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 rises above the conditional error to . 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, times, once for each parameter. 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 Figure 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 and ). To give a more concrete example -- a thin ridge of likelihood at 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.
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 . Bottom left: Best 320 modes from SVD application to modes optimised for and . Bottom right: Best 320 modes for optimising along the likelihood ridge axis. Likelihood contours are separated by 0.5 in natural log.
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 and the redshift distortion parameter , 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 second panel, top right, shows the single-parameter optimisation of the first part of this paper. The modes are optimised for , and only the best 320 modes are used. We see that the conditional error in the direction is not much worse than the full set, but the likelihood declines slowly along the ridge, and the marginal errors on both and have increased substantially. In the panel bottom left, the SVD procedure has been applied to the union of modes optimised for and , 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.