Upload
others
View
1
Download
0
Embed Size (px)
Citation preview
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
István Szapudi (IfA, Hawaii) Sufficient Statistics April 11, 2019 2 / 29
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
István Szapudi (IfA, Hawaii) Sufficient Statistics April 11, 2019 3 / 29
Summary of plateaux
István Szapudi (IfA, Hawaii) Sufficient Statistics April 11, 2019 4 / 29
Logarithmic mappingNeyrinck, Szapudi, & Szalay 2009
δ
log(1+δ)
δinitial
δGauss
István Szapudi (IfA, Hawaii) Sufficient Statistics April 11, 2019 5 / 29
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.
István Szapudi (IfA, Hawaii) Sufficient Statistics April 11, 2019 7 / 29
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
István Szapudi (IfA, Hawaii) Sufficient Statistics April 11, 2019 8 / 29
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
István Szapudi (IfA, Hawaii) Sufficient Statistics April 11, 2019 9 / 29
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
n=.012h3 Mpc−3 , Scale =8.1h−1 Mpc
E
z=0.0
n=.015h3 Mpc−3 , Scale =7.5h−1 Mpc
F
z=0.0
10-2 10-1 100100
101
102
103
104
n=.067h3 Mpc−3 , Scale =2.0h−1 Mpc
G
z=2.1
10-2 10-1 100
k [hMpc−1 ]
n=.059h3 Mpc−3 , Scale =2.0h−1 Mpc
H
z=2.1
10-2 10-1 100
n=.076h3 Mpc−3 , Scale =1.9h−1 Mpc
I
z=2.1
PredictedMill Sim
István Szapudi (IfA, Hawaii) Sufficient Statistics April 11, 2019 10 / 29
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)
István Szapudi (IfA, Hawaii) Sufficient Statistics April 11, 2019 11 / 29
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
István Szapudi (IfA, Hawaii) Sufficient Statistics April 11, 2019 12 / 29
Fitting Formulae
István Szapudi (IfA, Hawaii) Sufficient Statistics April 11, 2019 13 / 29
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
István Szapudi (IfA, Hawaii) Sufficient Statistics April 11, 2019 14 / 29
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
István Szapudi (IfA, Hawaii) Sufficient Statistics April 11, 2019 15 / 29
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
István Szapudi (IfA, Hawaii) Sufficient Statistics April 11, 2019 16 / 29
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
István Szapudi (IfA, Hawaii) Sufficient Statistics April 11, 2019 17 / 29
AccuracyRepp & Szapudi 2019
0
1
2
3
A
Millennium Simulation Cosmology
z=0.0
D
Millennium Simulation Cosmology
z=0.0
Millennium Simulation Cosmology
z=0.0
Millennium Simulation Cosmology
z=0.0
Millennium Simulation Cosmology
z=0.0
Millennium Simulation Cosmology
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
István Szapudi (IfA, Hawaii) Sufficient Statistics April 11, 2019 18 / 29
Forcast for A∗ in 3D (preliminary)Realistic density but real space no bias
István Szapudi (IfA, Hawaii) Sufficient Statistics April 11, 2019 19 / 29
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
István Szapudi (IfA, Hawaii) Sufficient Statistics April 11, 2019 20 / 29
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
István Szapudi (IfA, Hawaii) Sufficient Statistics April 11, 2019 21 / 29
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
István Szapudi (IfA, Hawaii) Sufficient Statistics April 11, 2019 22 / 29
Idea: compactify the infinite volumeIllustrated in 2D
O
Q
T
Δω
Δr
Δφ
ΔφΔφ
S
PRS
István Szapudi (IfA, Hawaii) Sufficient Statistics April 11, 2019 23 / 29
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
István Szapudi (IfA, Hawaii) Sufficient Statistics April 11, 2019 24 / 29
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]
István Szapudi (IfA, Hawaii) Sufficient Statistics April 11, 2019 25 / 29
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
István Szapudi (IfA, Hawaii) Sufficient Statistics April 11, 2019 26 / 29
István Szapudi (IfA, Hawaii) Sufficient Statistics April 11, 2019 27 / 29
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
István Szapudi (IfA, Hawaii) Sufficient Statistics April 11, 2019 28 / 29
RSD (preiminary)
István Szapudi (IfA, Hawaii) Sufficient Statistics April 11, 2019 29 / 29