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From Blackwell Dominance in Large Samples to Renyi Divergences and Back Again

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  • Xiaosheng Mu
  • Luciano Pomatto
  • Philipp Strack
  • Omer Tamuz

Abstract

We study repeated independent Blackwell experiments; standard examples include drawing multiple samples from a population, or performing a measurement in different locations. In the baseline setting of a binary state of nature, we compare experiments in terms of their informativeness in large samples. Addressing a question due to Blackwell (1951), we show that generically an experiment is more informative than another in large samples if and only if it has higher Renyi divergences. We apply our analysis to the problem of measuring the degree of dissimilarity between distributions by means of divergences. A useful property of Renyi divergences is their additivity with respect to product distributions. Our characterization of Blackwell dominance in large samples implies that every additive divergence that satisfies the data processing inequality is an integral of Renyi divergences.

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  • Xiaosheng Mu & Luciano Pomatto & Philipp Strack & Omer Tamuz, 2019. "From Blackwell Dominance in Large Samples to Renyi Divergences and Back Again," Papers 1906.02838, arXiv.org, revised Sep 2020.
    • Handle: RePEc:arx:papers:1906.02838
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    Cited by:

    1. Andrew Kosenko, 2021. "Algebraic Properties of Blackwell's Order and A Cardinal Measure of Informativeness," Papers 2110.11399, arXiv.org.
    2. Mira Frick & Ryota Iijima & Yuhta Ishii, 2021. "Learning Efficiency of Multi-Agent Information Structures," Cowles Foundation Discussion Papers 2299R, Cowles Foundation for Research in Economics, Yale University, revised Dec 2021.
    3. Xiaosheng Mu & Luciano Pomatto & Philipp Strack & Omer Tamuz, 2021. "Monotone Additive Statistics," Working Papers 2021-36, Princeton University. Economics Department..
    4. Mira Frick & Ryota Iijima & Yuhta Ishii, 2021. "Welfare Comparisons for Biased Learning," Cowles Foundation Discussion Papers 2274R, Cowles Foundation for Research in Economics, Yale University, revised Mar 2021.
    5. Xiaosheng Mu & Luciano Pomatto & Philipp Strack & Omer Tamuz, 2021. "From Blackwell Dominance in Large Samples to Rényi Divergences and Back Again," Econometrica, Econometric Society, vol. 89(1), pages 475-506, January.
    6. Mira Frick & Ryota Iijima & Yuhta Ishii, 2021. "Learning Efficiency of Multi-Agent Information Structures," Cowles Foundation Discussion Papers 2299R2, Cowles Foundation for Research in Economics, Yale University, revised Jul 2022.
    7. Xiaosheng Mu & Luciano Pomatto & Philipp Strack & Omer Tamuz, 2021. "Monotone additive statistics," Papers 2102.00618, arXiv.org, revised Apr 2024.

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