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 et al. 1997, hereafter TTH). 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, θ, is given by a hypothesis, Hx, 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(θ|x,Hx) = (2π) −n/2 |C(θ) | −1/2 [ exp − 1 2 x † C(θ) −1] x, (1) where θ = (Q,h,Ω0,ΩΛ,Ωb,...) are the usual cosmological parameters we would like to determine. Examples of data are x = ∆T or aℓ,m and the statistics of the n data are fully parametrised by the data covariance matrix, C(θ) = 〈xx † 〉. 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 best-fitting 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 τ ∝ q m ×n
