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**1 - 1**of**1**### High-Dimensional Density Ratio Estimation with Extensions to Approximate Likelihood Computation

"... The ratio between two probability density functions is an important component of var-ious tasks, including selection bias correc-tion, novelty detection and classification. Re-cently, several estimators of this ratio have been proposed. Most of these methods fail if the sample space is high-dimensio ..."

Abstract
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The ratio between two probability density functions is an important component of var-ious tasks, including selection bias correc-tion, novelty detection and classification. Re-cently, 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 geome-try 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 situ-ations where that function cannot be well-approximated by an analytical form. One is often faced with this situation when per-forming 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 dimen-sionality of the sample space, but where our spectral series method yields a reasonable es-timate of the likelihood function. We provide theoretical guarantees and illustrate the ef-fectiveness of our proposed method with nu-merical experiments.