Durham 102208

of 35

Content

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

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

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
●●

● ●●● ●

● ●
● ● ●

● ●
●●

●●
●●

●● ● ●●

● ●●●● ●●

●●

●●●
● ●
●●● ● ●

● ● ● ●●

● ● ●

● ●● ●●●

●● ●

● ●● ●●●

●● ●
●●●

● ●

●● ●● ● ●
●● ●●

● ● ● ●●● ●
● ●
●● ●
● ●
● ● ●●●
● ●●

●●

●●

●●

●●● ●
●●

●●
●●

● ●
●●

●●● ● ●

● ● ●●

● ●● ●● ●● ●
● ● ●● ●
● ●
● ●●

● ●● ● ●●●

●●

● ●●● ●

● ● ●● ●

param 2

● ● ● ●● ● ● ●

● ●●
●●
●● ●
● ●● ●
●● ●●●● ● ●

● ● ● ●●

●● ● ●●

● ●● ●

● ● ●●
● ●
● ●●

● ● ●
●●●

● ●● ● ●
●●

●●
●●
●● ●●
●● ● ●

●●
● ● ●
● ● ●●●

●●● ● ●●● ● ●

● ● ● ●● ●
●●

●●

●●● ●●

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

●●
● ●
●● ●●●

● ● ●

● ● ●●●
●● ● ●

●●

●● ●●●

● ● ●

● ●●● ●●
●● ●

●● ●
● ●
●● ●●
●●
● ●●

● ● ●●

● ●

● ●●
●●
●●●● ●
● ● ●● ●●
●●

● ●● ●

● ●●

● ●
● ● ●
●●
●● ● ●

●●
●●

●● ●
● ● ●

● ● ●
●● ●

● ● ●●
●● ● ●
●●
● ● ●
● ● ●● ●
●●

● ●● ● ●
● ● ●

●●●
●●●
● ● ● ●
●●

●●● ●●●

● ● ●● ●
● ●●
● ●●

●● ● ● ●

● ●● ●
●● ●
●● ● ● ●

●●

● ●
● ● ●●

● ● ● ● ●● ●

●●
● ●

● ●
● ● ●
● ●

● ●●●●● ●

●●●● ●

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

●●
●●
●● ●● ●

●●
●●
● ●● ●

● ● ●●

● ●●●●
●●
● ●●●
●●

●● ●
●●
●●●

●● ● ●●●
●● ●
●●
●● ● ●

●● ● ● ●

● ●●
●●
●● ● ●

●●● ● ● ●● ● ●

● ● ●● ● ●●

●● ●●

●●
● ●●
●●

●●

● ● ● ●

● ● ●

●● ● ●

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

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

●●
● ●
● ●
●●● ●

●●

● ●● ●
●●●
●●● ●

● ●
● ● ● ●●●

● ●● ●
●●

● ●●
● ●
●● ●●
● ● ●●●● ●
●●

param 3

● ●● ●
●●
●●

●● ●● ● ●
● ● ●●●

●● ●● ●

● ●●
● ●●
● ●● ●
● ●● ● ● ● ●

●●
● ●
●●● ● ●
● ●
●● ● ●●● ●

● ● ● ● ●●

● ● ●● ●

●●
●● ● ● ● ●●● ●
● ●
● ● ● ●● ●

● ●

● ●●
● ● ●●●

● ●

●●

●● ●
●●
●●●●

●●●

● ●●●●● ● ●●

●●
●●●

●● ●

●● ●●

● ●
●● ●

●●●
● ●● ●
●●
●●
●●

● ●●

●●● ● ● ●
●●
●●

● ● ● ●●
●●●
●● ●

●●●

●● ● ● ● ● ●
●●
● ● ●●

●● ● ●

●●

●●●●● ●● ● ●
● ●

●●●
● ●●●
●●
● ● ●●●

● ● ●●

●●● ●
● ●●●

● ● ●● ●
●● ●
● ●●
●● ●
● ● ●●
●●

● ●
● ● ● ●●●
●●

● ● ●●

● ● ● ●●
● ●● ● ● ●● ● ● ●

●●
●●

●●●
●●●

●●● ● ● ●
●●

● ●● ●
●●
●● ●

●●
●●
● ●●
●●●●●● ●
● ●● ●

●●
●●

●●● ●

●●
● ●

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

●● ●

● ●

●●
● ●●● ● ●●● ●

● ●
● ●●● ● ● ●● ●
●● ●
● ● ●● ●

●●● ●
● ● ● ●●
●●
● ●● ● ● ●
●● ●

● ● ●
● ●
●●●●

● ●

●●
● ●

● ●●
●●
●● ● ● ●

●● ●●

●●
●●
●●
●●●
● ●●

●● ●
● ●

● ●● ● ●

● ● ●●

●● ● ● ●● ● ●

●●● ●● ●
●● ● ● ●

●●

● ● ●● ● ●
●●
●●● ●
●●● ● ●
● ●●●●

● ●

● ●
● ● ●
●● ●●
● ●●

● ● ●●● ●
●●● ●
●●●●

● ●●

●●
● ●● ● ● ●●

● ●
●●●

● ● ●● ●●

●●

● ●● ●
● ●● ● ● ●
● ● ●

●●●● ● ●

param 4

● ●●
● ●
●● ● ●●
● ●
●● ●
● ● ●●

●●
●●● ● ●

●●
●● ●

● ● ●●● ●
●● ●

●●
● ● ●● ● ●
●●
●●

● ●
●●

●● ● ● ● ● ●

● ●● ● ● ●

●●

●● ●

●●● ●● ●● ●

● ●●

●●

● ● ● ●● ● ●●● ●

● ●

● ● ● ●● ● ● ●●

● ●●
● ● ● ●● ●●

● ● ●
●●
● ●●●● ●
● ●●

● ●●● ●

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

● ●● ●
●●

● ●●
● ●●●

●●●● ●●● ●
● ●

●●
● ● ● ●

● ●
●●●
●●

●●● ●●● ●

● ●●
● ●● ● ●
● ●
●● ● ●
●●
●●● ●
●●

●●

● ●●●
● ● ●

● ●

●●
●● ●
●●
● ● ●● ●

● ●

●● ● ●
●● ●●
● ●
● ●

●● ●

●●
●●
●●

●●

●● ●●
● ●●

● ●

● ●●
● ●●
● ●●●●●
● ● ●●
●●

● ● ● ●● ●

●●
● ● ●●●● ●
● ●
● ●●● ●●
● ●

●●● ●
●● ●
● ●●

●●
●●●● ● ●
●●●

●● ●● ● ●
●●
● ●●

●●●
●● ●

●● ●●
●● ●
● ●●
● ● ●● ●● ●●

● ●

● ● ●●

●●
●● ● ● ● ●
●● ●

●● ●
●● ●●

● ● ● ●● ●

● ●
●●

●●● ●

●● ●
●● ●
●●●
● ● ●●
● ●

●●

● ●

●●
●●

● ●
● ● ●● ● ●
●●

●●●

●●● ● ●

● ●

●● ●

●●
●●

● ●● ●

●●● ● ●●

●●

● ● ●●
● ●● ● ●●●●

●●

● ●●

●●

●● ● ●
●●
●● ●● ● ●

●● ●●●
● ●
●●●

●●

● ● ● ●● ●●● ●

●●
● ●● ●

●● ●

●●
● ●

●● ●●
●●
●● ●

●● ●

●● ●

●●
●●

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

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:

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

Recommended

Bull Durham Screenplay

Or use your account on DocShare.tips

Hide