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Sufficient Statistics

István Szapudi

Institute for AstronomyUniversity of Hawaii

April 11, 2019

István Szapudi (IfA, Hawaii) Sufficient Statistics April 11, 2019 1 / 29

Outline

1 Introduction to Sufficient Statistics

2 Prediction: PDF, the Log and A∗ Power Spectrum, and Ising bias

3 Compactified Simulations


Cosmological Information in LSS Surveys

The standard power spectrum extract a small fraction of theavailable information from LSS surveys, (unlike CMB)Non-Gaussianity, linear modes are used k ' 0.2Good news: many (∝ k3) high k modes are availableBad news: plateau in the power spectrum due to SS and IS modecouplingStandard idea: use higher order statistics (weakly non-linear)Worst news: information content of higher order statisticsvanishing in the non-linear regime


Summary of plateaux


Logarithmic mappingNeyrinck, Szapudi, & Szalay 2009

δ

log(1+δ)

δinitial

δGauss


Sufficient Statistics: All Information on a ParameterCarron & Szapudi (2013 MNRAS 434, 2961; 2014, MNRAS 439, L11)

∂α ln p(δ) ' τ2(δ) '(

(1 + δ)(n+1)/3 − 1(n + 1)/3

)2

' ln(1 + δ)2

Diagonal covariances matrix.István Szapudi (IfA, Hawaii) Sufficient Statistics April 11, 2019 6 / 29

Info. in the Millenium simulation density field

Analogus to lognormal fields.


Discreteness effectsSo we have now a good understanding at the level of the δ field. Howto apply these transformation to the observed discrete galaxy field ?

Now two layers of non-Gaussian statistics : underlying matterfield, and discrete sampling effects.

P(N|θ) =

∫ ∞0

dρ p(ρ|θ)P(N|ρ)︸︷︷︸∝P(ρ|N)

.

Saddle-point (Laplace) approximation (ρ∗ maximum of posteriorfor ρ) :

∂α ln P(N|θ) = ∂α ln p(ρ∗|θ)︸︷︷︸original suff. observable

− 12∂α ln′′ P(ρ∗|N,θ)︸︷︷︸

sensitivity of curv. of posterior

.

For Poisson sampling it all boils down to extract the mean andvariance of A∗ = ln ρ∗

Carron & Szapudi (2014), MNRAS 439, L11


Forecasting dark energy EoS:Lognormal is a great approximation in 2D

0.2 0.4 0.6 0.8 1.0z

0.5

1.0

1.5

2.0

2.5

Gai

n m

8

CFHTLSHSC

EUCLIDDES

LSSTWFIRST

0.2 0.4 0.6 0.8 1.0z

0.5

1.0

1.5

2.0

2.5

Gai

n w

0

CFHTLSHSC

EUCLIDDES

LSSTWFIRST

Survey Max. GainEuclid 1.86

WFIRST-AFTA 1.98HSC 1.83LSST 1.86DES 1.76


Prediction of the A∗ power spectrumRepp & Szapudi 2018b, 2019

103

104

105

n=.0013h3 Mpc−3 , Scale =31h−1 Mpc A

Millennium Simulation Cosmology

z=0.0

PMg (k)

PMA ∗ (k)

n=.0012h3 Mpc−3 , Scale =33h−1 Mpc B

WMAP7 Cosmology

z=0.0

n=.0015h3 Mpc−3 , Scale =30h−1 Mpc C

Planck 2013 Cosmology

z=0.0

102

103

104

Pow

er

[h−

3M

pc3

]

n=.013h3 Mpc−3 , Scale =7.8h−1 Mpc

D

z=0.0


E

z=0.0


F

z=0.0

10-2 10-1 100100

101

102

103

104


G

z=2.1

10-2 10-1 100

k [hMpc−1 ]


H

z=2.1

10-2 10-1 100


I

z=2.1

PredictedMill Sim


Prediction of the A∗ power spectrumRepp & Szapudi 2018b, 2019

10 A. Repp & I. Szapudi

Cosmology

Camb

Plin(k)

Equations 29–32

PA(k)

Equations 3–7, 34–38

P(A)

Equation 33

PMA (k)

P(N |A)

Equations 5–8, 16

Equations 21, 22

A, A

hAi, hAi, 2A , 2

A

Equations 23, 26, 27, 28

b2A , C, D, B

Equation 25

PMA

(k)

Equation 20

PMA (k)

Figure 6. Block diagram of procedure for calculating the optimal galaxy power spectrum PA (k). The required inputs (besides survey

parameters such as redshift) are the cosmology and the discretization scheme P(N |A).

These four parameters – along with P MA (k) – then yield

the spectrum of A:

P MA (k) =

hb2A B

1 eCk

+ Dk

i· P M

A (k). (25)

Finally, one must add the discreteness plateau to obtainthe power spectrum of A itself:

P MA (k) = P M

A (k) + V ·2

A 2A

. (20)

6 ACCURACY

It remains to evaluate the accuracy of the prescription inSections 3–5.

To do so, we obtain (as described previously in Sec-tion 5) snapshots of the Millennium Simulation at z = 0,1.0, and 2.1. These snapshots comprise 2563 cubical pix-els with side lengths 1.95h1Mpc. We then Poisson-samplethe dark matter density to obtain discrete realizations ofeach snapshot for mean number of particles per pixel N =0.01, 0.1, 0.5, 1.0, 3.0, and 10.0. For N > 0.5, we generate 10

MNRAS 000, 1–16 (0000)


Non-lognormality in 3DGEV distribution: Repp & Szapudi 2018a

4 3 2 1 0 1 2 3 4(xx>)/(x)0.0

0.10.20.30.40.5

P[(x

x)/

(x)] log(1+)Gaussian

4

1 0 1 2 3 4 5 6

δ

3.5

3.0

2.5

2.0

1.5

1.0

0.5

0.0

0.5

log 1

0P(δ)

Side length: 2.0h−1 Mpc

Side length: 15.6h−1 Mpc


Fitting Formulae


Numerical model for PARepp & Szapudi 2017, motivated by Szapudi and Kaiser 2003

The shape of the A = log(1 + δ) power spectrum is approximatelythe same as the linear δ

PA(k) = N C(k)σ2

A

σ2lin

Plin(k),

where

C(k) =

1 if k < 0.15h Mpc−1

(k/0.15)α if k ≥ 0.15h Mpc−1 ,

and

N =

∫dk k2Plin(k)∫

dk k2C(k)Plin(k), α(z) ' [0.02− 0.14]

where σ2lin is calculated with CAMB and

σ2A = 0.73 ln

(1 +

σ2lin

0.73

).

checked for several cosmologies and redshifts between0 ≤ z ≤ 2.1


Precision Prediction for the Log Power Spectrum

10-2 10-1 100

101

102

103

104

105

z=0.010-2 10-1 100

z=1.5

k [hMpc−1 ]

Pow

er

[Mpc3

h−

3]

P(k) ST PA (k) This work, Eqns 3 & 4 This work, Eqn 6


Prediction errors

1010

2

4

6

8

10

Original Mill Sim

101

WMAP7

101

Planck 2013

Smoothing Scale [Mpc/h]

RM

S P

er

Cent

Diffe

rence

z=0.0 z=0.1 z=0.5 z=1.0 z=1.5 z=2.1

This work, Eqns 3 & 4 This work, Eqn 6 ST


From A to A∗Repp & Szapudi 2019

10-2 10-1 100

k [hMpc−1 ]

100

101

102

103

104

Measu

red P

ow

er

[h−

3M

pc3

]

PMA (k)

PMA ∗ (k)

PMA

(k)

PMδA ∗ (k)

10-2 10-1 100

k [hMpc−1 ]

0.3

0.4

0.5

0.6

0.7

0.8

Rati

o o

f M

easu

red P

ow

er

PMA

(k)/PMA (k)

b 2A ∗

b 2A ∗ +Dk

b 2A ∗−B(1−e−Ck ) +Dk


AccuracyRepp & Szapudi 2019

0

1

2

3

A


z=0.0

D


z=0.0


z=0.0


z=0.0


z=0.0


z=0.0

n=0.0013n=0.013

n=0.067n=0.13

n=0.40n=1.3

B

WMAP7 Cosmology

E

WMAP7 Cosmology WMAP7 Cosmology WMAP7 Cosmology WMAP7 Cosmology WMAP7 Cosmology

n=0.0012n=0.012

n=0.059n=0.12

n=0.35n=1.2

C

Planck 2013 Cosmology

F

Planck 2013 Cosmology Planck 2013 Cosmology Planck 2013 Cosmology Planck 2013 Cosmology Planck 2013 Cosmology

n=0.0015n=0.015

n=0.076n=0.15

n=0.45n=1.5

0

1

2

3

Rati

o o

f M

S P

er

Cent

Diffe

rence

(A∗ -p

redic

tion c

om

pare

d t

o S

T)

z=1.0z=1.0z=1.0z=1.0z=1.0z=1.0

102 3 5 7 20 300

1

2

3

z=2.1z=2.1

G

z=2.1z=2.1z=2.1z=2.1

102 3 5 7 20 30

Smoothing Scale [h−1 Mpc]

H

102 3 5 7 20 30

I


Forcast for A∗ in 3D (preliminary)Realistic density but real space no bias


Bias in SimulationsRepp & Szapudi 2019

M ≡ 〈Ngal〉A · (1 + δ)−1 =bN

1 + exp(

At−AT

)

2 0 2 4ln(1 + δ)

0.0

0.5

1.0

1.5

2.0

2.5

3.0

M≡⟨ N ga

l⟩ /(1+δ)

4h−1Mpc

Mill Sim CatalogIsing Model

2 1 0 1 2ln(1 + δ)

50

75

100

125

150

175

200

M≡⟨ N ga

l⟩ /(1+δ)

16h−1Mpc


Bias in SDSSRepp & Szapudi 2019

1.0 0.5 0.0 0.5 1.0ln(1 + δ) [Log mass units]

0.0

0.5

1.0

1.5

2.0

2.5

3.0M≡⟨ N ga

l⟩ /(1+δ)

[Gal

axie

s pe

r m

ass

unit] SDSS Galaxies

Ising modellinear biasquadratic biaslog bias


Problems with cosmological simulations

Forces are wrong. (High Precision Cosmology?)Spherical symmetry brokenT 3 Topology contrary to observationsNumber of modes ' k3, too many high kDoes not match observational geometry


Idea: compactify the infinite volumeIllustrated in 2D

O

Q

T

Δω

Δr

Δφ

ΔφΔφ

S

PRS


Millennium resimulationsRacz etal 2019

Millennium Cosmology using the original scripts with StePSH0 = 73km/s/Mpc, Ωm = 0.25, ΩΛ = 0.75, σ8 = 0.927 volumes to simulate periodic B.C.zin = 127Original: 512 CPU × 683 hours 0.321 Gpc3

StEPS: 12 GPU × 106 hours 1.35 Gpc3

N log N vs N2

multiresolution


Results:Millennium

−400 −200 0 200 400

x[Mpc]

−400

−300

−200

−100

0

100

200

300

400

y[M

pc]

−400 −200 0 200 400

x[Mpc]


ResultsRacz etal 2019, Millennium resimulation

−20 −10 0 10 20

x[Mpc]

−20

−10

0

10

20

y[M

pc]

Millennium simulation

−20 −10 0 10 20

x[Mpc]

StePS simulation


Summary

Sufficient statistics are well approximated with the log transformThey are well understood in Fisher information theory, and can bepredicted and estimated from dataThe log and A∗ power spectra are now predicted to percent levelaccuracy in real spaceThe information gain for Euclid like surveys & 2×.Accurate bias fit inspired by the Ising modelCompactified cosmological simulation have unprecedenteddynamic range for given resourcesStatus: tested on 48 GPUs and recovered Millennium as a testFast prediction for parameter estimationSuper-survey modes/covariance matrices/high dynamic range(e.g., local group, galaxy simulations in cosmological background)Next: redshift distortions


RSD (preiminary)


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