cmb power spectrum estimation with hamiltonian...
TRANSCRIPT
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CMB Power Spectrum Estimation withHamiltonian Sampling
J.F. Taylor, M.A.J. Ashdown, M.P. Hobson
March 17, 2008
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Outline
I Aim of power spectrum estimation
I Standard methodsI Sampling
I GibbsI Hamiltonian Monte CarloI Tests on simulated data
I Extension to polarisationI What are the new challenges?I Preliminary results
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The aim of power spectrum estimation
Figures WMAP science team
I Pixelised CMB map t (xp)
t (xp) =`=`max∑`=0
∑m=−`
a`mY`m (xp)
I For an isotropic Gaussian CMB
〈a`ma∗`′m′〉 = C`δ``′δmm′
I Observed data d = s + n wheres = Rt = RYa
I pixelised map R = WBI time-ordered data R = AB
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Aim of power spectrum estimation
I non-stationary, correlatednoise
I beams
I cut-sky / partial coverage
I foregrounds
I systematicsI large data-sets
I WMAP : Npix ≈3× 106 `max ≈ 1000
I Planck : Npix ≈ 5× 107
`max ≈ 2500 ∼ 3000
Figures WMAP science team
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pseudo-C` estimators
I Frequentist approach (Peebles 1973; Hivon et al. 2002)
I compute sphericalharmonic coefficientsof data map a`m
C` =1
2`+ 1
∑m
|a`m|2
I Clearly different fromtrue C` applycorrections for the
1. cut2. noise3. beams4. filtering . . .
(Hivon et al. 2002)
I Fast
I but sub-optimal, particularly at low-`
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Maximum-likelihood
I likelihood L = Pr (d|C`) =∫
Pr (d|s) Pr (s|C`) ds
I Gaussian noise and signal
L ∝∫e−
12(d−s)T N−1(d−s)e−
12sT S−1sds
where N = 〈nnT 〉 and S = 〈ssT 〉 non-sparse
I Complete square and integrate
lnL = constant− 12{
ln |S + N|+ dT (S + N)−1d}
I Obtain ML estimate using an iterative algorithm, e.g.Newton–Raphson
I Basic method requires storage O(N2pix), operations O(N3
pix)
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more ML
There are a number of shortcuts
I Solve matrix equations with conjugate gradient
O(NiterN2pix)
I Compute traces with Monte–Carlo
O(NMCNiterN2pix)
I Ring torus method, for some (i.e. Planck) scanning strategiesS, N same block diagonal form
O(N2pix)
Feasible only up to Npix ∼ 104 or 105
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Sampling as an alternative approach
We would actually like to know about the posterior distribution
Pr (C`|d)
It is possible to sample from the joint density
Pr (C`,a|d)
and we could marginalize over a.
I Need to sample in extremely high dimensional space
I Most conventional Monte Carlo methods move throughparameter space by a random walk
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Gibbs sampling
I For sampling multi-dimensionaldistributions
I Joint distribution Pr ({xi}) hard tosample
I Conditional distribtionsPr (xi|{xj}j 6=i) tractable
I Sample from each conditionaldistribution in turn to build upjoint density
I No parameters to tune
I Explores parameter space by random walk
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Gibbs sampling
Collect samples by alternately drawing fromthe conditional distributions for C` and a
ai+1 ← Pr(a|Ci`,d
)∝ Pr (d|a) Pr
(a|Ci`
)Ci+1` ← Pr
(C`|ai,d
)∝ Pr
(ai|C`
)Pr (C`)
I C` step is simple. . . but slow for low signal to noise . . . so bin
I a step is hard
I Limited to Gaussian noise and signal
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Gibbs sampling
Signal sample
I distribution is multivariate Gaussian in a space
a =(S−1 + RtN−1R
)−1RtN−1d
V =(S−1 + RtN−1R
)−1
I performed using transformed white noise sampling
I solved using conjugate gradient method
Computational cost
I write matrix equations in form of SHTs
storage O(Npix), operations O(N3/2pix )
I need to construct preconditioner
I hence approach requires ∼ 100− 200 SHTs per a sample
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Hamiltonian Monte Carlo
I Proposed (Duane et al. 1987)
I Draws parallels between sampling and classical dynamics
I Introduces persistent motion into the Markov Chain → movesthrough high dimensional spaces efficiently.
I For each parameter xi we introduce a momentum pi anddefine the Hamiltonian
H =∑i
p2i
2mi+ ψ (x)
where ψ (x) = − log{Pr (x)} and mi is a fictional massassociated with each variable.
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Hamiltonian Monte Carlo
I Draw momenta pi fromGaussian with variance mi
I Move x,p along a trajectoryaccording to Hamilton’sequations using simpleiterative scheme.
I After random time testcandidate point withMetropolis rule.
I If using exact Hamiltonian and trajectory is accurate theacceptance rate will be 100 %.
I Explores correlations and degeneracies with relative ease.
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Hamiltonian Sampling
I Draw samples for C` and a simultaneously.
I ‘Potential’
ψ =12
(d− Ra)tN−1 (d− Ra) +∑`
(`+
12
)(lnC` +
σ`C`
)I Gradient
∇aψ = RtN−1 (d− Ra) +(l +
12
)a
C`
∇C`ψ =
(l +
12
)1C`
(1− σ`
C`
)where σ` = 1
2`+1
∑m |a`m|2 the spectrum of the signal
I recall R = BYI one SHT for potentialI two SHTs for gradient
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WMAP
I Simulated W-bandmap
I Nside = 512 i.e.3× 106 pixels
I Noise and beams asfor a combinedW-band map
I Kp2 mask (∼ 15% ofthe sky
I Currently takes a day on a workstation to process up to`max = 512
I Initially we had problems with long correlation lengths but wenow better understand how to keep these low.
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Polarisation
I Small signal
I Low multipoles of particularinterest
I Dominant foregrounds. . . large mask
I Ambiguity between E and B
Possible to separate E/B inmaps but we consider estimatingspectra directly from the data.
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PolSpice
Estimates power spectrum using correlation functions.(Chon et al, 2004, Szapudi, Prunet, Colombi 2001)
BB spectrum with full sky (top) and with WMAP cut (bottom)
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Sampler
BB spectrum with full sky (blue) and with WMAP cut (red). Mean spectrum rather than maximum likelihood.
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Conclusions
I Sampling provides a fast and optimal framework forperforming power spectrum estimation
I Hamiltonian Monte Carlo is a good candidate for performingthe sampling ... fast and flexible
I Optimal estimates are needed for polarisation
next. . .
I incorporate component separation?
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Scaling with problem size
Supressing random walk → increased sampling efficiency for largedimensional problems
Metropolis–HastingsGibbsHamiltonianMC
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Blackwell-Rao
We can use the a samples to form a fast likelihood code.(Wandelt et al. 2004; Chu et al. 2004)
I Allows us to compute Pr (C`|d) for arbitrary values of C`given our samples
Pr (C`|d) ≈ 1Nsamples
Nsamples∑i
Pr(C`|σi`
)For a Gaussian field
Pr (C`|σ`) ∝∞∏`=0
1σ`
(σ`C`
) 2`+12
exp(−2`+ 1
2σ`C`
)
I Require large numbers of samples to analyse high resolutiondata exactly
I But certainly useful at low `
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A convergence diagnostic
Hanson (2001) proposed the following diagnostic that makes use ofgradient information.Compare two estimates of the variance that depend differently onwhere our samples lie in the distribution.
var1(x) =∫ ∞−∞
(x− x)2 Pr (x) dx
var2(x) =13
∫ ∞−∞
(x− x)3 Pr (x)∂ψ(x)∂x
dx +13
∣∣∣(x− x)3 Pr (x)∣∣∣∞−∞
For most ‘interesting’ distributions the second term is zeroSo we compute the ratio from a set of samples {xk}
R =
∑k
(xk − x
)3 ∂ψ(x)∂x
∣∣xk
3∑
k (xk − x)2
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Hamiltonian Monte Carlo
Proposed (Duane et al. 1987)
I Draws parallels between sampling and classical dynamics
I Introduces persistent motion through the parameter space
For each parameter xi we introduce a momentum pi and definethe Hamiltonian
H =∑i
p2i
2mi+ ψ (x)
where ψ (x) = − log{Pr (x)} and mi is a fictional mass associatedwith each variable. (In general we can have a mass matrix M)
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Hamiltonian Monte Carlo
1. draw new momenta pi from Gaussian with variance mi
2. propagate x,p along a trajectory in the (x,p) space fromHamilton’s equations
∂p
∂t= −∂H
∂x
∂x
∂t=∂H
∂p
3. after some (randomised) length of time halt and accept thenew point according to the Metropolis rule
4. discard p variables, x sample Pr (x)
I We can use any Hamiltonian we like to define our trajectoryas long as we use the correct Hamiltonian to make theaccept/reject decision
I If we use the true Hamiltonian and simulate the dynamicsexactly then every proposed point will be accepted.Conservation of energy.
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The leapfrog method
I Simple method for followingdynamics
I Robust to numerical errors
I Reversible
I iterate for T = nτ
I randomise τ and n to avoidresonance conditions
p (t+ τ/2) = p (t)−τ2∇xψ (x) |x=x(t)
x (t+ τ) = x (t)+τ
mp (t+ τ/2)
p (t+ τ) = p (t+ τ/2)−τ2∇xψ (x) |x=x(t+τ)