LLNL-PRES-733055This work was performed under the auspices of the U.S. Department of Energy by Lawrence Livermore National Laboratory under contract DE-AC52-07NA27344. Lawrence Livermore National Security, LLC

HierarchicalProbabilisticInferenceofCosmicShearAstronomyinthe2020s:SynergieswithWFIRST

MichaelD.SchneiderwithJoshMeyersandWillDawson

June 27, 2017

Collaborators:D. Bard, D. Hogg, D. Lang, P. Marshall, K. Ng

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§ WFIRST&LSSTareideallysuitedforajointcosmicshearmeasurementtoconstraincosmologicalparametersanddarkenergy

§ NewshearinferencemethodsarerequiredtofullyexploitthesensitivitiesofWFIRST&LSST

§ Ahierarchicalprobabilisticforwardmodelapproachshowsthemostpromiseformeetingshearbiasrequirements

§ Theseprobabilisticalgorithmsenableboth:— Exploitationofinformationinnewcosmologicalstatistics— Moreflexiblecomputingpipelinestoingestandinterpretnewdata

Summary

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Introduction

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Cosmicshear measurement

§ Thelensingbylargescalestructure§ Lookingforverysmallsignalunderverylarge

amountofnoise§ Wedon’tknow“unsheared”shapes,butcan

(roughly)assumetheyareisotropicallydistributed

§ Cosmicsheardistortsstatisticalisotropy;galaxyellipticitiesbecomecorrelated

§ ExquisiteprobeofDE,ifsystematicscanbecontrolled

§ LSST:willmeasurefewbilliongalaxyellipticities.ExcellentsensitivitytobothDEandsystematics!

4

Cosmic shearsignal iscomparableto ellipticity of the Earth, ~0.3%

- D. Wittman

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WhatwilltheWFIRSTHLSaddtocosmicshear?Improvedtomography,reducedbiasmeanstighterdarkenergyconstraints

§ Deblending— 30– 50%ofLSSTgalaxydetectionswillbemultipleresolvedobjects

asseenbyWFIRST— Shearbias:Mostblendsarechancealignmentsofgalaxiesat

differentredshifts— Photo-zbias:mixedcolors/biasedphotometry— Assertnumber&propertiesofblendcomponentsgivenHLS

overlapwithLSST— TrainstatisticalcalibrationforentireLSSTfootprint

§ Improvedphoto-z’sfromLSSToptical+WFIRSTNIRbands

§ HigherfidelityLSScross-correlations(fromgrism survey)— Breakmanysystematicsandcosmologicalparameterdegeneracies

§ Reducedshearbias?

5

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Weaklensingofgalaxies:theforwardmodel

Unknown & dominates signal

Want this

Marginalize

Constrained by

Image credit: GREAT08, Bridle et al.

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Wewon’tjusthavemoredatawith‘StageIV’surveys.- We’reinanerawithqualitativelynewcomputingcapabilities

Dijkstra’s Law

A quantitative difference is also a qualitative difference if the quantitative difference is greater than an order of magnitude.

Figure credit: Wikipedia

> 2 orders of magnitude since SDSS era

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Qualitativechangesincomputingenablenewscientificmethods

- Mark Seager, CTO for the HPC Ecosystem at Intel(interview in Inside HPC on June 6, 2016)

“…predictive simulation has brought together theory and experiment in such a compelling way that it’s fundamentally extended the scientific method for the first time since Galileo Galilei invented the telescope in 1609…”

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§ We’refacingsystematics-limitedmeasurements— End-to-endsimulationsoftheexperimentarethebestapproachtoimproveaccuracy&

precision

— Tiesdataandsimulationmoreintricatelythaninpastcosmologypipelines

§ Imageandcatalogsummarystatisticsarenolongergoodenoughtomeetnextgenerationsciencerequirements— Probabilistichierarchicalmodelsandrelatedmachine-learningapproachesshowpromise

butaremuchmorecomputationallyintensive

— Potentialchangestothetraditional‘facility’/‘user’separateanalysisstages

Data+Computeconvergenceincosmology– DOEASCRinitiative,April2016

Removing the line between ‘analysis’ and ‘simulation’.

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Catalog cross-matchingbetweenspaceandgroundisconfusedbysignificantobjectblendingasseenbyLSST

Ground:(Subaru(Suprime0Cam(

Space:(Hubble(ACS(

LSST blend fractions estimated from Subaru & HST overlapping imaging

Dawson+2015

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Shear bias

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ShapetoShear:NoiseBias

§ Ellipticity:

§ Ensembleaverageellipticity isanunbiasedestimatorofshear.

§ However,maximumlikelihoodellipticity inamodelfitisnotunbiased.

§ Ellipticity isanon-linearfunctionofpixelvalues.

e =a� ba+ b

exp(2i✓)

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1. Calibrateusingsimulations.(im3shape,sfit)

— Butcorrectionsareupto50xlargerthanexpectedsensitivity!

2. Propagateentireellipticity distributionfunctionP(ellip |data)

— UseBayes’theorem:P(ellip |data)∝ P(data|ellip)P(ellip)

— MeasureP(ellip)indeepfields.(lensfit,ngmix,FDNT).

— Infersimultaneouslywithshearinahierarchicalmodel.(MBI).

MitigatingNoiseBias– atleast2strategies

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Ahierarchicalmodelforthegalaxydistribution

§ σe =intrinsicellipticity dispersion

§ eint =galaxyintrinsicellipticity

§ g=shear

§ esh =galaxyshearedellipticity

§ PSF=pointspreadfunction

§ D=modelimage

§ σn =pixelnoise

§ D=data:observedimage

arXiv:1411.2608

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Ourgraphicalmodeltellsushowtofactorthejointlikelihood

§ Useaprobabilisticgraphicalmodeltoencodethefactorizationofthejointprobabilitydistributionofvariablesinthemodel.

§ Wedon’tcareaboutesh forcosmology,sointegrateitout.

Huge complicated integral to compute for every posterior evaluation.

/Z Y

ij

P⇣D̂ij|PSFj, �n,j, eshi

⌘Y

i

P⇣eshi |g, �e

⌘P (g) P (�e) d{eshi }

Pr⇣g,�e| {PSF}j , {�n,j , {Dij}}

⌘

/Z

dngal�eshi

2

4Y

ij

Pr�Dij |PSFj ,�n,j , eshi

�3

5"Y

i

Pr�eshi |g,�e

�Pr(g)Pr(�e)

#

arXiv:1411.2608

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ImportanceSampling allowstractabledivide&computeWethusestimatethepseudo-marginallikelihood forshear

§ Don’tgobacktopixelsforeverytimewesampleanewgorσe.

§ Foreachgalaxy,drawimagemodelparametersamplesunderafixed“interim”prior.Thisisembarrassinglyparallelizable.

§ Usereweightedsamplestoapproximatetheintegral viaMonteCarlo.

1

ConditionalPrior

InterimPosterior

InterimPrior

Likelihood

Ongoing research question:How many interim samples are needed?

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Sourcecharacterizationviaprobabilisticimagemodeling

Infer image model parameters via MCMC under an interim prior distribution for the galaxy and PSF parameters.

MBI GREAT3 analysis with:The Tractor (Lang & Hogg)

Now use GalSim + MCMC

GalSim models inside an MCMC chain – Can it be made fast enough?

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Exampleinterimposteriorinferencesforgalaxystampimages

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ProbabilisticforwardmodelingcanmeetLSSTshearbiasrequirements… atleastwhentestedonsimulatedimages

§ GREAT3CGC-likesetup— 200’fields’withconstantshearperfield— 10kgalaxiesperfield

§ Marginalize7parameterspergalaxy:— e1,e2,HLR,flux,dx,dy,n— Notable:Sersic indexmarginalized

§ HaveNOTmarginalizedPSF(yet!)

Immediate takeaway: Hierarchical inference performs significantly better than ensemble average maximum likelihood ellipticity.

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Multi-epoch & multi-telescope data sets

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Howdowecombinemultipleobservationsofthesamegalaxy?Naïvelywemustjointfitallepochssimultaneously

arXiv:1511.03095Generalized Multiple Importance SamplingElvira, Martino, Luengo, & Bugallo

Problem: Imagine we have fit pixel data from LSST year 1. How do we incorporate year 2 observations without redoing (expensive) calculations?

Solution: Consider single-epoch samples as draws from a multi-modal importance sampling distribution:

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‘cross-pollination’ needed:

Evaluate the likelihood of epoch i given model parameter samples

from epoch j, for all combinations of i, j.

A standard scatter / gather operation

Multipleimportancesampling(MIS)viaweightedpseudo-marginals

1. Sample from the conditional posterior for each epoch individually

2. Evaluate the ratio of the conditional posterior for each epoch i to that of the MIS sampling distribution

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MultipleimportancesamplingenablesstreamingdataanalysisEfficiencyissignificantlyenhancedbyusingolddataasasampling‘prior’

§ Drawparametersamplesfromfirstepochunderanominalinterimprior

§ Drawsamplesfromsubsequentepochswithapriorinformedbypreviousepochsamples

§ Simulationstudiesshow:— ~10%ofsampleshavesignificantweight

whencombining200epochsinstreamingfashion

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PSF marginalization

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MarginalizingPSFs:MISmakesthistractable

§ LSSTwillhave~200epochsperobjectperfilter— WeaimtomarginalizethePSF∏n,i inevery

epoch— Themarginalizationisconstrainedby:

• ConsistencyofPSFrealizationsoverthefocalplaneforeachepoch

• Consistencyoftheunderlyingsourcemodelacrossepochs

§ Simplestapproach(statistically,notcomputationally):Infergalaxymodelsgivenallepochimagingsimultaneously— “Interim”samplesareofsize:~10galaxy

params +200*~4PSFparams =~1kparameters!

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1. Fitstarfootprintsinallepochsviaprobabilisticforwardmodels

2. MarginalizestarimageparameterstoconstraintheglobalfieldPSFmodelforeachepoch— Stateoftheopticsaberrations,and— Distributionofatmosphereturbulencestatistics

3. Fitallgalaxyfootprintsineachepochviaforwardmodels— UsePSFmodelsdrawnfromthemarginalposteriorgiventhestarimages

4. RunThresherontheinterimgalaxysamplesforallepochs(via‘cross-pollinator’)

ThepipelineforPSFmarginalization

One approximation needed: Marginalize PSF model independently for each field location

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Probabilistic cosmological one-point statistics

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Probabilisticcosmologicalmassmapping

Zero E/B mode mixing by construction

Objective: infer the 3D gravitational potential of the initial conditions

arXiv:1610.06673

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Hierarchicalinferenceofcosmologicallensingmassdistributions

Validation with simulations A real merging galaxy cluster New:• Linear and

nonlinear scales reconstructed in one framework

• No E/B mode mixing by construction

Schneider+2017, ApJ

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Applicationtodata:WeaklensingmassmapsforAbel781merginggalaxyclustersasseenbytheDeepLensSurvey

Our method Previously published method

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Latent features of the galaxy distribution

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Pr(eint)isnotGaussian!

§ WouldrathernotassertaparticularparametricformforP(eint).

§ Usea“non-parametric”distribution:aDirichletProcessMixtureModel

3

Ellipticities from COSMOS

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Hierarchicalinferenceofintrinsic galaxyproperties

Specify a Dirichlet Process (DP) for the distribution of intrinsic galaxy property hyper-parameters

The DP is a ‘non-parametric’ distribution with discrete support

The DP distribution allows clustering of data points (e.g., galaxies) to infer latent structure in the data.

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GibbsupdatesintheDirichlet Processmodel

Pr(cn = c`|c�n,!n,↵,X ) = bN�n,c Pr(dn|↵c` ,X ), 8` 6= n

Pr(cn 6= c`8` 6= n|c�n,!n,X ) = bZ

Pr(dn|↵,X )G0(↵) d↵,

↵cn ⇠ G0 (↵cn)NcnY

`=1

Pr(d`|↵cn ,X ))

Latent class assignments are updated with different conditional distributions depending on whether any other observations are assigned to the current class.

The DP mixture parameters are simply updated with the posterior given all observations currently associated with the given latent class.

ZPr(dn|↵,X )G0(↵) d↵ =

ZnN

NX

k=1

Prmarg(!nk|a)Pr(!nk|I0)

Prmarg(!nk|a) ⌘Z

d↵cn G0(↵cn |a)Pr(!nk|↵cn)

Pr(cn 6= c`8` 6= n|c�n,!n,X ) = bZ

Pr(dn|↵,X )G0(↵) d↵

Highlighted integral is expensive to compute in general.

With importance sampling we only require the DP base distribution to be conjugate to the distribution of galaxy properties – NOT the likelihood.

Neal (2000)

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Asimulationstudywith100galaxiesvalidatestheDPmodel

100 galaxies drawn from 1 of 2 Gaussian ellipticity distributions

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Simulationstudy:Wecanbeatthetraditional‘shapenoise’statisticalerrorboundbyinferringlatentstructureinthedata

100 galaxies drawn from 1 of 2 Gaussian ellipticity distributions

3x improvement in cosmic shear precision

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GREAT3results

§ TestedhierarchicalapproachusingsimulationsfromthethirdGRavitationallEnsingAccuracyTest(GREAT3).

§ Hierarchicalinferenceperformssignificantlybetterthanensembleaveragemaximumlikelihoodellipticity.

§ TheDPMMellipticitypriorperformsbetterthanthesingleGaussianellipticityprior.

Mean ML ellipticityHierarchical Inference

: 13% shear calibration errorsH.I. : 4% shear calibration errors

Dirichlet Process Inference

DP : 1-2% shear calibration errors

Input shearSh

ear r

esid

uals

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Multi-variateDPmixturemodel (inprogress):“standardizable”ellipticities.

§ Ellipticalgalaxieshaveanarrowerintrinsicellipticitydistributionthanlate-type.Highersensitivitytoshear!

§ Ellipticals/spiralsalsodistinguishablebycolorandmorphology(e.g.,Sersicindex,Ginicoefficient,asymmetry),potentiallyprovidingadditionalvariableswithwhichtocluster.

§ Othercorrelationstoexploit?

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ApplicationtotheDeepLensSurvey:realgalaxiesrequireatleast2latentclasses(ignoringlensing)

We infer 2 latent classes given only an ellipticity catalog

Preliminary: The marginal posterior distribution of ellipticity variance from the Deep Lens Survey

photo-z: [0.65, 0.7]

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TheprobabilisticweaklensingworkflowplanforLSST

Probabilistic pipeline

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§ Cosmic shear is systematics limited & signal is dominated by PSF and astrophysics

— A probabilistic approach is warranted to infer a small signal and mitigate biases

§ A hierarchical probabilistic model for cosmic shear can trade bias for variance, but also can increase precision by learning latent structure in the galaxy distribution.

§ Importance sampling methods allow tractable approaches to a probabilistic forward model of LSST & WFIRST imaging

— With billions of galaxies and hundreds of epochs per galaxy modeling LSST or WFIRST imaging requires an approach to separating analyses of data subsets, even though statistically correlated

§ We are able to sample from a probabilistic model with multiple hierarchies to marginalize both correlated image systematics and astrophysical properties of galaxies.

Summary

Probabilistic