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The ratio between two probability density functions is an important componentof various tasks, including selection bias correction, novelty detection andclassification. Recently, several estimators of this ratio have been proposed.Most of these methods fail if the sample space is high-dimensional, and hencerequire a dimension reduction step, the result of which can be a significantloss of information. Here we propose a simple-to-implement, fully nonparametricdensity ratio estimator that expands the ratio in terms of the eigenfunctionsof a kernel-based operator; these functions reflect the underlying geometry ofthe data (e.g., submanifold structure), often leading to better estimateswithout an explicit dimension reduction step. We show how our general frameworkcan be extended to address another important problem, the estimation of alikelihood function in situations where that function cannot bewell-approximated by an analytical form. One is often faced with this situationwhen performing statistical inference with data from the sciences, due thecomplexity of the data and of the processes that generated those data. Weemphasize applications where using existing likelihood-free methods ofinference would be challenging due to the high dimensionality of the samplespace, but where our spectral series method yields a reasonable estimate of thelikelihood function. We provide theoretical guarantees and illustrate theeffectiveness of our proposed method with numerical experiments.

High-Dimensional Density Ratio Estimation with Extensions to Approximate Likelihood Computation