  Density ratio estimation (DRE) is a fundamental machine learning technique
for comparing two probability distributions. However, existing methods struggle
in high-dimensional settings, as it is difficult to accurately compare
probability distributions based on finite samples. In this work we propose
DRE-\infty, a divide-and-conquer approach to reduce DRE to a series of easier
subproblems. Inspired by Monte Carlo methods, we smoothly interpolate between
the two distributions via an infinite continuum of intermediate bridge
distributions. We then estimate the instantaneous rate of change of the bridge
distributions indexed by time (the "time score") -- a quantity defined
analogously to data (Stein) scores -- with a novel time score matching
objective. Crucially, the learned time scores can then be integrated to compute
the desired density ratio. In addition, we show that traditional (Stein) scores
can be used to obtain integration paths that connect regions of high density in
both distributions, improving performance in practice. Empirically, we
demonstrate that our approach performs well on downstream tasks such as mutual
information estimation and energy-based modeling on complex, high-dimensional
datasets.
