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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
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Cross-Identification
What is the right question?How good…What is the probability…What is the observational evidence… ??
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Cross-Identification
What is the right question?How good…What is the probability…What is the observational evidence…
Bayesian hypothesis testingIntroducing the Bayes factor
??
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Bayesian View of Astrometry
Astrometric precision
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Bayesian View of Astrometry
Astrometric precision
Where is the object?
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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
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Astrometry:
Analytic results:
For the typical large weights and small separations
Normal Distribution
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Tw
o-W
ay M
atch
ing
1-1 1-2 2-2
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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
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Uniform Prior
Partial overlap on sky
Footprint intersection
Radial selection fnSubset of sources
11
XX
22
21
)(NN
NHP X
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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
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Other Physical Input
Multi-color photometry commonModel for SEDs and filter transmissionsModel for photometric accuracy
Can fold in other measurementsStraightforward and completely separated
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Efficient Incremental Evaluation
Recycle fast two-way matching tools
Recursive computation
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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
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