Papers
Topics
Authors
Recent
Gemini 2.5 Flash
Gemini 2.5 Flash
194 tokens/sec
GPT-4o
7 tokens/sec
Gemini 2.5 Pro Pro
46 tokens/sec
o3 Pro
4 tokens/sec
GPT-4.1 Pro
38 tokens/sec
DeepSeek R1 via Azure Pro
28 tokens/sec
2000 character limit reached

Robust Statistical Comparison of Random Variables with Locally Varying Scale of Measurement (2306.12803v2)

Published 22 Jun 2023 in stat.ML, cs.LG, math.ST, and stat.TH

Abstract: Spaces with locally varying scale of measurement, like multidimensional structures with differently scaled dimensions, are pretty common in statistics and machine learning. Nevertheless, it is still understood as an open question how to exploit the entire information encoded in them properly. We address this problem by considering an order based on (sets of) expectations of random variables mapping into such non-standard spaces. This order contains stochastic dominance and expectation order as extreme cases when no, or respectively perfect, cardinal structure is given. We derive a (regularized) statistical test for our proposed generalized stochastic dominance (GSD) order, operationalize it by linear optimization, and robustify it by imprecise probability models. Our findings are illustrated with data from multidimensional poverty measurement, finance, and medicine.

Definition Search Book Streamline Icon: https://streamlinehq.com
References (61)
  1. Increasing diversity in random forest learning algorithm via imprecise probabilities. Expert Syst Appl, 97:228–243, 2018.
  2. T. Augustin and G. Schollmeyer. Comment: On focusing, soft and strong revision of Choquet capacities and their role in statistics. Stat Sci, 36(2):205–209, 2021.
  3. Introduction to Imprecise Probabilities. Wiley, 2014a.
  4. Statistical inference. In T. Augustin, F. Coolen, G. de Cooman, and M. Troffaes, editors, Introduction to Imprecise Probabilities, pages 135–189. Wiley, 2014b.
  5. G. Barrett and S. Donald. Consistent tests for stochastic dominance. Econometrica, 71(1):71–104, 2003.
  6. J. Bauer. Selection errors of random route samples. Sociol Method Res, 43(3):519–544, 2014.
  7. J. Bauer. Biases in random route surveys. Journal of Survey Statistics and Methodology, 4(2):263–287, 2016.
  8. Pitfalls of epistemic uncertainty quantification through loss minimisation. In Advances in Neural Information Processing Systems, 2022.
  9. Depth functions for partial orders with a descriptive analysis of machine learning algorithms. In International Symposium on Imprecise Probabilities: Theories and Applications, 2023. PMLR (to appear).
  10. B. Breyer and D. Danner. Skala zur Erfassung des Lebenssinns (ALLBUS). In Zusammenstellung sozialwissenschaftlicher Items und Skalen (ZIS) (GESIS – Leibniz-Institut für Sozialwissenschaften), volume 10, 2015.
  11. An unbiased test for the bioequivalence problem. Ann Stat, 25(6):2345 – 2367, 1997.
  12. CREDICI: A Java library for causal inference by credal networks. In M. Jaeger and T. Nielsen, editors, International Conference on Probabilistic Graphical Models, volume 138 of PMLR, pages 597–600, 2020.
  13. Y. Carranza and S. Destercke. Imprecise Gaussian discriminant classification. Pattern Recogn, 112:107739, 2021.
  14. A stochastic dominance approach to financial risk management strategies. J Econometrics, 187:472–485, 2015.
  15. An adaptive test of stochastic monotonicity. Econometric Theory, 37(3):495–536, 2021.
  16. G. Corani and M. Zaffalon. Learning reliable classifiers from small or incomplete data sets: The naive credal classifier 2. J Mach Learn Res, 9(4), 2008.
  17. Statistical comparison of classifiers through Bayesian hierarchical modelling. Mach Learn, 106(11):1817–1837, 2017.
  18. Learning to optimize for stochastic dominance constraints. In F. Ruiz, J. Dy, and J. van de Meent, editors, Artificial Intelligence and Statistics, volume 206 of PMLR, pages 8991–9009, 2023.
  19. Learning differential diagnosis of Eryhemato-Squamous diseases using voting feature intervals. Artif Intell Med, 13:147–165, 1998.
  20. J. Demsar. Statistical comparisons of classifiers over multiple data sets. J Mach Learn Res, 7:1–30, 2006.
  21. D. Denneberg. Non-additive Measure and Integral. Kluwer Academic Publishers, 1994.
  22. Processing distortion models: A comparative study. Int J Approx Reason, 145:91–120, 2022.
  23. D. Dua and C. Graff. UCI machine learning repository, 2017. http://archive.ics.uci.edu/ml.
  24. European Commission. Knowledge service: Competence centre on composite indicators and scoreboards, 2023. URL https://knowledge4policy.ec.europa.eu/composite-indicators_en. (Febr. 16, 2023).
  25. A review of stochastic dominance methods for poverty analysis. J Econ Surv, 33(5):1437–1462, 2019.
  26. GESIS. Allgemeine Bevölkerungsumfrage der Sozialwissenschaften ALLBUS 2014. GESIS Datenarchiv, Köln. ZA5240 Datenfile Version 2.2.0, https://doi.org/10.4232/1.13141, 2018.
  27. E. Hüllermeier and W. Waegeman. Aleatoric and epistemic uncertainty in machine learning: An introduction to concepts and methods. Mach Learn, 110(3):457–506, 2021.
  28. Quantification of credal uncertainty in machine learning: A critical analysis and empirical comparison. In J. Cussens and K. Zhang, editors, Uncertainty in Artificial Intelligence, volume 180 of PMLR, pages 548–557, 2022.
  29. C. Jansen and T. Augustin. Decision making with state-dependent preference systems. In Information Processing and Management of Uncertainty in Knowledge-Based Systems, pages 729–742. Springer, 2022.
  30. Concepts for decision making under severe uncertainty with partial ordinal and partial cardinal preferences. Int J Approx Reason, 98:112–131, 2018.
  31. Information efficient learning of complexly structured preferences: Elicitation procedures and their application to decision making under uncertainty. Int J Approx Reason, 144:69–91, 2022a.
  32. Statistical comparisons of classifiers by generalized stochastic dominance, 2022b. URL https://arxiv.org/abs/2209.01857. arXiv preprint.
  33. Quantifying degrees of E-admissibility in decision making with imprecise probabilities. In T. Augustin, F. Cozman, and G. Wheeler, editors, Reflections on the Foundations of Probability and Statistics: Essays in Honor of Teddy Seidenfeld, pages 319–346. Springer, 2022c.
  34. Multi-target decision making under conditions of severe uncertainty. In V. Torra and Y. Narukawa, editors, Modeling Decisions for Artificial Intelligence, pages 45–57. Springer, 2023.
  35. I. Levi. On indeterminate probabilities. The Journal of Philosophy, 71:391–418, 1974.
  36. J. Lienen and E. Hüllermeier. Credal self-supervised learning. Advances in Neural Information Processing Systems, 34:14370–14382, 2021.
  37. R. Luce. Semiorders and a theory of utility discrimination. Econometrica, 24:178–191, 1956.
  38. A. Malinin and M. Gales. Predictive uncertainty estimation via prior networks. Advances in Neural Information Processing Systems, 31, 2018.
  39. D. Maua and F. Cozman. Thirty years of credal networks: Specification, algorithms and complexity. Int J Approx Reason, 126:133–157, 2020.
  40. D. Maua and C. de Campos. Editorial to: Special issue on robustness in probabilistic graphical models. Int J Approx Reason, 137:113, 2021.
  41. D. McFadden. Testing for stochastic dominance. In T. Fomby and T. Seo, editors, Studies in the Economics of Uncertainty, pages 113–134. Springer, 1989.
  42. Unifying neighbourhood and distortion models: Part II – new models and synthesis. Int J Gen Syst, 49:636–674, 2020.
  43. K. Mosler. Testing whether two distributions are stochastically ordered or not. In H. Rinne, B. Rüger, and H. Strecker, editors, Grundlagen der Statistik und ihre Anwendungen: Festschrift für Kurt Weichselberger, pages 149–155. Physica-Verlag, 1995.
  44. K. Mosler and M. Scarsini. Some theory of stochastic dominance. Lecture Notes-Monograph Series, 19:261–284, 1991.
  45. Tikhonov, Ivanov and Morozov regularization for support vector machine learning. Mach Learn, 103:103–136, 2016.
  46. M. Pivato. Multiutility representations for incomplete difference preorders. Math Sco Sci, 66:196–220, 2013.
  47. J. Pratt and J. Gibbons. Concepts of Nonparametric Theory. Springer, 2012.
  48. T. Range and L. Østerdal. First-order dominance: stronger characterization and a bivariate checking algorithm. Math Program, 173:193––219, 2019.
  49. J. Rodemann and T. Augustin. Accounting for Gaussian process imprecision in Bayesian optimization. In International Symposium on Integrated Uncertainty in Knowledge Modelling and Decision Making (IUKM), pages 92–104. Springer, 2022.
  50. In all likelihoods: Robust selection of pseudo-labeled data. In International Symposium on Imprecise Probabilities: Theories and Applications, 2023. PMLR (to appear).
  51. G. Schollmeyer. A short note on the equivalence of the ontic and the epistemic view on data imprecision for the case of stochastic dominance for interval-valued data. In International Symposium on Imprecise Probabilities: Theories and Applications, pages 330–337. PMLR, 2019.
  52. Detecting stochastic dominance for poset-valued random variables as an example of linear programming on closure systems, 2017. URL https://epub.ub.uni-muenchen.de/40416/13/TR_209.pdf. Technical Report 209, Department of Statistics, LMU Munich.
  53. A. Sen. Commodities and Capabilities. Elsevier, 1985.
  54. G. Shafer. A Mathematical Theory of Evidence. Princeton University Press, 1976.
  55. M. Shaked and G. Shanthikumar. Stochastic orders. Springer, 2007.
  56. M. Shaker and E. Hüllermeier. Ensemble-based uncertainty quantification: Bayesian versus credal inference, 2021. URL https://arxiv.org/abs/2107.10384. arXiv preprint.
  57. M. Timonin. Maximization of the Choquet integral over a convex set and its application to resource allocation problems. Ann Oper Res, 196:543–579, 2012.
  58. UNECE. Guidelines on producing leading, composite and sentiment indicators, 2019. URL https://unece.org/DAM/stats/publications/2019/ECECESSTAT20192.pdf. (Febr. 16, 2023).
  59. L. Utkin. An imprecise deep forest for classification. Expert Syst Appl, 141:112978, 2020.
  60. L. Utkin and A. Konstantinov. Attention-based random forest and contamination model. Neural Networks, 154:346–359, 2022.
  61. P. Walley. Statistical Reasoning with Imprecise Probabilities. Chapman and Hall, London, 1991.
Citations (12)
View on Semantic Scholar

Summary

We haven't generated a summary for this paper yet.

Ai Sparkles Streamline Icon: https://streamlinehq.com Summarize Now