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Density-ratio matching under the Bregman divergence: a unified framework of density-ratio estimation

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  • Masashi Sugiyama
  • Taiji Suzuki
  • Takafumi Kanamori

Abstract

Estimation of the ratio of probability densities has attracted a great deal of attention since it can be used for addressing various statistical paradigms. A naive approach to density-ratio approximation is to first estimate numerator and denominator densities separately and then take their ratio. However, this two-step approach does not perform well in practice, and methods for directly estimating density ratios without density estimation have been explored. In this paper, we first give a comprehensive review of existing density-ratio estimation methods and discuss their pros and cons. Then we propose a new framework of density-ratio estimation in which a density-ratio model is fitted to the true density-ratio under the Bregman divergence. Our new framework includes existing approaches as special cases, and is substantially more general. Finally, we develop a robust density-ratio estimation method under the power divergence, which is a novel instance in our framework. Copyright The Institute of Statistical Mathematics, Tokyo 2012

Suggested Citation

  • Masashi Sugiyama & Taiji Suzuki & Takafumi Kanamori, 2012. "Density-ratio matching under the Bregman divergence: a unified framework of density-ratio estimation," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 64(5), pages 1009-1044, October.
    • Handle: RePEc:spr:aistmt:v:64:y:2012:i:5:p:1009-1044
    DOI: 10.1007/s10463-011-0343-8
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    References listed on IDEAS

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    1. Masashi Sugiyama & Taiji Suzuki & Shinichi Nakajima & Hisashi Kashima & Paul Bünau & Motoaki Kawanabe, 2008. "Direct importance estimation for covariate shift adaptation," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 60(4), pages 699-746, December.
    2. Sugiyama Masashi & Müller Klaus-Robert, 2005. "Input-dependent estimation of generalization error under covariate shift," Statistics & Risk Modeling, De Gruyter, vol. 23(4/2005), pages 249-279, April.
    3. B. W. Silverman, 1978. "Density Ratios, Empirical Likelihood and Cot Death," Journal of the Royal Statistical Society Series C, Royal Statistical Society, vol. 27(1), pages 26-33, March.
    4. Fujisawa, Hironori & Eguchi, Shinto, 2008. "Robust parameter estimation with a small bias against heavy contamination," Journal of Multivariate Analysis, Elsevier, vol. 99(9), pages 2053-2081, October.
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    Cited by:

    1. Abdolnasser Sadeghkhani & Yingwei Peng & Chunfang Devon Lin, 2019. "A Parametric Bayesian Approach in Density Ratio Estimation," Stats, MDPI, vol. 2(2), pages 1-13, March.
    2. Gaëlle Chagny & Claire Lacour, 2015. "Optimal adaptive estimation of the relative density," TEST: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 24(3), pages 605-631, September.
    3. Yukitoshi Matsushita & Taisuke Otsu & Keisuke Takahata, 2022. "Estimating density ratio of marginals to joint: Applications to causal inference," STICERD - Econometrics Paper Series 619, Suntory and Toyota International Centres for Economics and Related Disciplines, LSE.
    4. David Bruns-Smith & Oliver Dukes & Avi Feller & Elizabeth L. Ogburn, 2023. "Augmented balancing weights as linear regression," Papers 2304.14545, arXiv.org, revised Jun 2024.

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