  This article focuses on estimating distribution elements over a
high-dimensional binary hypercube from multivariate binary data. A popular
approach to this problem, optimizing Walsh basis coefficients, is made more
interpretable by an alternative representation as a "Fourier-Walsh"
diagonalization. Allowing monotonic transformations of the resulting matrix
elements yields a versatile binary density estimator: the main contribution of
this article. It is shown that the Aitchison and Aitken kernel emerges from a
constrained exponential form of this estimator, and that relaxing these
constraints yields a flexible variable-weighted version of the kernel that
retains positive-definiteness. Estimators within this unifying framework mix
together well and span over extremes of the speed-flexibility trade-off,
allowing them to serve a wide range of statistical inference and learning
problems.
