Bayesian f lux reconstruction in one and two bands

Bayesian flux reconstruction in one and two bands

Statistical Challenges in Modern Astronomy V – June 13, 2011

Eric R. Switzer (KICP)T. M. Crawford, E. R. Switzer, W. L. Holzapfel, C. L. Reichardt, D. P. Marrone, and J. D. Vieira, “A Method for Individual Source Brightness Estimation in Single- and Multi-band Data” ApJ, 718:513–521, July 2010.

Intrinsically faint sources are much more probable than bright sources.

It is more likely that the observed flux is a dimmer source and a positive noise fluctuation: “deboost” the measured flux.

The source problem

Goal: posterior parameters {xi, Si,υ} given d and prior information.

SPT 2mm; J. Vieira

~1’ resolution, 1 deg.: normal galaxy is unresolved for z>0.05 (200 Mpc), “point sources”

The source problem

SPT 2mm; J. Vieira

Apply a matched filter, measure the flux Sm, but:

The single-band, multi-source posterior distribution

Goal:

Data model:

MeasurementLikelihood:

Where:

Prior:

Distinction here: posterior of the brightest individual source of flux in a resolution element.

Or, P(Ssmax) P(S>Ssmax)

(Scheuer 1957)

The two-band problem

Suggesting the posterior:

If background sources approximate a Gaussian distribution:

Optionally either Flux1, Flux2, or e.g. Flux1 and alpha.

(For experiments where the Gaussian assumption is not accurate, the full PDF can be developed as a 2D version of the single-band argument.)

The two-band posterior

4.2σ4.9σ

0.5σ

Deboosting in the flux plane

Deboosting in the flux plane

Region of large flux errors (not shown for simplicity)

Dust locu

s

Sync. l

ocus

Uses and extensions• Population counts, spectral

energy distributions, categorized source catalogs

• P(α > αd) > Pd

• Method to determine population counts from the flux PDFs of the catalog.

• A rigorous multi-band, multi-experiment counts method with appropriate prior information.

AGN-powered, synchrotron-dominated, falling spectrum (in frequency). Dust emission-dominated, rising spectrum.

Uses and extensions• Population counts, spectral

energy distributions, categorized source catalogs

• P(α > αd) > Pd

• Method to determine population counts from the flux PDFs of the catalog.

• A rigorous multi-band, multi-experiment counts method with appropriate prior information.

Uses and extensions• Population counts, spectral

energy distributions, categorized source catalogs

• P(α > αd) > Pd

• Method to determine population counts from the flux PDFs of the catalog.

• A rigorous multi-band, multi-experiment counts method with appropriate prior information.

Right way: P(D) – what is the PDF of underlying counts model parameters which explains the PDF of pixel fluxes?

References• This talk: T. M. Crawford, E. R. Switzer, W. L. Holzapfel, C. L. Reichardt, D. P. Marrone, and J.

D. Vieira, “A Method for Individual Source Brightness Estimation in Single- and Multi-band Data,” ApJ, 718:513–521, July 2010.

• Bayesian catalogs: P. Carvalho, G.Rocha, and M. P. Hobson, “A fast Bayesian approach to discrete object detection in astronomical data sets - PowellSnakes I,” MNRAS, 393:681–702, March 2009.

• P(D) method: Patanchon et al., “Submillimeter Number Counts from Statistical Analysis of BLAST Maps,” ApJ, 707:1750–1765, December 2009.

• PDF of source fluxes: P. A. G. Scheuer, “A statistical method for analysing observations of faint radio stars,” In Proceedings of the Cambridge Philosophical Society, volume 53 pages 764–773, 1957.

• Method here applied to SPT data: J. D. Vieira, T. M. Crawford, E. R. Switzer, et al. “Extragalactic Millimeter-wave Sources in South Pole Telescope Survey Data: Source Counts, Catalog, and Statistics for an 87 Square- degree Field,” ApJ, 719:763–783, August 2010.

• Deboosting in literature: K. Coppin, M. Halpern, D. Scott, C. Borys, and S. Chapman, “An 850-μm SCUBA map of the Groth Strip and reliable source extraction” MNRAS, 357:1022–1028, March 2005.

• https://github.com/eric-switzer/bayes_flux