  The ratio between two probability density functions is an important component
of various tasks, including selection bias correction, novelty detection and
classification. Recently, several estimators of this ratio have been proposed.
Most of these methods fail if the sample space is high-dimensional, and hence
require a dimension reduction step, the result of which can be a significant
loss of information. Here we propose a simple-to-implement, fully nonparametric
density ratio estimator that expands the ratio in terms of the eigenfunctions
of a kernel-based operator; these functions reflect the underlying geometry of
the data (e.g., submanifold structure), often leading to better estimates
without an explicit dimension reduction step. We show how our general framework
can be extended to address another important problem, the estimation of a
likelihood function in situations where that function cannot be
well-approximated by an analytical form. One is often faced with this situation
when performing statistical inference with data from the sciences, due the
complexity of the data and of the processes that generated those data. We
emphasize applications where using existing likelihood-free methods of
inference would be challenging due to the high dimensionality of the sample
space, but where our spectral series method yields a reasonable estimate of the
likelihood function. We provide theoretical guarantees and illustrate the
effectiveness of our proposed method with numerical experiments.
