The density ratio of two probability distributions is one of the fundamental
tools in mathematical and computational statistics and machine learning, and it
has a variety of known applications. Therefore, density ratio estimation from
finite samples is a very important task, but it is known to be unstable when
the distributions are distant from each other. One approach to address this
problem is density ratio estimation using incremental mixtures of the two
distributions. We geometrically reinterpret existing methods for density ratio
estimation based on incremental mixtures. We show that these methods can be
regarded as iterating on the Riemannian manifold along a particular curve
between the two probability distributions. Making use of the geometry of the
manifold, we propose to consider incremental density ratio estimation along
generalized geodesics on this manifold. To achieve such a method requires Monte
Carlo sampling along geodesics via transformations of the two distributions. We
show how to implement an iterative algorithm to sample along these geodesics
and show how changing the distances along the geodesic affect the variance and
accuracy of the estimation of the density ratio. Our experiments demonstrate
that the proposed approach outperforms the existing approaches using
incremental mixtures that do not take the geometry of the

基于增量混合模型的密度比率估计方法通过在勒贝格流形上进行几何解释和蒙特卡洛采样，显著改善了稳定性和准确度。