![Likelihood of Correlated Gaussian fields
CMBR temperature anisotropy field is represented as
T(ˆn) =
∞
l=0
l
m=−l
almYlm(ˆn) (1)
where
alm = dˆnT(ˆn)Y ∗
lm(ˆn) and < alma∗
l m >= δll δmm Cl (2)
where the distribution of alm is Guassian with mean zero and
variance given by the angular power spectrum Cl , which
follows χ2 distribution.
f (x; n) =
1
2n/2Γ(n/2)
x(n/2)−1
exp[−x/2] for x > 0;
0, otherwise
(3)
note that
χ
2
=
n
i=1
yi − y(xi )
σi
2
(4)
which is the “square sum of Gaussian errors”.](https://image.slidesharecdn.com/likelihoodp2-180616050217/85/CMB-Likelihood-Part-2-3-320.jpg)
![CMBR temperature and polarization fields, which individually
follow Gaussian distribution, are correlated.
alm = (aT
lm, aE
lm, aB
lm) (5)
In this case we have six power spectra, two of these BB and
BE are zero for a “parity-invariant ensemble” so we are left
with CTT
l , CEE
l , CBB
l , CTE
l .
The covariance matrix Cl and estimator ˆCl are given as:
Cl ≡ alm a†
lm and ˆCl ≡
1
2l + 1 m
alm a†
lm (6)
The probability P({alm| Cl )of a set alm at a given l is given
by:
−2ln [P({alm| Cl )] =
m=l
m=−l
almC−1
l a†
lm + ln|2π Cl |
= (2l + 1) Tr[ ˆCl C−1
l ] + ln| Cl | + const (7)](https://image.slidesharecdn.com/likelihoodp2-180616050217/85/CMB-Likelihood-Part-2-5-320.jpg)
![Integrating out all the alm with the same ˆCl (or normalizing
with respect to ˆCl )gives a Wishart distribution for ˆCl :
P(ˆCl | Cl ) ∝
| ˆCl |(2l−n)/2
| Cl |(2l+n)/2
exp −(2l + 1)Tr[ ˆCl C−1
l ]/2 ∝ L( Cl |ˆCl )
(8)
It can be shown that the likelihood has a maximum Cl = ˆCl ,
so ˆCl is the maximum likelihood estimator.
For only temperature (n = 1) the Wishart distribution reduces
to χ2 distribution with (2l + 1) degrees of freedom:
−2lnP(ˆCl | Cl ) = (2l +1)
ˆCl
Cl
+ ln|Cl | −
2l − 1
2l + 1
ln(ˆCl ) +const (9)](https://image.slidesharecdn.com/likelihoodp2-180616050217/85/CMB-Likelihood-Part-2-6-320.jpg)
![Karhunen-Loveve Techniques
In any experiment there are always some modes which are
heavily contaminated by noise and somehow if we can identify
and discard those, we can speed up computation. If we can
keep only 10% of the modes, computational can speed up by
a factor of 1000 !
Karhunen-Loveve (KL) technique provides a prescription for
that.
[Tegmark et al. (1997); Dodelson (2003)]](https://image.slidesharecdn.com/likelihoodp2-180616050217/85/CMB-Likelihood-Part-2-7-320.jpg)
CMB Likelihood Part 2
- 1. Cosmological Microwave Background Radiation Likelihood Analysis - Part 2 Jayanti Prasad Inter-University Centre for Astronomy & Astrophysics (IUCAA) Pune, India (411007) September 13, 2011
- 2. Plan of the talk Cut-sky likelihood function Likelihood of the Polarization field Karhunen-Loveve Techniques
- 3. Likelihood of Correlated Gaussian fields CMBR temperature anisotropy field is represented as T(ˆn) = ∞ l=0 l m=−l almYlm(ˆn) (1) where alm = dˆnT(ˆn)Y ∗ lm(ˆn) and < alma∗ l m >= δll δmm Cl (2) where the distribution of alm is Guassian with mean zero and variance given by the angular power spectrum Cl , which follows χ2 distribution. f (x; n) = 1 2n/2Γ(n/2) x(n/2)−1 exp[−x/2] for x > 0; 0, otherwise (3) note that χ 2 = n i=1 yi − y(xi ) σi 2 (4) which is the “square sum of Gaussian errors”.
- 5. CMBR temperature and polarization fields, which individually follow Gaussian distribution, are correlated. alm = (aT lm, aE lm, aB lm) (5) In this case we have six power spectra, two of these BB and BE are zero for a “parity-invariant ensemble” so we are left with CTT l , CEE l , CBB l , CTE l . The covariance matrix Cl and estimator ˆCl are given as: Cl ≡ alm a† lm and ˆCl ≡ 1 2l + 1 m alm a† lm (6) The probability P({alm| Cl )of a set alm at a given l is given by: −2ln [P({alm| Cl )] = m=l m=−l almC−1 l a† lm + ln|2π Cl | = (2l + 1) Tr[ ˆCl C−1 l ] + ln| Cl | + const (7)
- 6. Integrating out all the alm with the same ˆCl (or normalizing with respect to ˆCl )gives a Wishart distribution for ˆCl : P(ˆCl | Cl ) ∝ | ˆCl |(2l−n)/2 | Cl |(2l+n)/2 exp −(2l + 1)Tr[ ˆCl C−1 l ]/2 ∝ L( Cl |ˆCl ) (8) It can be shown that the likelihood has a maximum Cl = ˆCl , so ˆCl is the maximum likelihood estimator. For only temperature (n = 1) the Wishart distribution reduces to χ2 distribution with (2l + 1) degrees of freedom: −2lnP(ˆCl | Cl ) = (2l +1) ˆCl Cl + ln|Cl | − 2l − 1 2l + 1 ln(ˆCl ) +const (9)
- 7. Karhunen-Loveve Techniques In any experiment there are always some modes which are heavily contaminated by noise and somehow if we can identify and discard those, we can speed up computation. If we can keep only 10% of the modes, computational can speed up by a factor of 1000 ! Karhunen-Loveve (KL) technique provides a prescription for that. [Tegmark et al. (1997); Dodelson (2003)]
- 8. How Karhunen-Loveve Technique ? We carry out a coordinate transformation on the data vector ∆ ∆i ≡ Rij ∆j , (10) and the new covariance matrix Cij = (R∆)i (R∆)j or C = RCRT (11) where the old covariance matrix is given by C = ∆i ∆j ≡ CS,ij + CN,ij (12) Note that CN and CS are real symmetric matrices so can be easily diagonalized.
- 9. How Karhunen-Loveve Technique works ? This technique needs the following three rotations: R1 : To diagonalized the noise covariance matrix CN. R2 : To make the diagonalized CN, i.e., CN unity. R3 : To diagonalized the new signal covariance matrix CS . Example: CN = σ2 n 0 0 σ2 n , CS = σ 2 s 1 1 R1 = I, R2 = 1 σ2 n I and R3 = 1 √ 2 1 1 −1 1 which gives ∆1 = 1 √ 2σN (∆1 + ∆2), ∆2 = 1 √ 2σN (∆1 − ∆2) and CS = σ2 s σ2 n 1 + 0 0 1 −
- 10. Summary & Discussion I have discussed the difficulties and approximations used to compute the CMBR likelihood function. My aim was to understand why and how the likelihood function is computed differently for different values of l and different type of power spectra. I have also discussed various estimators related to angular power spectrum and the likelihood function. I was not able to discuss the actual computation of the various components of the likelihood function in the WMAP likelihood code. It has been suggested (by Dodelson) that the Nelder Mead method or downhill simplex method (commonly known amoeba) can be be used for finding the best fit parameters. Another method called NEWUOA which is used for unconstrained optimization without derivatives, is also suggested (by Antony Lewis ) to find the best fit parameters, optimization of the likelihood function.
- 11. Dodelson, S. 2003, Modern cosmology (San Diego, U.S.A.: Academic Press) Tegmark, M., Taylor, A. N., & Heavens, A. F. 1997, Astrophys. J. , 480, 22