Optimal estimation of the binned mask-free power spectrum, bispectrum, and trispectrum on the full sky: Scalar edition

We derive optimal estimators for the two-, three-, and four-point correlators of statistically isotropic scalar fields defined on sphere, such as Cosmic Microwave Background temperature fluctuations, allowing arbitrary (linear) masking inpainting schemes. In each case, we give unwindowed estimator (obtained via a maximum-likelihood prescription, with an associated Fisher deconvolution matrix), idealized form, pay close attention to their efficient computation. For trispectrum, include both parity-even parity-odd contributions, allowed by symmetry. The can weighting data (and remain unbiased), but are shown be in limit inverse-covariance Gaussian statistics. normalization is computed Monte Carlo methods, rate-limiting steps (involving spherical harmonic transforms) scaling linearly number bins. An accompanying code package, PolyBin, implements these Python, demonstrate estimators' efficacy suite validation tests.