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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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    1. Antonio Cabrales & Olivier Gossner & Roberto Serrano, 2013. "Entropy and the Value of Information for Investors," American Economic Review, American Economic Association, vol. 103(1), pages 360-377, February.
    2. Lones Smith & Peter Sorensen, 2000. "Pathological Outcomes of Observational Learning," Econometrica, Econometric Society, vol. 68(2), pages 371-398, March.
    3. Kimball, Miles S, 1990. "Precautionary Saving in the Small and in the Large," Econometrica, Econometric Society, vol. 58(1), pages 53-73, January.
    4. Ross, Stephen A., 1999. "Adding Risks: Samuelson's Fallacy of Large Numbers Revisited," Journal of Financial and Quantitative Analysis, Cambridge University Press, vol. 34(3), pages 323-339, September.
    5. Yaron Azrieli, 2014. "Comment on “The Law of Large Demand for Information”," Econometrica, Econometric Society, vol. 82(1), pages 415-423, January.
    6. Dean P. Foster & Sergiu Hart, 2009. "An Operational Measure of Riskiness," Journal of Political Economy, University of Chicago Press, vol. 117(5), pages 785-814.
    7. Robert J. Aumann & Roberto Serrano, 2008. "An Economic Index of Riskiness," Journal of Political Economy, University of Chicago Press, vol. 116(5), pages 810-836, October.
    8. 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.
    9. Christian Gollier, 1996. "Repeated Optional Gambles and Risk Aversion," Management Science, INFORMS, vol. 42(11), pages 1524-1530, November.
    10. Kimball, Miles S, 1993. "Standard Risk Aversion," Econometrica, Econometric Society, vol. 61(3), pages 589-611, May.
    11. Fishburn, Peter C., 1980. "Continua of stochastic dominance relations for unbounded probability distributions," Journal of Mathematical Economics, Elsevier, vol. 7(3), pages 271-285, December.
    12. Miles S. Kimball, 1991. "Precautionary Motives for Holding Assets," NBER Working Papers 3586, National Bureau of Economic Research, Inc.
    13. Sergiu Hart, 2011. "Comparing Risks by Acceptance and Rejection," Journal of Political Economy, University of Chicago Press, vol. 119(4), pages 617-638.
    14. Yuichi Kitamura & Michael Stutzer, 1997. "An Information-Theoretic Alternative to Generalized Method of Moments Estimation," Econometrica, Econometric Society, vol. 65(4), pages 861-874, July.
    15. Yongmiao Hong & Halbert White, 2005. "Asymptotic Distribution Theory for Nonparametric Entropy Measures of Serial Dependence," Econometrica, Econometric Society, vol. 73(3), pages 837-901, May.
    16. Matthew Rabin & Richard H. Thaler, 2013. "Anomalies: Risk aversion," World Scientific Book Chapters, in: Leonard C MacLean & William T Ziemba (ed.), HANDBOOK OF THE FUNDAMENTALS OF FINANCIAL DECISION MAKING Part I, chapter 27, pages 467-480, World Scientific Publishing Co. Pte. Ltd..
    17. Pratt, John W & Zeckhauser, Richard J, 1987. "Proper Risk Aversion," Econometrica, Econometric Society, vol. 55(1), pages 143-154, January.
    18. Shlomo Benartzi & Richard H. Thaler, 1999. "Risk Aversion or Myopia? Choices in Repeated Gambles and Retirement Investments," Management Science, INFORMS, vol. 45(3), pages 364-381, March.
    19. Ullah, Aman, 2002. "Uses of entropy and divergence measures for evaluating econometric approximations and inference," Journal of Econometrics, Elsevier, vol. 107(1-2), pages 313-326, March.
    20. Christian Gollier, 2004. "The Economics of Risk and Time," MIT Press Books, The MIT Press, edition 1, volume 1, number 0262572249, December.
    21. Luciano Pomatto & Philipp Strack & Omer Tamuz, 2020. "Stochastic Dominance under Independent Noise," Journal of Political Economy, University of Chicago Press, vol. 128(5), pages 1877-1900.
    22. Giuseppe Moscarini & Lones Smith, 2002. "The Law of Large Demand for Information," Econometrica, Econometric Society, vol. 70(6), pages 2351-2366, November.
    23. White, Halbert, 1982. "Maximum Likelihood Estimation of Misspecified Models," Econometrica, Econometric Society, vol. 50(1), pages 1-25, January.
    24. Sawa, Takamitsu, 1978. "Information Criteria for Discriminating among Alternative Regression Models," Econometrica, Econometric Society, vol. 46(6), pages 1273-1291, November.
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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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