Putting the precision in precision cosmology: How accurate should your data covariance matrix be?
MNRAS 432, 1928–1946 (2013)
doi:10.1093/mnras/stt270
Advance Access publication 2013 May 10
Putting the precision in precision cosmology: How accurate should your
data covariance matrix be?
Andy Taylor,1‹ Benjamin Joachimi1 and Thomas Kitching1,2
1 Scottish Universities Physics Alliance (SUPA), Institute for Astronomy, School of Physics and Astronomy, University of Edinburgh, Royal Observatory,
Blackford Hill, Edinburgh EH9 3HJ, UK
2 Mullard Space Science Laboratory, University College London, Holmbury St. Mary, Surrey RH5 6NT, UK
Accepted 2013 February 11. Received 2013 February 9; in original form 2012 October 25
Cosmological parameter estimation requires that the likelihood function of the data is accurately known. Assuming that cosmological large-scale structure power spectra data are
multivariate Gaussian distributed, we show that the accuracy of parameter estimation is limited by the accuracy of the inverse data covariance matrix – the precision matrix. If the data
covariance and precision matrices are estimated by sampling independent realizations of the
data, their statistical properties are described by the Wishart and inverse-Wishart distributions,
respectively. Independent of any details of the survey, we show that the fractional error on a
parameter variance, or a figure of merit, is equal to the fractional variance of the precision
matrix. In addition, for the only unbiased estimator of the precision matrix, we find that the
fractional accuracy of the parameter error depends only on the difference between the number
of independent realizations and the number of data points, and so can easily diverge. For a 5
per cent error on a parameter error and ND 102 data points, a minimum of 200 realizations
of the survey are needed, with 10 per cent accuracy in the data covariance. If the number of data
√
points ND 102 , we need NS > ND realizations and a fractional accuracy of < 2/ND in
the data covariance. As the number of power spectra data points grows to ND > 104 –106 , this
approach will be problematic. We discuss possible ways to relax these conditions: improved
theoretical modelling, shrinkage methods, data compression, simulation and data resampling
methods.
Key words: methods: statistical – cosmological parameters – cosmology: theory – large-scale
structure of Universe.
1 I N T RO D U C T I O N
A central part of modern cosmology is the measurement of the
parameters that characterize cosmological models of the Universe.
These can be the set that constitutes the standard cosmological
model (m , b , , H0 , σ8 , ns , τ ), or an extended set that characterizes, for example, more complex dark energy models (see e.g.
Copeland, Sami & Tsujikawa 2006; Amendola et al. 2012 for reviews), deviations from Einstein gravity (e.g. Amendola et al. 2012;
Clifton et al. 2012 for recent reviews), more detail about the inflationary epoch (e.g. Amendola et al. 2012), isocurvature density and
velocity modes (e.g. Bucher, Moodley & Turok 2001), or massive
neutrinos and their abundance (e.g. Bird, Viel & Haehnelt 2012
and references therein). Furthermore, if we want to differentiate
between theoretical models in a Bayesian framework, as well as
estimate their parameter value, we also need to accurately inte-
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grate over the model parameter space (e.g. Liddle, Mukherjee &
Parkinson 2006; Trotta 2007; Taylor & Kitching 2010).
To carry out these tasks, we need both accurate theoretical predictions of the physical properties of the model to compare to the
data, and sufficiently accurate models of their statistical properties.
Ideally, we would like to be able to accurately predict the full multivariate probability distribution of the data for each model. If, as
is commonly assumed, the data can be modelled as a multivariate
Gaussian distribution, all of the statistical properties of the model
reside in the mean and covariance of the model. Attention has been
focused on the accuracy of the predictions of the mean value – e.g.
the model power spectra – and the effect of biases or errors in the
mean (e.g. Huterer & Takada 2005; Huterer et al. 2006; Taylor et al.
2007). But to fully specify the distribution of the data, we also need
accurate predictions of the data covariance matrix and the inverse
of the data covariance – the precision matrix.
If we assume that the mean is well known, the accuracy of
the probability distribution of the data, and hence the likelihood
function in parameter space, is determined by the accuracy of the
precision matrix. However, as yet there is no unique approach to
C 2013 The Authors
Published by Oxford University Press on behalf of the Royal Astronomical Society
Downloaded from http://mnras.oxfordjournals.org/ at Acquisitions DeptHunt Library on July 24, 2016
ABSTRACT
Putting the precision in precision cosmology
2 PA R A M E T E R E S T I M AT I O N
To begin with, we shall assume that the cosmological parameters,
θ, being measured are estimated from maximizing a posterior parameter distribution, p(θ| D, M), given a data set, D, and some
theoretical model, M (see e.g. Sivia 1996). From Bayes theorem,
p(θ| D, M) =
L( D|θ , M)π(θ |M)
,
E( D|M)
(1)
we can determine the posterior parameter distribution from the likelihood function for the data, L( D|θ, M), predicted by the model, a
prior, π(θ |M), which is the probability distribution of the parameters before the data are analysed, and normalized by the evidence,
E( D|M), which marginalizes over the likelihood and prior in parameter space. If we restrict our study to parameter estimation for
a given model, we can ignore this term. We shall assume that the
prior on the parameters is flat.
If we model the data distribution as a multivariate Gaussian, then
the likelihood function can be written as
L( D|μ, M, M) =
1
(2π)ND /2
√
1
exp − Tr W ,
2
|M|
(2)
where
W = DDt ,
(3)
a superscript, t, indicates a transpose,
D = D − D
(4)
is the variation in the data vector, μ = D is the mean of the data
and ND is the length of the data vector. The data covariance matrix
is given by
M = W = DDt .
(5)
We define |M| = det M as the determinant. Comparing with a multivariate Gaussian, we see that the matrix, , is the inverse of the
data covariance matrix;
= M−1 .
(6)
As we shall find that this matrix is central to our analysis, we
shall define the inverse data covariance as the precision matrix. The
model dependence on cosmological model parameters, θ , may lie
in either the mean, μ = μ(θ), or the data covariance matrix, M =
M(θ ), or both. Throughout, we shall assume that the cosmological
parameter dependence lies only in the mean. In Appendix A, we
describe the data vectors commonly used in cosmological largescale structure analysis: galaxy redshift surveys, CMB experiments
and weak lensing surveys. Throughout, we shall assume that the
data are a set of power spectra estimated from the data, although our
results hold for correlation functions and are general (...truncated)
