Durham 102208

SIMULATIONS AND COSMOLOGICAL INFERENCE Michael D. Schneider Durham In collaboration with Lloyd Knox (UC Davis), Salman Habib, Katrin Heitmann, David Higdon (Los Alamos National Laboratory), Charles Nakhleh (Sandia National Laboratories)

October 22, 2008

Overview Question: How do we estimate cosmological parameters when theoretical models are only known via forward simulation? Answer: Use statistical model to interpolate outputs of select simulation runs. 1. Simulation design 2. Emulator Simultaneously learn the error distribution for the data. Applicable to CMB, galaxy, and weak lensing surveys (or really anywhere that uses simulations for parameter inference).

arXiv:0806.1487

Technical motivation: simulations are costly! Most astrophysical systems can only be modeled with numerical simulations Even when the physics is easily understood, accurate noise modeling can require large simulations (e.g. the CMB) Constraining dark energy via BAO and cosmic shear provides formidable computational challenges in predicting both the model and the error distributions

Parameter estimation requires many simulations Use Monte Carlo algorithms to integrate the joint probability distribution of the data and model: P(model | data) = P(model, data) / P(data) Requires many calculations of the model at different parameter settings (~10,000 evaluations for ~5 parameters) This is computationally prohibitive for many applications

Likelihood model Multivariate Gaussian model for the Likelihood: θ ≡ model parameters

x≡d T

¯ (θ)) C −1 (θ) (x − x ¯ (θ)) + log(det(C(θ))) −2 log (P (x|θ)) = (x − x

For galaxy surveys or CMB, “data” = power spectrum model dependence of covariance usually neglected

Framework identical for N-point correlations Gaussian distribution can be extended using mixture models

EXAMPLE: NONLINEAR MATTER POWER SPECTRUM

Non-Gaussian errors in the cosmic shear power spectrum Fisher matrix constraints from Halo Model calculation of power spectrum covariance (Cooray & Hu (2000))

non-Gaussian effects can dominate at scales < 10 arcmin. (even when apparently shape noise dominated) (Semboloni et al. (2006))

Full sky weak lensing survey (limiting mag in R~25)

Clusters + weak lensing Takada & Bridle (2007)

Consider cross-covariance between cluster number counts and cosmic shear power spectrum

Power spectrum covariance from N-body simulations 32 realizations of N-body cube 450 Mpc/h on a side Chop into 64 sub-cubes Window has large impact on covariance Not explained by simple convolution with the power spectrum

0.6 0.4

1e!03 1e!05

0.0

0.2

1e!04

500 1000 200

450 Mpc/h periodic box 112.5 Mpc/h windowed box

!0.2

100

0.8

Gaussian 450 Mpc/h periodic box 112.5 Mpc/h windowed box

1e!02

5000

450 Mpc/h periodic box 112.5 Mpc/h windowed box

Correlation coefficients 1.0

Normalized variance 1e!01

20000

Mean power spectra

0.02

0.05

0.10

0.20 k [h/Mpc]

0.50

1.00

2.00

0.02

0.05

0.10

0.20 k [h/Mpc]

0.50

1.00

2.00

0.05

0.10

0.20

0.50 k [h/Mpc]

1.00

2.00

Parameter dependence of the power spectrum covariance

5e!04

5e!03

5e!02

Gaussian HM !8 = 0.6 HM !8 = 1 PT !8 = 0.6 PT !8 = 1 sim. !8 = 0.6 sim. !8 = 1

1e!04

Normalized variance of power spectrum

Normalized variance

0.05

0.10

0.20

0.50 k [h/Mpc]

1.00

2.00

Correlation coefficients (Halo model)

Parameterization of the power spectrum error distribution Multivariate Normal distribution: P (k) ∼ N (! µ(θ), Σ(θ))

Consider “shell-averaged” estimates of power spectrum bands Central limit theorem guarantees a Gaussian distribution for band powers except for a few k-bins on the largest scales of the survey Correlations in power spectrum captured in this model

SIMULATION DESIGN

Choosing which simulations to run Simulation design (OALH) 1.0

Orthogonal Array Latin Hypercube

!

Optimize with distance criterion

0.6 0.4

!

0.2

!

!

0.0

Orthogonal array: each quadrant has a sample

parameter 2

Latin square: one point per row and column

0.8

Specify hypercube parameter bounds (rescaled to unit interval)

0.0

0.2

0.4

0.6

parameter 1

0.8

1.0

Example design ●● ● ● ●●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ●● ●● ● ●● ● ●● ● ● ● ●●●● ●● ● ● ● ● ●● ● ●●● ● ● ●●● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ●● ●●● ● ● ●● ● ● ● ●● ●●● ● ●● ● ●●● ● ● ● ● ● ●● ●● ● ● ●● ●● ● ● ● ● ●

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param 2

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param 3

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param 4

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param 5

● ●● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ●● ●● ● ●●● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ●●● ● ●● ● ● ● ● ● ● ● ● ● ● ● ●● ●● ●● ●● ● ● ● ● ● ● ● ●●●● ● ● ● ●● ● ● ● ●● ●● ● ● ●● ● ● ● ● ● ● ●● ●●●● ● ●● ● ● ● ● ●● ● ● ●● ● ● ● ● ● ●

● ● ●●● ●●●● ●●●● ● ● ●●● ● ●● ● ● ● ●● ●● ● ●● ● ● ● ● ● ● ●● ● ● ● ●● ●●● ● ● ● ● ● ●● ● ● ● ● ● ● ●● ● ● ●● ●● ● ● ●● ●● ● ●●● ● ●●● ● ● ● ●●● ●●●● ● ● ● ● ●● ● ● ● ● ● ● ●●● ● ● ● ● ● ● ● ●●●● ●● ●●● ● ● ●

● ●●● ● ● ● ● ●● ● ●● ● ● ● ●●● ● ● ●● ● ● ● ● ●● ● ● ●● ● ●●●●● ● ● ●● ● ●● ● ● ● ● ●●● ●● ● ● ●● ● ● ●● ● ● ●● ●● ● ● ●● ●● ● ● ● ●● ●● ● ●● ● ●● ● ● ● ●● ● ● ● ●● ● ● ●● ●● ● ● ●● ● ● ●● ● ● ●●● ●● ● ● ● ● ● ●● ●

●● ● ● ● ●● ●● ● ●● ●●● ● ●● ● ●● ● ● ●●● ●● ● ● ●● ● ● ● ● ● ● ● ● ● ● ●●●● ● ● ● ● ● ●● ● ● ● ● ● ● ●● ●● ● ● ● ●● ● ● ● ● ● ●● ● ● ● ● ●● ● ● ●● ●●● ● ● ● ● ● ● ●● ● ● ● ● ● ● ●● ● ● ● ● ● ●● ●●● ● ● ● ● ●●● ● ● ● ● ● ●

● ●● ●● ●●● ● ● ● ●● ●● ●● ●● ● ●● ●● ● ● ● ● ●● ● ● ● ● ● ● ●●● ● ●● ● ● ● ● ● ● ● ●● ● ● ● ●● ● ● ● ● ● ● ●● ● ●● ●● ● ●● ● ● ● ● ●● ● ● ●● ●● ● ●● ● ●● ● ● ●● ● ●●● ● ● ● ● ●●●●● ● ●●● ● ●● ●● ● ● ● ●● ● ● ● ●●

● ● ● ●● ● ●● ●● ● ●●● ● ● ●● ● ● ●● ● ● ● ● ● ●● ● ●● ●● ●● ● ● ● ● ● ● ●● ● ● ● ●● ● ● ● ● ●●● ●● ●● ● ● ● ●● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ●● ●● ● ● ● ●● ●● ● ●● ●● ● ● ● ●● ● ●● ● ● ● ●● ●●● ● ● ● ●●

param 6

0.0

0.4

0.8

0.0

0.4

0.8

0.0

0.4

0.8

0.4

● ● ●● ● ● ● ● ● ●● ● ● ●●● ● ● ●●● ● ●● ● ● ● ● ● ● ● ●● ● ● ● ●● ● ●● ● ●●● ● ●● ● ● ● ● ● ● ● ●● ● ● ● ●● ●● ● ● ● ● ●●●● ● ● ● ● ● ● ● ●●●● ● ● ● ● ● ● ●● ●● ●●● ● ● ● ● ● ● ● ●● ● ● ● ●● ● ● ● ●● ● ● ●●● ● ●● ●●

0.0

● ●● ●● ● ● ● ● ● ● ●● ● ●● ●● ●● ● ● ●● ● ● ● ● ●● ● ● ●● ●●● ●● ● ●● ●●●● ● ●● ● ● ●● ● ● ● ● ● ● ● ● ●●● ● ●●● ● ● ● ● ●● ● ●● ●●● ● ● ● ● ● ● ●●● ●● ● ● ● ● ●● ● ● ● ● ● ● ●● ●● ●●● ●● ● ● ● ● ●● ● ● ● ●● ● ●●

0.8

0.8

0.8

0.4

0.4

0.0

0.0

0.8

0.8

0.8 0.4 0.0 0.8 0.4

0.4

● ● ●● ● ● ● ●●● ● ● ● ●● ●● ●●● ● ● ● ● ●●● ● ● ●● ● ● ●●● ● ●● ● ● ● ●● ● ● ● ● ●●● ● ● ●● ●● ● ● ●● ● ●● ● ●● ● ● ● ●● ● ●● ● ●● ● ● ●● ● ●● ● ● ● ● ● ● ● ● ● ●● ● ●● ● ● ●● ● ● ● ● ●● ● ● ● ● ● ● ● ● ●● ● ● ● ●● ● ●

● ●

0.0

Each 1-D projection has an equal spacing of points

0.0

0.4

0.8 0.4 0.0

parameters rescaled to interval [0,1]

0.8

● ●●● ● ● ●● ● ● ● ●● ● ● ●● ●●●●● ● ● ● ● ● ● ● ● ● ● ●● ● ● ●●●●● ● ● ● ●●● ● ● ● ● ● ● ●● ● ●● ●● ● ● ● ● ● ● ● ●● ● ●● ● ●● ● ● ● ● ● ●● ● ● ●● ●●●● ● ● ●●● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ●● ● ● ●● ●●● ● ● ● ● ●

param 1

128 points in 6 dimensions

0.4

0.0

0.0

Intelligent design

!!

!! ! !! ! ! ! ! !! ! ! !! !! !! !! ! !!! ! ! ! ! !! !! ! ! ! !! ! ! ! !! !!!! !! !!!! ! ! !! ! !! !! ! !! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! !! !! !! ! ! !! ! !!! ! !! ! ! ! ! !! ! ! !! !! ! !! !! ! !! ! !! ! ! ! ! ! !! ! ! ! ! !! ! ! !! ! ! ! !! ! !! ! ! ! ! !! ! !! ! ! !! ! ! !! !! ! !! !! !! ! !! !! ! ! !! ! ! ! !! !

0.980

0.995

! !! ! ! !! ! ! !! !! ! ! ! !!!! !!!!!! !!!! !!!! !!! !! ! ! !! !! ! ! !!! ! ! !! !! ! ! ! !!!! !! ! !!!! !! ! ! ! !! ! ! !! !! ! ! ! ! ! ! ! !! !

theta

! ! ! ! ! ! ! !! ! !!! ! !!! ! ! ! ! !! !! ! ! ! !!!! ! !!! ! !!!! ! ! ! ! ! !! !!!! ! ! !! ! ! !! ! ! ! !! !! ! !! ! ! !! !! ! ! ! ! !! ! ! ! ! !! ! !! ! ! ! ! !

! !! ! ! ! !! !!! ! ! ! ! !!! ! ! !! ! !! !! ! !!!! ! !! !! ! ! !!! !! ! ! ! !!! ! ! ! ! !! ! !!! ! ! ! ! ! !! !! ! ! ! ! !! !! !! ! ! ! ! !! !

!! ! ! !! ! !! ! ! ! ! !! ! ! ! !!!!! !!! ! !! ! ! !! ! !!! ! ! !!!!!!! !! ! ! !! !! !! ! ! !! !! ! !! ! !! ! ! !! ! !! ! !! ! ! !! ! ! ! ! ! ! ! !! ! ! !!

!

! !! !! ! ! !! !!! ! !!! ! !!!! ! ! !! !! !! ! ! !!! ! ! ! ! !! ! !! !! ! ! ! !! ! ! ! ! ! !! !! ! ! !! ! !! ! !!! ! ! !! ! ! ! ! ! !!

! !

! ! ! ! !!! ! ! ! !!!!! ! !! ! ! !!! !! !! !! ! !! ! ! ! ! ! ! !!! ! !! ! ! ! !!! ! ! ! ! ! !!!! !!!!! ! ! ! !!! ! ! ! !! !! ! ! !! ! ! ! ! ! ! ! ! !! ! !! ! !

! ! ! ! ! !! !!! !!! !! ! !! ! ! !! ! ! ! ! !! ! ! ! ! ! ! !!!!!!!! !! ! !! ! ! ! ! ! !! ! ! ! !!! ! !! ! ! ! ! ! !!! ! ! !! !! ! ! ! ! !!! !! !! ! !! ! ! ! ! ! !

! ! !! ! ! ! ! !! !! !! ! ! !! ! ! !! ! ! ! ! ! ! !! !!! ! !! ! ! ! ! !! ! !!! !! ! ! ! ! !! !! ! ! !!! !!! !! ! ! ! ! !! !!! !!! ! ! ! ! ! ! ! ! !! ! ! ! ! ! ! ! ! !

! ! !! ! ! ! ! ! !! !! ! ! ! !! ! !!!!! !! ! !! ! ! ! ! ! !! !!! ! !!! ! !! ! !!!!! ! ! ! ! ! ! ! !! !! ! ! ! ! !! !! !!! ! ! ! ! ! !! ! ! ! ! ! !! ! ! ! ! ! ! ! !!

! ! ! !! ! ! !! !!!!! !! ! ! !! !! ! ! ! ! ! ! ! ! ! ! !! ! !! ! ! !! ! ! ! !! ! !!! ! ! !! !! ! !! !! ! ! ! !! !! !!! ! ! ! !! ! ! ! !! !!

!

! ! ! ! !! ! ! ! ! ! !! ! !!! ! ! !! !!!! ! ! ! !!!! ! ! !! ! ! ! ! ! ! ! ! ! ! !!!! ! ! !! ! ! !! !! !! ! !! ! ! !!!!! ! !! ! ! ! !! ! ! !! ! ! ! ! ! ! ! ! ! ! !

!! ! ! ! ! ! !! ! ! ! ! ! ! ! !! !! !!!! ! ! ! !!!! ! !! ! !! !!!! ! ! ! !! !! ! ! !!! !! !! !! ! ! !! ! ! ! !! ! ! ! !! ! ! ! ! ! !! ! ! ! ! ! ! ! !! ! ! ! ! !

!!

! ! ! ! !! ! !! ! !! ! !! ! ! ! ! !!! ! !! ! !! ! ! ! !!! ! ! ! ! !! ! ! ! ! ! ! !! !! !! !!!! ! ! ! !!!! ! !! ! ! ! ! ! ! !! !!! ! ! !!! ! ! ! ! ! !!!! ! !! ! ! ! !

! ! ! ! ! !! ! ! !! ! !! ! !!! ! ! ! !! !! ! ! ! !!! ! ! ! !!! !!!! ! ! ! ! !! ! !!!! ! ! !! ! ! ! ! ! ! ! !! ! ! !! !! ! ! ! ! ! ! ! ! ! !! ! ! !!! ! ! !! ! ! ! ! !

! ! ! !!! !!! ! !! ! !! ! ! ! ! ! !! ! !! ! ! ! ! ! ! ! ! ! !!!! ! ! ! !! ! ! !!!! !! ! ! !!! ! ! ! ! !! ! ! ! ! ! !! ! ! ! !! ! ! ! ! ! ! ! ! ! ! !! !! ! ! !! ! !!

! !! ! ! !! ! ! ! !! !! !! !! ! ! ! ! ! ! ! ! ! !! ! !!! !! !! !! ! !! !!! ! ! ! ! !! ! ! ! ! ! ! !! !! ! !!!! ! ! ! !!! ! !!! ! !! ! ! ! !! ! ! ! ! ! !! ! ! !! ! !

! ! ! !! ! !! ! ! ! !! !! !! !!! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! !! ! !! ! ! ! ! ! ! ! !! !! ! ! !! ! ! !! !!! ! ! ! ! !! ! ! ! ! ! ! ! ! ! ! ! ! !!! ! !! ! !!

omegam

! ! !! ! !! ! !!!!! ! ! !! ! ! ! !! ! !! ! ! ! ! !!!!! !! ! ! ! ! !! ! ! ! ! ! ! ! ! ! !!! ! ! !! ! !! !! ! !! ! ! ! ! ! ! !! !!! !! !!!! !! !! !! ! ! ! !! ! ! !

! ! ! ! !!! ! ! ! ! !! !! ! !! !! !!! ! ! !! ! ! !! ! ! ! ! ! ! ! ! !! !!! ! ! ! ! ! ! ! ! !! ! ! ! !!! ! !!!!! ! ! ! !!! ! ! ! !!!! ! ! !! ! ! ! ! ! ! ! ! !! ! !! ! !

! ! ! ! !! ! !! ! !! !! !! !!! !! ! !! ! ! !! ! ! !! ! !! !! !! !!! !! ! ! ! ! ! ! ! ! ! !! ! ! !!! ! ! ! ! ! ! ! !! ! ! ! ! ! !! ! ! ! !! ! ! ! ! ! ! !! ! ! ! ! !

omegab

!! !! ! ! ! !! ! !! ! ! ! ! ! ! ! !!!! ! ! !! ! ! ! ! ! !! ! ! ! !! ! !! !! ! ! ! !! ! ! !! ! ! ! ! ! !! !! ! ! !! ! ! ! ! ! !! !!!! !! ! !! ! ! !! ! ! ! ! !! !! ! ! ! ! !!

!!! ! !! !! ! ! ! ! ! ! ! !! ! !! ! ! ! ! !! ! ! ! ! ! ! !!! ! ! !! !!! !! ! ! ! !! ! !! ! ! !! ! !! ! ! ! ! ! !! ! ! ! ! ! ! ! ! ! ! ! !! ! !!!! ! ! !! ! !! ! !! !!

! ! ! ! !! ! ! ! ! ! ! !!!! ! ! !! ! ! !!! ! ! !!! !! ! ! ! ! !! ! ! !! ! ! ! !!! !!! ! !! ! !!! !! ! !! ! !! ! ! ! !! !! ! !! ! !!! ! ! ! ! !! ! ! !! ! ! ! !! !! ! ! !

tau

!

sigma8

0.70 0.80 0.90

0.10

!

!

0.0436

0.0442

(using CMB Fisher matrix)

!

0.995

!! ! ! ! ! ! !! ! ! ! ! ! !!! ! !! ! ! ! !! ! !! ! ! !! ! ! ! !! !! !!! ! !! ! ! ! ! ! ! ! ! ! ! ! ! ! ! !! ! !! !! ! !!! ! ! ! ! ! ! ! !! ! ! !! !! !! ! ! !! ! ! !

!0.05

0.980

0.276

0.70 0.80 0.90

0.7102 0.7096 0.276 0.270 0.264

!! !!! ! !! ! !! !! !!! ! !! ! !! ! !! ! !! ! !! ! ! ! !! ! ! !! ! ! !! ! ! !!!!!! ! !! ! !!! !! ! ! !!! ! !! !!!! ! ! !! ! ! !!! ! ! ! !!! ! ! ! !

!

0.10

Huge volume reduction over hypercube design:

!!! ! ! !! ! ! !! ! !! ! !! ! !! !! !! ! ! !! ! ! ! ! ! ! ! ! ! !! ! !! !! ! !! !! ! ! ! ! ! ! ! ! ! ! ! ! !! !! ! ! !! !!! ! !! ! ! !! !! !! !! ! ! ! ! ! ! !!

0.270

! ! ! ! ! ! ! ! ! !! !! ! ! !! ! ! ! ! !! !! !! ! ! !! !! !! ! ! ! ! !! ! ! ! ! ! ! !! ! ! !! !! ! ! ! !!!!!!! !! !! !! ! ! !!! ! ! ! ! ! !! !! ! ! ! !! ! ! !!

!! ! ! ! ! ! !! ! !! !! ! ! ! ! !!!! ! ! ! ! !! !! ! ! ! ! !!!!! !! ! ! ! ! ! ! !! ! ! !! !!! ! ! ! !!!!!! ! ! ! !!!! ! ! !! ! !! ! ! ! !!!!! ! ! ! ! ! !

!

!0.05

Build design inside a hypersphere of constant probability

ns

0.264

0.0442

Use Fisher matrix to rotate and rescale parameter space

0.7102

0.0436

0.7096

GAUSSIAN PROCESS MODELS FOR INTERPOLATION

How to do interpolation in high dimensions We need to interpolate multivariate simulation output as a function of large (~ 10) numbers of parameters Power spectrum mean and covariance components modeled as Gaussian processes (GPs) (following Habib et. al 2007) Interpolation error propagated within Bayesian framework GP determined by correlation parameters for the interpolated surface GPs scale well for interpolation in high dimensions

Gaussian process models for spatial phenomena 2

z(s)

1 0 !1 !2

0

1

2

3

4

5

6

7

s

An example of z(s) of a Gaussian process model on s1, . . . , sn 32











z(s1)   0        .    . .  ∼ N  .  ,  z= 0 z(sn)      

Σ

      

, with Σij = exp{−||si − sj ||2},

where ||si − sj || denotes the distance between locations si and sj . z has density π(z) = (2π)

− n2

|Σ|

− 12

1 T −1 exp{− 2 z Σ z}.

Higdon, Williams, Gattiker (LANL)

n 1 − − Realizations from π(z) = (2π) 2 |Σ| 2 exp{− 21 z T Σ−1z} 2

z(s)

1 0 !1 !2 20

1

2

3

4

5

6

7

1

2

3

4

5

6

7

1

2

3

4

5

6

7

z(s)

1 0

33

!1 !2 20

z(s)

1 0 !1 !2

0

s

model for z(s) can be extended to continuous s

Higdon, Williams, Gattiker (LANL)

Conditioning on some observations of z(s) 2

z(s)

1 0 !1 !2

0

1

2

3

4

5

6

38

We observe z(s2) and z(s5) – what do we now know about {z(s1), z(s3), z(s4), z(s6), z(s7), z(s8)}? 



  z(s )  2     0       z(s )    '    0   5 '     ' 1 .0001    0  '  z(s1 )     '      '      '      .0001 1  z(s )   0   ' 3      '      .3679 ' 0 ∼ N ,      '  z(s )   0   '      4 '      ' . . . . . .      '  z(s )   0   '      6  '    ' 0 .0001        0   z(s7 )          0 z(s8 )

.3679 0 1 .. 0

··· 0 · · · .0001 ··· 0 .. ... ··· 1

7



                        

Higdon, Williams, Gattiker (LANL)

Conditioning on some observations of z(s)   











Σ12  z1  −1 −1  0   Σ11  , z2 |z1 ∼ N (Σ21 Σ ,  ∼ N  z , Σ − Σ Σ 22 21 11 Σ12) 11 1 Σ21 Σ22 0 z2 conditional mean 2

z(s)

1 0 !1 !2

0

1

2

3

4

6

7

5

6

7

39

5

contitional realizations 2

z(s)

1 0 !1 !2 0

1

2

3

4 s

Higdon, Williams, Gattiker (LANL)

A 2-d example, conditioning on the edge Σij = exp{−(||si − sj ||/5)2} mean conditional on Y=1 points

Z -2 -1 0 1 2 3 4

Z -2 -1 0 1 2 3 4

a realization

20

20 15 10 Y

15

5

5

15

20

10 Y

10 X

15 5

5

20

10 X

42

realization conditional on Y=1 points

Z -2 -1 0 1 2 3 4

Z -2 -1 0 1 2 3 4

realization conditional on Y=1 points

20

20

15 10 Y

15

5

5

10 X

20

15 10 Y

15

5

5

20

10 X

Higdon, Williams, Gattiker (LANL)

Limitations of Gaussian Processes

z(s)

s

. mode amp ha alp

alp

ha

. mode amp

A

A

EMULATOR

Power spectrum emulator Multivariate power spectrum output decomposed into incomplete orthogonal basis (achieves dimension reduction): µ(k, θ) = Φµ (k) w(θ) + "µ

!µ ∼ N (0, λ−1 ! )

Model basis weights as independent Gaussian Processes w(θ) ∼ GP (0, Σw (θ; λw , ρw ))

Do MCMC to calibrate GP parameters given the design runs !−1/2 ! −1 P (wdesign |λ! , λw , ρw ) ∝ !λ! + Σw !

" % # −1 $−1 1 T exp − wdesign λ! + Σw wdesign 2

Example: 2-parameter matter power spectrum emulator !

! !

!

!

!

!

!

!

!

!

!

!

! ! !

!

!

200

!

!

! !

!

!

!

!

!

!

!

!

ns

!

!

!

! ! !

! !

! !

! !

! !

!

!

!

!

!

!

!

100

1.00

!

!

!

!

0.90

!

!

!

!

0.95

! !

50

1.05

!

!

!

! !

!

!

20

1.10

!

!

!

! !

!

!

!

!

!

!

!

!

!

!

!

!

!

!

!

!

!

!

0.80

! !

!

!

!

!

!

!

! !

!

sigma_8

!

2000

!

500

! !

!

!

!

P(k)

!

!

!

!

!

20000

!

!

5000

!

!

1.10

0.85

!

1.05

0.90

1.00

0.75

0.95

0.70

0.90

!

0.001 0.70

0.75

0.80

0.85

0.005

0.050

0.90

!1.0 !1.5 !2.0 !2.5

!

!0.5

0.0

0.5

k

0.001

0.005

0.050 k

0.500

0.500

Covariance matrix parameterization Generalized Cholesky decomposition (Pouramahdi et. al 2007) T −1 Σ−1 (θ) = T (θ) D (θ) T(θ) y

Components of T are unconstrained:

ϕij ≡ −Tij

2 ≤ i ≤ ny ,

j = 1, . . . , i − 1

Impose prior structure on covariance with a ( θ independent) conjugate Gaussian prior on ϕ (allows “shrinking” to constant T)

ϕ ∼ N (ϕ, ¯ Cϕ )

Prior mean can be set from sample covariance of design runs Model ϕ as GP just like mean and “variance”

ny (ny − 1) ϕi (θ) ∼ GP (i , Σϕ (θ; λϕ,i ,ϕ,i )) i = 1, . . . , 2

Estimate covariance at each design point simultaneously - fewer realizations needed

Simplified emulator Simulation outputs reduced to mean and covariance estimates at ∗ ˜∗ µ ˜ each design point, , D Approximation: neglect error in sample mean and covariance Model “variance” as a GP just like the mean Sampling model for the data: y|w(θ), v(θ) ∼ N (Φµ w(θ), Σy (ΦD v(θ)))

The joint likelihood for parameter estimation breaks into: ˜ ∗ |θ0 , λ! , λ, ρ) = L(y, µ ˜∗ , D

!

dpD v L(w ˆy , w|v, ˆ θ0 , λ!µ , λw , ρw ) · π(v, vˆ|θ0 , λv , ρv )

Validation: toy power-law model 9

P (k) = A k

−α

var(P (k)) ∝ P (k)

Black: N-body Red: model Blue: mock data

7

8

2

! !

6

!

!

! !

!

5

!

! !

! !! ! !! !! ! !

!!! ! !

!3

!2

!1 log(k)

0

! !

! !

3

This gives more noticeable differences in posteriors for later validation tests

!

!

4

Assume the same number of modes are used to estimate P(k) in each band

log(P(k))

Covariance is diagonal

!

1

0.0

0.2

0.4

amplitude

!

PC2

!

PC3

!

PC4

!

PC5

!

0.2

0.4

0.6

0.8

1.0

slope

PC1

0.0

0.6

0.8

!

!

!

!

!

1.0

!

Emulator correlations Marginal posterior samples given design runs

0.2

0.4

0.6

30 pt. design amplitude

30 pt. design slope

7 pt. design amplitude

7 pt. design slope

0.8

5

4

3

2

1

Marginal distributions for the 2 “cosmological parameters”

5

4

3

Density

Parameter posteriors

0

2

1

0

30 pt. design: sample cov. amplitude

30 pt. design: sample cov. slope

5

4

3

2

1

0 0.2

0.4

0.6

0.8

Scaled model parameters

!5

PC weight 1

0

5

PC weight 2

Density

0.3

0.2

0.1

0.0

!5

0

5

PC weights of variance

Variance parameters Marginal posterior distributions of PC weights for the power spectrum variance

Summary Our method uses limited numbers of simulations to calibrate a model for the power spectrum sample variance distribution. Obtaining precise estimates of the power spectrum covariance is a challenge - full formulation may make this feasible Our framework can be readily applied to general parameter inference problems using simulations Plan to release an R package implementing these methods Next: demonstrate covariance matrix emulator using N-body simulations of the matter power spectrum

Gaussian process model formulation for the mean power spectrum Principal component weights of mean are modeled as independent Gaussian processes: pµ

µ(!k, θ) =

!

φµ,i (!k) wi (θ) + !$µ

wi (θ) ∼ GP(0, Σw (θ; λw , ρw ))

i=1 Design outputs also have Gaussian sampling model (from error term)

µ |w , λ!µ ∼ N (Φµ w ∗

∗

∗

−1 , λ!µ I),

λ!µ ∼ Γ(aµ , bµ )

After marginalization over GP realizations:

ΦTµ µ∗ ∼ complicated Normal distribution,

λ!µ ∼ modified Gamma prior

Emulator outputs at new designs points can be drawn from: ∗ (w , w(θ)) ∼ N (0, Σw,w(θ) (λw , ρw )) draws from posterior