Optimized Large-Scale CMB Likelihood And Quadratic Maximum Likelihood
  Power Spectrum Estimation
  We revisit the problem of exact CMB likelihood and power spectrum estimation
with the goal of minimizing computational cost through linear compression. This
idea was originally proposed for CMB purposes by Tegmark et al.\ (1997), and
here we develop it into a fully working computational framework for large-scale
polarization analysis, adopting \WMAP\ as a worked example. We compare five
different linear bases (pixel space, harmonic space, noise covariance
eigenvectors, signal-to-noise covariance eigenvectors and signal-plus-noise
covariance eigenvectors) in terms of compression efficiency, and find that the
computationally most efficient basis is the signal-to-noise eigenvector basis,
which is closely related to the Karhunen-Loeve and Principal Component
transforms, in agreement with previous suggestions. For this basis, the
information in 6836 unmasked \WMAP\ sky map pixels can be compressed into a
smaller set of 3102 modes, with a maximum error increase of any single
multipole of 3.8\% at $\ell\le32$, and a maximum shift in the mean values of a
joint distribution of an amplitude--tilt model of 0.006$\sigma$. This
compression reduces the computational cost of a single likelihood evaluation by
a factor of 5, from 38 to 7.5 CPU seconds, and it also results in a more robust
likelihood by implicitly regularizing nearly degenerate modes. Finally, we use
the same compression framework to formulate a numerically stable and
computationally efficient variation of the Quadratic Maximum Likelihood
implementation that requires less than 3 GB of memory and 2 CPU minutes per
iteration for $\ell \le 32$, rendering low-$\ell$ QML CMB power spectrum
analysis fully tractable on a standard laptop.
