Density Ratio Estimation via Sampling along Generalized Geodesics on Statistical Manifolds

Masanari Kimura, Howard Bondell
Proceedings of The 28th International Conference on Artificial Intelligence and Statistics, PMLR 258:253-261, 2025.

Abstract

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.

Cite this Paper

BibTeX
@InProceedings{pmlr-v258-kimura25a, title = {Density Ratio Estimation via Sampling along Generalized Geodesics on Statistical Manifolds}, author = {Kimura, Masanari and Bondell, Howard}, booktitle = {Proceedings of The 28th International Conference on Artificial Intelligence and Statistics}, pages = {253--261}, year = {2025}, editor = {Li, Yingzhen and Mandt, Stephan and Agrawal, Shipra and Khan, Emtiyaz}, volume = {258}, series = {Proceedings of Machine Learning Research}, month = {03--05 May}, publisher = {PMLR}, pdf = {https://raw.githubusercontent.com/mlresearch/v258/main/assets/kimura25a/kimura25a.pdf}, url = {https://proceedings.mlr.press/v258/kimura25a.html}, abstract = {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.} }
Endnote
%0 Conference Paper %T Density Ratio Estimation via Sampling along Generalized Geodesics on Statistical Manifolds %A Masanari Kimura %A Howard Bondell %B Proceedings of The 28th International Conference on Artificial Intelligence and Statistics %C Proceedings of Machine Learning Research %D 2025 %E Yingzhen Li %E Stephan Mandt %E Shipra Agrawal %E Emtiyaz Khan %F pmlr-v258-kimura25a %I PMLR %P 253--261 %U https://proceedings.mlr.press/v258/kimura25a.html %V 258 %X 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.
APA
Kimura, M. & Bondell, H.. (2025). Density Ratio Estimation via Sampling along Generalized Geodesics on Statistical Manifolds. Proceedings of The 28th International Conference on Artificial Intelligence and Statistics, in Proceedings of Machine Learning Research 258:253-261 Available from https://proceedings.mlr.press/v258/kimura25a.html.

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