Exact joint likelihood of pseudo-$C_\ell$ estimates from correlated
  Gaussian cosmological fields
  We present the exact joint likelihood of pseudo-$C_\ell$ power spectrum
estimates measured from an arbitrary number of Gaussian cosmological fields.
Our method is applicable to both spin-0 fields and spin-2 fields, including a
mixture of the two, and is relevant to Cosmic Microwave Background, weak
lensing and galaxy clustering analyses. We show that Gaussian cosmological
fields are mixed by a mask in such a way that retains their Gaussianity,
without making any assumptions about the mask geometry. We then show that each
auto- or cross-pseudo-$C_\ell$ estimator can be written as a quadratic form,
and apply the known joint distribution of quadratic forms to obtain the exact
joint likelihood of a set of pseudo-$C_\ell$ estimates in the presence of an
arbitrary mask. Considering the polarisation of the Cosmic Microwave Background
as an example, we show using simulations that our likelihood recovers the full,
exact multivariate distribution of $EE$, $BB$ and $EB$ pseudo-$C_\ell$ power
spectra. Our method provides a route to robust cosmological constraints from
future Cosmic Microwave Background and large-scale structure surveys in an era
of ever-increasing statistical precision.
