16. 10/26/2007Tamás Budavári16 Other Physical Input Multi-color photometry common Model for SEDs and filter transmissions Model for photometric accuracy Can fold in other measurements Straightforward and completely separated

17. 10/26/2007Tamás Budavári17 Efficient Incremental Evaluation Recycle fast two-way matching tools Recursive computation

18. 10/26/2007Tamás Budavári18 Summary Theoretically any astrometric model Bayesian hypothesis testing w/ generic PDFs Probabilistic interpretation of results Spherical normal distribution is easy Analytical formula for the observational evidence Straightforward to fold in the physics For example, SED modelling and photometric errors Efficient evaluation via fast 2-way tools Recursive algorithm for high performance apps

19. 10/26/2007Tamás Budavári19

20. 10/26/2007Tamás Budavári20

21. 10/26/2007Tamás Budavári21 Astronometry: Analytic results: For the typical large weights and small separations Normal Distribution

22. 10/26/2007Tamás Budavári22 Astronometry: Analytic results: For the typical large weights and small separations Normal Distribution

23. 10/26/2007Tamás Budavári23 Astronometry: Analytic results: For the typical large weights and small separations Normal Distribution

24. 10/26/2007Tamás Budavári24 In case of 2 catalogs In case of 3 catalogs Normal Distribution