Optimal Polyspectra Estimation
  We show that full inverse covariance weighting can be naturally incorporated
into modal estimation methods making them optimal for the CMB power spectrum,
bispectrum and trispectrum, as well as in other 3D applications. Modal
estimation methods are highly efficient requiring inversion of only an nmax x
nmax covariance matrix, where nmax is the limited number of modes needed to
describe both the theoretical model and the noise which affects it (in contrast
to the full lmax^2 x lmax^2 inverse covariance weighting). A WMAP resolution
implementation for modal bispectrum estimation is discussed and shown to yield
optimal constraints for local and other models at lmax =500 and nmax <= 50. We
also discuss further improvements by characterising and preconditioning the
covariance matrix using a single bispectrum variance shape induced by the noise
and mask. We describe optimal modal estimation in general terms applicable to
polyspectra of any order in a formalism that can be extended beyond the
isotropic case. We use the example of anisotropic noise in bispectrum
estimation to show the completeness of the isotropic modal basis and the
necessity of subtracting a linear estimator term for optimality, removing
cross-terms in the variance. We briefly discuss applications to the CMB power
spectrum and trispectrum, as well as large-scale structure.
