STATISTICAL CHALLENGES IN MODERN ASTRONOMY VI Approximate Bayesian computation: an application to weak-lensing peak counts Chieh-An Lin & Martin Kilbinger SAp, CEA Saclay Carnegie Mellon University, Pittsburgh June 6 th, 2016
Outline linc.tw 1 Weak-lensing peak counts 2 Approximate Bayesian computation 3 Data application ABC: an application to weak-lensing peak counts SCMA6, Pittsburgh Jun 6th, 2016 2
linc.tw Weak gravitational lensing in 3 minutes Source: ALMA ABC: an application to weak-lensing peak counts SCMA6, Pittsburgh Jun 6th, 2016 3
linc.tw Weak gravitational lensing in 3 minutes Unlensed sources Weak lensing ABC: an application to weak-lensing peak counts SCMA6, Pittsburgh Jun 6th, 2016 4
linc.tw Weak gravitational lensing in 3 minutes ABC: an application to weak-lensing peak counts SCMA6, Pittsburgh Jun 6th, 2016 5
linc.tw Weak gravitational lensing in 3 minutes κ map γ map ABC: an application to weak-lensing peak counts SCMA6, Pittsburgh Jun 6th, 2016 6
Weak-lensing peak counts linc.tw κ map and peaks Halo number density dn/dlogm [(Mpc/h) 3 ] 10-2 10-3 10-4 10-5 10-6 10-7 σ 8 = 0.76 σ 8 = 0.82 σ 8 = 0.88 Halo mass function 10-8 12.5 13.0 13.5 14.0 14.5 15.0 15.5 16.0 Halo mass [log(m/m h 1 )] Local maxima of the projected mass Probe the mass function Non-Gaussian information ABC: an application to weak-lensing peak counts SCMA6, Pittsburgh Jun 6th, 2016 7
linc.tw Dealing with selection function Early studies Count only the true clusters with high S/N Recent studies Include the selection effect into the model Analytical formalism N-body simulations Fast stochastic model (this work) ABC: an application to weak-lensing peak counts SCMA6, Pittsburgh Jun 6th, 2016 8
linc.tw A new model to predict weak-lensing peak counts Sample halos from a mass function PDF Sampled mass Assign density profiles, randomize their positions Compute the projected mass, add noise Make maps, create peak catalogues Public code in C: Camelus@GitHub See also Lin & Kilbinger (2015a) ABC: an application to weak-lensing peak counts SCMA6, Pittsburgh Jun 6th, 2016 9
Validation linc.tw Peak number density n peak [deg 2 ν 1 ] 10 2 10 1 10 0 10-1 Noise-only Peak abundance histogram Full N-body runs N-body: halos NFW N-body: halos NFW + random position Our model 2 3 4 5 6 7 S/N ν Lin & Kilbinger (2015a) ABC: an application to weak-lensing peak counts SCMA6, Pittsburgh Jun 6th, 2016 10
linc.tw Approximate Bayesian computation Likelihood ABC P(x π) L(π x obs ) P(x π) x x obs ɛ x obs x obs x x ABC: an application to weak-lensing peak counts SCMA6, Pittsburgh Jun 6th, 2016 11
linc.tw Approximate Bayesian computation ABC Requirements: Stochastic model P ( π) Distance x x Tolerance level ɛ P(x π) x x obs ɛ x obs x ABC: an application to weak-lensing peak counts SCMA6, Pittsburgh Jun 6th, 2016 11
linc.tw Approximate Bayesian computation ABC Accept-reject process: Draw a π from the prior P( ) Draw a x from the model P ( π) Accept π if x x obs ɛ Reject otherwise P(x π) x x obs ɛ One-sample test x obs x ABC: an application to weak-lensing peak counts SCMA6, Pittsburgh Jun 6th, 2016 11
linc.tw Approximate Bayesian computation Parameter space Data space prior P(π) ABC posterior P(x π i ) x x obs ɛ π i x obs π Distribution of accepted π = x prior green areas prior 2ɛ likelihood posterior ABC: an application to weak-lensing peak counts SCMA6, Pittsburgh Jun 6th, 2016 12
linc.tw Combined with population Monte Carlo Population Monte Carlo (PMC) Iterative solution for ɛ Set ɛ = + for the first iteration Do ABC Update ɛ and the prior based on results from the previous iteration Repeat until satifying a stop criterion See also Lin & Kilbinger (2015b) ABC: an application to weak-lensing peak counts SCMA6, Pittsburgh Jun 6th, 2016 13
linc.tw Comparison with the likelihood 1.1 Ω m -σ 8 constraints 1.0 abd5, ABC, credible 1-σ, 68.3% 2-σ, 95.4% 0.9 0.8 abd, cg, credible 1-σ, 68.3% 2-σ, 95.4% Contours and dots: ABC Colored regions: likelihood σ 8 0.7 0.6 Gain of time cost for ABC: 2 orders of magnitude 0.5 0.4 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 Ω m Lin & Kilbinger (2015b) ABC: an application to weak-lensing peak counts SCMA6, Pittsburgh Jun 6th, 2016 14
Data from three surveys linc.tw Survey Field size Number of Effective density [deg 2 ] galaxies [deg 2 ] CFHTLenS 126 6.1 M 10.74 KiDS DR1/2 75 2.4 M 5.33 DES SV 138 3.3 M 6.65 ABC: an application to weak-lensing peak counts SCMA6, Pittsburgh Jun 6th, 2016 15
Various settings linc.tw Model settings Compensated filter (suggested by Lin et al. 2016) Adaptive choice for pixel sizes and filtering scales No intrinsic alignment and baryons (not yet!) ABC settings Data vector (summary statistic): peak counts with S/N > 2.5 of all scales Distance: a χ 2 -like normalized sum ABC: an application to weak-lensing peak counts SCMA6, Pittsburgh Jun 6th, 2016 16
Preliminary result linc.tw Ω m -σ 8 constraints 1.6 1.4 PMC ABC 1-σ, 68.3% 2-σ, 95.4% 1.2 σ 8 1.0 Lin & Kilbinger in prep. 0.8 0.6 0.4 0.2 0.0 0.2 0.4 0.6 0.8 1.0 Ω m ABC: an application to weak-lensing peak counts SCMA6, Pittsburgh Jun 6th, 2016 17
Preliminary result linc.tw Ω m -σ 8 constraints 1.6 1.4 PMC ABC 1-σ, 68.3% 2-σ, 95.4% 1.2 σ 8 1.0 0.8 Width: Σ 8 = 0.13 Area: FoM = 5.2 0.6 0.4 0.2 0.0 0.2 0.4 0.6 0.8 1.0 Ω m ABC: an application to weak-lensing peak counts SCMA6, Pittsburgh Jun 6th, 2016 18
Preliminary result linc.tw σ 8 1.2 1.0 0.8 Parameter constraints with PMC ABC 1-σ, 68.3% 2-σ, 95.4% 0.6 0.4-0.6 w de 0-1.0-1.4-1.8 0.1 0.3 0.5 0.7 Ω m 0.4 0.6 0.8 1.0 1.2 σ 8 Lin & Kilbinger in prep. ABC: an application to weak-lensing peak counts SCMA6, Pittsburgh Jun 6th, 2016 19
linc.tw Ongoing improvements and perspectives S/N bin choice: less bias, more accuracy and precision Halo correlation Tomography Intrinsic alignment Baryonic effects ABC: an application to weak-lensing peak counts SCMA6, Pittsburgh Jun 6th, 2016 20
Summary A new model to predict WLPC Likelihood-free parameter inference: ABC Constraints with CFHTLenS, KiDS, DES Collaborators: Martin Kilbinger Austin Peel (poster talk on Wednesday) Sandrine Pires References: [1410.6955] [1506.01076] [1603.06773] Camelus@GitHub http://linc.tw
Backup slides
Advantages of our model linc.tw Fast Only few seconds for creating a 25-deg 2 field, without MPI or GPU programming Flexible Straightforward to include real-world effects (photo-z errors, masks, intrinsic alignment, baryonic effects, etc.) Full PDF information Allow more flexible constraint methods (varying covariances, copula, p-value, approximate Bayesian computation, etc.) ABC: an application to weak-lensing peak counts SCMA6, Pittsburgh Jun 6th, 2016 B 1
Map examples linc.tw Noise-free map Noisy map Smoothed map Mask ABC: an application to weak-lensing peak counts SCMA6, Pittsburgh Jun 6th, 2016 B 2
linc.tw Cosmology-dependent covariance 1.1 Ω m -σ 8 constraints 1.1 Ω m -σ 8 constraints 1.0 abd, cg, confidence 1-σ, 68.3% 2-σ, 95.4% 1.0 abd, vg, confidence 1-σ, 68.3% 2-σ, 95.4% 0.9 0.8 abd, svg, confidence 1-σ, 68.3% 2-σ, 95.4% 0.9 0.8 abd, svg, confidence 1-σ, 68.3% 2-σ, 95.4% σ 8 σ 8 0.7 0.7 0.6 0.5 1.00 2.50 4.00 7.00 5.50 0.6 0.5 12.5 15.0 17.5 20.0 22.5 0.4 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 Ω m 0.4 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 Ω m cg svg vg FoM 46 57 56 Lin & Kilbinger (2015b) ABC: an application to weak-lensing peak counts SCMA6, Pittsburgh Jun 6th, 2016 B 3
True likelihood linc.tw 1.1 Ω m -σ 8 constraints 1.0 0.9 0.8 abd, true, confidence 1-σ, 68.3% 2-σ, 95.4% abd, vc, confidence 1-σ, 68.3% 2-σ, 95.4% σ 8 0.7 0.6 0.5 0.4 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 Ω m Lin & Kilbinger (2015b) ABC: an application to weak-lensing peak counts SCMA6, Pittsburgh Jun 6th, 2016 B 4
Degeneracy with w de 0 linc.tw σ 8 1.2 1.0 0.8 Parameter constraints with likelihood Starlet, θ ker =2, 4, 8 1-σ, 68.3% 2-σ, 95.4% 0.6 0.4-0.6 w de 0-1.0-1.4-1.8 0.1 0.3 0.5 0.7 Ω m 0.4 0.6 0.8 1.0 1.2 σ 8 Lin et al. (2016) ABC: an application to weak-lensing peak counts SCMA6, Pittsburgh Jun 6th, 2016 B 5
Technical detail linc.tw Mass function from Jenkins et al. (2001) M-c relation from Takada & Jain (2002) Source redshift fitted from surveys Random source position, not catalogue Used raw galaxy densities, not effective Derive σ ɛ from the emperical total variance Pixel size: n gal θpix 2 7 Kaiser-Squires inversion Filtering with the starlet function Scale = 2, 4, 8 pixels Locally determined noise Dimension of x = 75 (= 36 + 15 + 24) ABC: an application to weak-lensing peak counts SCMA6, Pittsburgh Jun 6th, 2016 B 6
Field examples linc.tw -3 CFHTLenS W1 field 58 CFHTLenS W3 field CFHTLenS W4 field 4-6 Declination [deg] -9 Declination [deg] 55 Declination [deg] 2 52 0-12 39 36 33 30 219 216 213 210 336 334 332 330 Right ascension [deg] Right ascension [deg] Right ascension [deg] KiDS G09a field KiDS G12 field DES SPT-E field 4 2-45 Declination [deg] 2 Declination [deg] 0 Declination [deg] 0-2 -60 134 132 130 128 186 184 182 90 75 60 Right ascension [deg] Right ascension [deg] Right ascension [deg] ABC: an application to weak-lensing peak counts SCMA6, Pittsburgh Jun 6th, 2016 B 7
Covariance linc.tw Correlation matrix CFHTLenS KiDS DES 1.0 DES KiDS CFHTLenS 0.8 0.6 0.4 0.2 0.0 0.2 0.4 0.6 0.8 1.0 Lin & Kilbinger in prep. ABC: an application to weak-lensing peak counts SCMA6, Pittsburgh Jun 6th, 2016 B 8
linc.tw Constrain concentration paramters σ 8 1.1 0.8 Parameter constraints with PMC ABC 1-σ, 68.3% 2-σ, 95.4% 0.5 0.5 w de 0 1.0 1.5 16 c 0 10 4 0.6 β NFW 0.0 0.6 0.2 0.5 0.8 Ω m 0.5 0.8 1.1 σ 8 1.5 1.0 0.5 w de 0 4 10 16 c 0 Lin & Kilbinger in prep. ABC: an application to weak-lensing peak counts SCMA6, Pittsburgh Jun 6th, 2016 B 9
linc.tw Definition of Σ 8 Ω m -Σ 8 constraint 1.1 1.0 Σ 8 0.9 0.8 0.7 0.6 0 1 2 3 4 5 6 7 2 ln(l/l max ) 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 Ω m 0 5 10 15 20 25 pdf(σ 8 ) Definition 1 Σ 8 = σ 8(Ω m/0.27) α Σ 8 = 0.825 Σ 8 = 0.16 α = 0.48 ( ) 1 α ( ) α Ω Definition 2 Σ 8 = m+β σ 8 1 α α Σ 8 = 1.935 Σ 8 = 0.13 α = 0.38 β = 0.82 ABC: an application to weak-lensing peak counts SCMA6, Pittsburgh Jun 6th, 2016 B 10
PMC ABC algorithm linc.tw set t = 0 for i = 1 to Q do ( ) generate θ (0) i ( from P( ) and x from P θ (0) i set δ (0) i = D x, x obs) and w (0) i = 1/Q end for set ɛ (1) = median ( δ (0) i ) and C (0) = cov ( π (0) i, w (0) i while success rate r stop do t t + 1 for i = 1 to Q do repeat { generate j from {1,..., Q} with weights w (t 1) 1,..., w (t 1) generate π (t) i set δ (t) i until δ (t) i set w (t) i from N ( π (t 1) j = D ( x, x obs) ɛ (t) P ( π (t) i ) / Q j=1 w(t 1) ) Q, C (t 1)) ( and x from P π (t) i j K ( π (t) i end for ( ) ( ) set ɛ (t+1) = median δ (t) i and C (t) = cov π (t) i, w (t) i end while π (t 1) j, C (t 1)) } ) ABC: an application to weak-lensing peak counts SCMA6, Pittsburgh Jun 6th, 2016 B 11
Peaks v.s. power spectrum linc.tw Taken from Liu J. et al. (2015) ABC: an application to weak-lensing peak counts SCMA6, Pittsburgh Jun 6th, 2016 B 12
Public code linc.tw Fast weak-lensing peak counts modelling in C with PMC ABC Camelus@GitHub ABC: an application to weak-lensing peak counts SCMA6, Pittsburgh Jun 6th, 2016 B 13