  We apply the Karhunen-Lo\'eve methods to cosmic microwave background (CMB)
data sets, and show that we can recover the input cosmology and obtain the
marginalized likelihoods in $\Lambda$ cold dark matter cosmologies in under a
minute, much faster than Markov chain Monte Carlo methods. This is achieved by
forming a linear combination of the power spectra at each multipole $l$, and
solving a system of simultaneous equations such that the Fisher matrix is
locally unchanged. Instead of carrying out a full likelihood evaluation over
the whole parameter space, we need evaluate the likelihood only for the
parameter of interest, with the data compression effectively marginalizing over
all other parameters. The weighting vectors contain insight about the physical
effects of the parameters on the CMB anisotropy power spectrum $C_l$. The shape
and amplitude of these vectors give an intuitive feel for the physics of the
CMB, the sensitivity of the observed spectrum to cosmological parameters, and
the relative sensitivity of different experiments to cosmological parameters.
We test this method on exact theory $C_l$ as well as on a Wilkinson Microwave
Anisotropy Probe (WMAP)-like CMB data set generated from a random realization
of a fiducial cosmology, comparing the compression results to those from a full
likelihood analysis using CosmoMC. After showing that the method works, we
apply it to the temperature power spectrum from the WMAP seven-year data
release, and discuss the successes and limitations of our method as applied to
a real data set.
