Probabilistic Catalogs and beyond...

Transcription

1 Department of Statistics The University of Auckland brewer/

2 Probabilistic catalogs So, what is a probabilistic catalog? And what s beyond?

3 Finite Mixture Models

4 Finite Mixture Models These are what statisticians call finite mixture models. They have a lot of potential in astronomy.

5 Finite Mixture Models These are what statisticians call finite mixture models. They have a lot of potential in astronomy. The data have something to do with a sum of an unknown number of components, whose properties we want to know.

6 Finite Mixture Models These are what statisticians call finite mixture models. They have a lot of potential in astronomy. The data have something to do with a sum of an unknown number of components, whose properties we want to know. I calculate the posterior distribution for the number of components N, and their parameters.

7 Probabilistic catalogs Brewer, Foreman-Mackey, and Hogg, 2013, AJ, 146, 7. arxiv: (a) Data (b) Nine plausible catalogs

8 The hypothesis space N=0 N=1 N=2 N=3

9 Computation I use DNest4 (Diffusive Nested Sampling) Brewer, Pártay, and Csányi, 2011, Statistics and Computing, 21, 4, arxiv: The target distribution is the mixture of constrained priors: j max w j 1 [ ] L(θ) > l j p(θ) = π(θ). (1) X(l j ) j=0

10 Computation

11 Probabilistic catalogs Brewer, Foreman-Mackey, and Hogg, 2013, AJ, 146, 7. arxiv: (c) Data (d) Nine plausible catalogs

12 Results: Hyperparameters 5 Posterior Probability h α Number of Stars (N) h α1

13 Results: Luminosity Function Number (Flux > f) Test Case 1 Inference SExtractor True Number (Flux > f) Test Case 2 Inference SExtractor True 10 0 f f

14 First real application! A candidate dwarf around a nearby galaxy (all other information suppressed by me not being a proper astronomer)..

15 First real application! A candidate dwarf around a nearby galaxy (all other information suppressed by me not being a proper astronomer).. Four bands, unknown (5-parameter) PSF in each band, fluxes different in each band, hierarchical prior for fluxes.

16 First real application! A candidate dwarf around a nearby galaxy (all other information suppressed by me not being a proper astronomer).. Four bands, unknown (5-parameter) PSF in each band, fluxes different in each band, hierarchical prior for fluxes. Play movie.mkv

17 Gravitational lensing and dark substructures Simulated Dataset Brewer, Huijser, and Lewis, MNRAS, 455, arxiv:

18 The prior information α src σ α lens N src θ src i D θ lens i N lens Source Blobs i = 1,..., N src Lens Blobs i = 1,..., N lens θ SIE

19 Yes we can! Substructure Positions

20 Sí, se puede! 50 Number of posterior samples µ lens

21 The Jackpot Was discovered in the Sloan Digital Sky Survey [spectrum had two redshifts]. HST follow-up as part of SLACS (SLoan Lens ACS Survey). Has two Einstein rings! Figure: Sonnenfeld et al, 2012, ApJ 752, 163. arxiv:

22 The Jackpot Sadly, the redshift of the second source couldn t be measured. But we can do plenty with the first. 1.5 The "Jackpot" Gravitational Lens 1.0 y (arcseconds) x (arcseconds)

23 The Jackpot Previous authors found a dark substructure with mass M sub = (3.51 ± 0.15) 10 9 M. Vegetti et al, 2010, MNRAS, 408, arxiv:

24 Samples of lenses Show movie2.mkv

25 Measuring the mass? Mass (M ) Width (arcseconds)

26 A challenge for the radial velocity technique 3 2 A challenge... Keplerian exoplanet signal Stellar rotation signal Radial Velocity (m/s) Time (periods)

27 A challenge for the radial velocity technique Can we distinguish these? Depends. In the limit (eccentricity 0, stellar rotation signal sinusoid), there is a genuine ambiguity. Away from this limit, it depends on how informative the data is about the signal shape. Inference methods should tell us about the ambiguity, not explain the stellar rotation signal with multiple planets.

28 The Prior Information µ P w P µ K P K e η 1 η 2 η 3 η 4 N p φ ω N p planets v sys Σ s t i v kep i v i σ i N data points Faria et al, 2016, A&A 588, A31 arxiv: Brendon J. Brewer

29 HARPS Data RV [m/s] RV [m/s] a BJD [days] b RV-GP-planet c orbital phase c RV-GP-planet b Faria et al, 2016, A&A 588, A31 arxiv:

30 How many planets? Number of posterior samples p(3) p(2) 4.49 p(2) p(1) Number of planets Faria et al, 2016, A&A 588, A31 arxiv:

31 Have a mixture modelling problem? Or any Bayesian computation for that matter (including ABC). Software paper on the way (draft in repo currently 30 pages!).