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Cosmological Parameter Estimation from the CMB
Andy Taylor 1 , Alan Heavens 2 , Bill Ballinger 3
Institute for Astronomy, University of Edinburgh, Royal Observatory, Blackford Hill, Edinburgh, EH9 3HJ, U.K.
Max Tegmark 4
Institute for Advanced Study, Princeton, U.S.A
We discuss the problems of applying Maximum Likelihood methods to the CMB and
how one can make it both efficient and optimal. The solution is a generalised eigenvalue
problem that allows virtually no loss of information about the parameter being estimated,
but can allow a substantial compression of the data set. We discuss the more difficult
question of simultaneous estimation of many parameters, and propose solutions. A much
fuller account of most of this work is available (Tegmark, Taylor & heavens (1997)).
1 Likelihood Analysis
The standard method for extracting cosmological parameters from the CMB is through the use of
Maximum Likelihood methods. In general the likelihood function, L, for a set of parameters, `, is
given by a hypothesis, H x , for the distribution function of the data set. In the case of uniform prior,
and assuming a multivariate Gaussian distributed data set consistent with Inflationary models, the a
posteriori probability distribution for the parameters is
L(`jx;H x ) = (2ú) \Gamman=2 jC(`)j \Gamma1=2 exp
Ÿ
\Gamma
1
2 x y C(`) \Gamma1 x
–
; (1)
where ` =
(Q;h;\Omega 0
;\Omega \Lambda
;\Omega b ; :::) are the usual cosmological parameters we would like to determine.
Examples of data are x = \Delta T or a `;m and the statistics of the n data are fully parametrised by the
data covariance matrix, C(`) = hxx y i. For simplicity here we assume the data have zero means.
2 Problems with the likelihood method
Two important questions we would like to settle about likelihood analysis are (a) is the method optimal
in the sense that we get the minimum variance (smallest error bars) for a given amount of data? and
(b) is the method efficient -- can we realistically find the best­fitting parameters? As an example of
this last point, if we have n data points (pixels, harmonic coefficients, etc), and m parameters to
estimate with a sampling rate of 1=q, we find that the calculation time scales as
Ü / q m \Theta n 3 (2)
where the first term is just the total number of points at which we need to calculate the likelihood,
and the second term is the time that it takes to calculate the inverse of C and its determinant. Of
course, in practice one would not find the maximum likelihood solution this way, but it serves to
illustrate the point. Note that the covariance matrix depends on the parameters and therefore must
be evaluated locally in parameter space. For MAP or Planck we have m ¸ 11, q ¸ 10 and n ¸ 10 7 ,
resulting in Ü ¸ H \Gamma1
0 , even for nanosecond technology. But before we give up in dismay, it is worth
looking a bit further at the theory of parameter estimation.
1 mailto:ant@roe.ac.uk
2 mailto:afh@roe.ac.uk
3 mailto:web@roe.ac.uk
4 mailto:max@ias.edu
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3 Parameter information and the solution to our problems
Suppose we have found the maximum likelihood solutions for each parameter, ` = ` 0 , then the likeli­
hood function can be approximated by another multivariate Gaussian about this point;
L(`j` 0 ; H ` ) = (2ú) \Gammam=2 jF j 1=2 exp
Ÿ
\Gamma
1
2 ffi ` y F ffi `
–
; (3)
where ffi ` = ` \Gamma ` 0 is the distance to the maximum in parameter space and the parameter covariance
matrix is given by the inverse of F , the Fisher Information matrix;
F ij = hffi` i ffi ` j i \Gamma1 = 1
2 Tr[A i A j ]; (4)
(if the means of the data are dependent on the parameters, this is modified -- see Tegmark, Taylor &
heavens (1997)). The far right hand side expression can be calculated for Gaussian distributed data
sets (ie equation 1), where A i j @ lnC(`)=@` i is the slope of the log of the data covariance matrix in
parameter space.
By considering the Fisher matrix as the information content contained in the data set about each
parameter, we see that the solution to our problem is to reduce the data set without changing the
parameter information content. Hence to solve the problem of efficiency, we need to make a linear
transformation of the data set
x 0 =Bx; (5)
where B is a n 0 \Theta n matrix where n 0 Ÿ n, and so x 0 may be a smaller data set than x. If n 0 ! n the
transformation is not invertible and some information about the data has been lost. To ensure that
the lost information does not affect the parameter estimation (requirement (a)), we also require
@F 0
@B
= 0; (6)
where F 0 =BFB y is the transformed Fisher matrix. In order to avoid learning the unhelpful fact that
no data is an optimal solution, we add in the constraint that data exists. Since we have the freedom
to transform the data covariance matrix, we add the constraint –(BCB y \Gamma I), where I is the unit
matrix and – is a Lagrangian multiplier.
It can be shown (Tegmark, Taylor & heavens (1997)) that this is equivalent to a generalised
Karhunen­Lo`eve eigenvalue problem, which has a unique solution B for each parameter. These solu­
tions have the property that
B(@ ` i
C)B y = – i I; (7)
where – i = 1=oe 0
i are the eigenvalues of the transformed data set and the inverse errors associated with
each eigenmode of the new data set.
The new, compressed data set, x 0 , can now be ordered by decreasing eigenvalue, so that the
first eigenmode contains the most information about the desired parameter, the second slightly less
information, and so on. The total error on the parameter is then simply given by the inverse of the
1 \Theta 1 Fisher matrix
F 0 = 1
2
n 0
X
i=1
` 1
oe 02
i
'
: (8)
We are now free to choose how many eigenmodes to include in the likelihood analysis. A compression
of 10 will lead to a time saving of 10 3 . However this is only exact if we know the true value of the
parameters used to calculate B. But if we are near the maximum likelihood solution then we can
iterate towards the exact solution.
This procedure is optimal for all parameters -- linear and nonlinear -- in the model. In the special
case of linear parameters that are just proportional to the signal part of the data covariance matrix
(for example the amplitude of C ` , if the data are the a`m ), the eigenmodes reduce to signal­to­noise
eigenmodes (Bond (1994)). Hence our eigenmodes are more general than signal­to­noise eigenmodes.
Furthermore, as our eigenmodes satisfy the condition that the Fisher matrix is a maximum, they are
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Figure 1: The 3 heavy lines show the error bars on 3 CMB parameters as a function of the number
of modes used. Each set of modes has been optimised for the parameter in question. Note that
approximately 400 modes are all that is required to get virtually all the information from the entire
4016 cut COBE dataset. The thin lines show the conditional errors from the SVD procedure outlined
in section 5: virtually all the (conditional) information on all 3 parameters is obtained from the best
500 SVD modes.
the optimal ones for data compression. Any other choice, including signal­to­noise eigenmodes, would
give a higher variance.
In Figure 1 we plot the uncertainty on 3 parameters for COBE­type data, the quadrupole, Q, the
spectral index of scalar perturbations, n and the re­ionization optical depth, Ü .
4 Estimating many parameters at once
The analysis presented so far is strictly optimal only for the conditional likelihood -- the estimation
of one parameter when all others are known. A far more challenging task is to optimise the data
compression when all parameters are to be estimated from the data. In this case, the marginal error
on a single parameter ` i rises above the conditional error 1= p
F ii to
q
F \Gamma1
ii . As far as we are aware,
there is no general solution known to this problem, but here we present some methods which have
intuitive motivation and appear successful in practice.
Suppose that we repeat the optimisation procedure, outlined above, m times, once for each para­
meter. The union of these sets should do well at estimating all parameters, but the size may be large.
However, many of the modes may contain similar information, and this dataset may be trimmed
further without significant loss of information. This is effected by a singular value decomposition of
the union of the modes, and modes corresponding to small singular values are excluded. Full details
are given in Tegmark, Taylor & heavens (1997), and an example from COBE is illustrated in Fig­
ure 1, which shows that for the conditional likelihoods at least, the data compression procedure can
work extremely well. However, this on its own may not be sufficient to achieve small marginal errors,
especially if two or more parameters are highly correlated. This is expected to be the case for high­
resolution CMB experiments such as MAP and Planck (e.g. for parameters Q rms and n). To give a
more concrete example -- a thin ridge of likelihood at 45 ffi to two parameter axes has small conditional
errors, but the marginal errors can be very large. This applies whether or not the likelihood surface
can be approximated well by a bivariate Gaussian.
This latter case motivates an alternative strategy, which recognises that the marginal error is
dominated not by the curvature of the likelihood in the parameter directions, but by the curvature
along the principal axis of the Hessian matrix with the smallest eigenvalue. Figure 2 shows how
various strategies fare with a simultaneous estimation of the amplitude of clustering \Delta and the redshift
distortion parameter fi, in a simulation of the PSCz galaxy redshift survey. The top left panel shows
the likelihood surface for the full set of 508 modes considered for this analysis (many more are used
in the analysis of the real survey). The modes used, and indeed the parameters involved, are not
important for the arguments here. We see that the parameter estimates are highly correlated. The
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Figure 2: Illustration of data compression with different algorithms. Top left: `Full' dataset of 508
modes (for details of parameters etc, see text). Top right: Best 320 modes optimised for measuring fi.
Bottom left: Best 320 modes from SVD application to modes optimised for fi and \Delta. Bottom right:
Best 320 modes for optimising along the likelihood ridge axis. Likelihood contours are separated by
0.5 in natural log.
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second panel, top right, shows the single­parameter optimisation of the first part of this paper. The
modes are optimised for fi, and only the best 320 modes are used. We see that the conditional error in
the fi direction is not much worse than the full set, but the likelihood declines slowly along the ridge,
and the marginal errors on both fi and \Delta have increased substantially. In the panel bottom left, the
SVD procedure has been applied to the union of modes optimised for fi and \Delta, keeping the best 320
modes. The procedure does reasonably well, but in this case the error along the ridge has increased.
The bottom right graph shows the result of diagonalizing the Fisher matrix and optimising for the
eigenvalue along the ridge. We see excellent behaviour for the best 320 modes, with almost no loss of
information compared with the full set. This illustrative example shows how data compression may
be achieved with good results by application of a combination of rigorous optimisation and a helping
of common sense.
5 Conclusions
We have shown that single­parameter estimation by likelihood analysis can be made efficient in the
sense that we can compress the original data set to make parameter estimation tractable, and it is
optimal in the sense that there is no loss of information about the parameter we wish to estimate. Our
eigenmodes are generalised versions of the signal­to­noise eigenmodes, and are optimal for parameters
entering the data covariance matrix in arbitrary ways.
As with all parameter estimation, this is a model­dependent method in the sense that we need only
to know the covariance matrix of the data and the assumption of Gaussianity. However we have not
had to introduce anything more than the standard assumptions of likelihood analysis. The dependence
on the initial choice of parameter values is minimal, and can be reduced further by iteration.
For many­parameter estimation, we have shown the effects of two algorithms for optimisation.
Optimising separately for several parameters by the single­parameter method, and trimming the
resulting dataset via an SVD step is successful in recovering the conditional likelihood errors. For
correlated parameter estimates, a promising technique appears to be to diagonalize the Fisher matrix
and optimise for the single parameter along the likelihood ridge.
References
Bond J.R., 1994, Phys. Rev. Lett., 72, 13.
Tegmark M., Taylor A., Heavens A., 1997, ApJ 480, 22 (astro­ph/9603021).
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