Inference of Galaxy Population Statistics Using Photometric Redshift Probability Distribution Functions
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1 Inference of Galaxy Population Statistics Using Photometric Redshift Probability Distribution Functions Alex Malz Center for Cosmology and Particle Physics, New York University 7 June 2016 Galaxy population statistics from photo-z PDFs Alex Malz NYU 1
2 Background Galaxy population statistics from photo-z PDFs Alex Malz NYU 2
3 Background Weak gravitational lensing probes cosmology. LSST Galaxy population statistics from photo-z PDFs Alex Malz NYU 3
4 Background Weak gravitational lensing requires the redshift distribution function N (z) = dng dz. Sheldon+2012 Galaxy population statistics from photo-z PDFs Alex Malz NYU 4
5 Background Photometric redshifts approximate spectroscopic redshifts. LSST-DESC Galaxy population statistics from photo-z PDFs Alex Malz NYU 5
6 Background Photometric redshifts are problematic. Jain+2015 Galaxy population statistics from photo-z PDFs Alex Malz NYU 6
7 Motivation Galaxy population statistics from photo-z PDFs Alex Malz NYU 7
8 Motivation Why use photo-z PDFs? Photo-z PDFs include all redshift uncertainties and biases. Galaxy population statistics from photo-z PDFs Alex Malz NYU 8
9 Motivation Why use photo-z PDFs? Photo-z PDFs include all redshift uncertainties and biases. How do we use photo-z PDFs? A probabilistic object necessitates a probabilistic approach. Galaxy population statistics from photo-z PDFs Alex Malz NYU 9
10 Motivation Why use photo-z PDFs? Photo-z PDFs include all redshift uncertainties and biases. How do we use photo-z PDFs? A probabilistic object necessitates a probabilistic approach. What does the fully probabilistic approach get us? It can yield the full posterior of parameters given data. Galaxy population statistics from photo-z PDFs Alex Malz NYU 10
11 Probabilistic Graphical Models and Hierarchical Inference N (z) z k d k K Galaxy population statistics from photo-z PDFs Alex Malz NYU 11
12 Probabilistic Graphical Models and Hierarchical Inference N (z) z k d k K p(z N ) = N (z) N (z) dz Galaxy population statistics from photo-z PDFs Alex Malz NYU 12
13 Probabilistic Graphical Models and Hierarchical Inference N (z) z k seek full posterior p(n { d k } K ) d k K p(z N ) = N (z) N (z) dz Galaxy population statistics from photo-z PDFs Alex Malz NYU 13
14 Probabilistic Graphical Models and Hierarchical Inference N (z) z k d k K seek full posterior p(n { d k } K ) from individual posteriors {p(z k d k )} K p(z N ) = N (z) N (z) dz Galaxy population statistics from photo-z PDFs Alex Malz NYU 14
15 Probabilistic Graphical Models and Hierarchical Inference N (z) z k d k K seek full posterior p(n { d k } K ) from individual posteriors {p(z k d k )} K p(z N ) = N (z) N (z) dz (SDSS, LSST,... ) Galaxy population statistics from photo-z PDFs Alex Malz NYU 15
16 Derivation seek full posterior p(n { d k } K ) Galaxy population statistics from photo-z PDFs Alex Malz NYU 16
17 Derivation seek full posterior p(n { d k } K ) posterior p(n { d k } K ) = prior p(n ) p({ d k } K ) }{{} evidence likelihood p({ d k } K N ) Galaxy population statistics from photo-z PDFs Alex Malz NYU 17
18 Derivation seek full posterior p(n { d k } K ) posterior p(n { d k } K ) = prior p(n ) p({ d k } K ) }{{} evidence Poisson { [ }} ]{ exp N (z)dz K k=1 }{{} independence p( d k N ) }{{} likelihood Galaxy population statistics from photo-z PDFs Alex Malz NYU 18
19 Derivation seek full posterior p(n { d k } K ) posterior p(n { d k } K ) = prior p(n ) p({ d k } K ) }{{} evidence K k=1 }{{} independence Poisson { [ }} ]{ exp N (z)dz p( d k z k )p(z k N )dz k } {{ } hierarchical model Galaxy population statistics from photo-z PDFs Alex Malz NYU 19
20 Derivation seek full posterior p(n { d k } K ) posterior p(n { d k } K ) = prior p(n ) p({ d k } K ) }{{} evidence K k=1 }{{} independence Poisson { [ }} ]{ exp N (z)dz p( d k z k )p(z k N )dz k } {{ } hierarchical model need individual likelihoods {p( d k z k )} K from individual posteriors {p(z k d k )} K Galaxy population statistics from photo-z PDFs Alex Malz NYU 20
21 Derivation need individual likelihoods {p( d k z k )} K from individual posteriors {p(z k d k )} K likelihood p( d k z k ) = Galaxy population statistics from photo-z PDFs Alex Malz NYU 21
22 Derivation need individual likelihoods {p( d k z k )} K from individual posteriors {p(z k d k )} K likelihood p( d k z k ) = likelihood p( d k z k ) Galaxy population statistics from photo-z PDFs Alex Malz NYU 22
23 Derivation need individual likelihoods {p( d k z k )} K from individual posteriors {p(z k d k )} K ; interim prior N 0 likelihood p( d k z k ) = likelihood p( d k z k ) 1 p(z k d k, N 0 ) p(z k d k, N 0 ) Galaxy population statistics from photo-z PDFs Alex Malz NYU 23
24 Derivation need individual likelihoods {p( d k z k )} K from individual posteriors {p(z k d k )} K ; interim prior N 0 likelihood p( d k z k ) = likelihood p( d k z k ) interim posterior p(z k d k, N 0 ) { Bayes Rule }} { p( d k N 0 ) p( d k z k, N 0 ) p(z k N 0 ) Galaxy population statistics from photo-z PDFs Alex Malz NYU 24
25 Derivation need individual likelihoods {p( d k z k )} K from individual posteriors {p(z k d k )} K ; interim prior N 0 likelihood p( d k z k ) = interim posterior p(z k d k, N 0 ) p(z k N 0 ) }{{} N 0 likelihood p( d k z k ) p( d k N 0 ) } p( d k z k, N 0 ) {{ } independence Galaxy population statistics from photo-z PDFs Alex Malz NYU 25
26 Derivation need individual likelihoods {p( d k z k )} K likelihood p( d k z k ) = from individual posteriors {p(z k d k )} K ; interim prior N 0 interim posterior p(z k d k, N 0 ) p(z k N 0 ) }{{} N 0 Galaxy population statistics from photo-z PDFs Alex Malz NYU 26
27 Derivation need individual likelihoods {p( d k z k )} K likelihood p( d k z k ) = from individual posteriors {p(z k d k )} K ; interim prior N 0 interim posterior p(z k d k, N 0 ) p(z k N 0 ) }{{} N 0 given interim prior N 0 have individual interim posteriors {p(z k d k, N 0 )} K Galaxy population statistics from photo-z PDFs Alex Malz NYU 27
28 Derivation [ p(n { d k } K ) p(n ) exp Sample this posterior! K k=1 ] N (z)dz p(z k d k, N 0 ) p(z k N ) p(z k N 0 ) dz k Galaxy population statistics from photo-z PDFs Alex Malz NYU 28
29 Preliminary Results: Fiducial Test Galaxy population statistics from photo-z PDFs Alex Malz NYU 29
30 Preliminary Results: Fiducial Test Fiducial True p(z) In simulations p(z) z Galaxy population statistics from photo-z PDFs Alex Malz NYU 30
31 Preliminary Results: Fiducial Test Fiducial PDFs Interim P(z) True z MLE z In simulations 25 p(z d) z Galaxy population statistics from photo-z PDFs Alex Malz NYU 31
32 Preliminary Results: Fiducial Test 11 Fiducial Malz ln N(z) True 1.0 Interim Mean of Samples In simulations the mean of the samples recovers the true N (z) well N(z) z Galaxy population statistics from photo-z PDFs Alex Malz NYU 32
33 Preliminary Results: Fiducial Test 11 Fiducial Malz ln N(z) 8 N(z) True Interim Stacked KLD=(0.0139, ) Mean of Samples KLD=(0.0052, 0.007) In simulations the mean of the samples recovers the true N (z) well (and better than popular alternatives) z Galaxy population statistics from photo-z PDFs Alex Malz NYU 33
34 Preliminary Results: Magnitude-cut BOSS Subsample Galaxy population statistics from photo-z PDFs Alex Malz NYU 34
35 Preliminary Results: Magnitude-cut BOSS Subsample Biased PDFs Interim P (z) MAP z With real data 20 p(z d) z Galaxy population statistics from photo-z PDFs Alex Malz NYU 35
36 Preliminary Results: Magnitude-cut BOSS Subsample 10 Biased Malz ln N(z) Interim Mean of Samples With real data the mean of the samples differs from the interim prior. N(z) z Galaxy population statistics from photo-z PDFs Alex Malz NYU 36
37 Preliminary Results: Magnitude-cut BOSS Subsample 10 Biased Malz ln N(z) Interim Stacked Mean of Samples With real data the mean of the samples differs from the interim prior. N(z) z Galaxy population statistics from photo-z PDFs Alex Malz NYU 37
38 Conclusion Galaxy population statistics from photo-z PDFs Alex Malz NYU 38
39 Conclusion Summary Cosmology needs a trustworthy N (z). Photo-z PDFs propagate all uncertainties and errors. Hierarchical inference correctly estimates N (z). Galaxy population statistics from photo-z PDFs Alex Malz NYU 39
40 Conclusion Summary Cosmology needs a trustworthy N (z). Photo-z PDFs propagate all uncertainties and errors. Hierarchical inference correctly estimates N (z). Recommendations Stacking photo-z PDFs yields a biased estimate of N (z). The mean of samples is an accurate point estimator. It s best to sample the full posterior. Galaxy population statistics from photo-z PDFs Alex Malz NYU 40
41 Conclusion Summary Cosmology needs a trustworthy N (z). Photo-z PDFs propagate all uncertainties and errors. Hierarchical inference correctly estimates N (z). Recommendations Stacking photo-z PDFs yields a biased estimate of N (z). The mean of samples is an accurate point estimator. It s best to sample the full posterior. Code is now publicly available on GitHub! Galaxy population statistics from photo-z PDFs Alex Malz NYU 41
42 Preliminary Results: Noisier Mock Data p(z d) Interim P(z) True z Observed z In simulations with lower S/N data z Galaxy population statistics from photo-z PDFs Alex Malz NYU 42
43 Preliminary Results: Noisier Mock Data Malz ln N(z) 9 8 In simulations with lower S/N data the marginalized MLE and mean of the samples recover the true N (z) True Interim Mean of Samples N(z) z Galaxy population statistics from photo-z PDFs Alex Malz NYU 43
44 Preliminary Results: Noisier Mock Data Malz ln N(z) 9 8 In simulations with lower S/N data the marginalized MLE and mean of the samples recover the true N (z) (with broader error bars) True Interim Stacked KLD= MMAP KLD= MExp KLD= MMLE KLD= Mean of Samples KLD= N(z) z Galaxy population statistics from photo-z PDFs Alex Malz NYU 44
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To be submitted to the Astrophysical Journal Preprint typeset using L A TEX style emulateapj v. 5/2/11 COSMOLOGICAL DEPENDENCE OF THE MEASUREMENTS OF LUMINOSITY FUNCTION, PROJECTED CLUSTERING AND GALAXY-GALAXY
arxiv: v1 [astro-ph] 24 Jan 2008
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