Papers
Topics
Authors
Recent
Gemini 2.5 Flash
Gemini 2.5 Flash
158 tokens/sec
GPT-4o
7 tokens/sec
Gemini 2.5 Pro Pro
45 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

Gambling-Based Confidence Sequences for Bounded Random Vectors (2402.03683v2)

Published 6 Feb 2024 in stat.ME, cs.IT, math.IT, math.ST, and stat.TH

Abstract: A confidence sequence (CS) is a sequence of confidence sets that contains a target parameter of an underlying stochastic process at any time step with high probability. This paper proposes a new approach to constructing CSs for means of bounded multivariate stochastic processes using a general gambling framework, extending the recently established coin toss framework for bounded random processes. The proposed gambling framework provides a general recipe for constructing CSs for categorical and probability-vector-valued observations, as well as for general bounded multidimensional observations through a simple reduction. This paper specifically explores the use of the mixture portfolio, akin to Cover's universal portfolio, in the proposed framework and investigates the properties of the resulting CSs. Simulations demonstrate the tightness of these confidence sequences compared to existing methods. When applied to the sampling without-replacement setting for finite categorical data, it is shown that the resulting CS based on a universal gambling strategy is provably tighter than that of the posterior-prior ratio martingale proposed by Waudby-Smith and Ramdas.

Definition Search Book Streamline Icon: https://streamlinehq.com
References (26)
  1. Time-uniform confidence spheres for means of random vectors. arXiv preprint arXiv:2311.08168, 2023.
  2. The use of confidence or fiducial limits illustrated in the case of the binomial. Biometrika, 26(4):404–413, 1934.
  3. Thomas M Cover. Universal portfolios. Math. Financ., 1(1):1–29, 1991.
  4. Universal portfolios with side information. IEEE Trans. Inf. Theory, 42(2):348–363, 1996.
  5. Elements of information theory. John Wiley & Sons, 2006.
  6. Confidence sequences for mean, variance, and median. Proc. Natl. Acad. Sci. U. S. A., 58(1):66, 1967.
  7. Evidently. How evidently calculates results, 2023. URL https://docs.aws.amazon.com/AmazonCloudWatch/latest/monitoring/CloudWatch-Evidently-calculate-results.html.
  8. Wassily Hoeffding. Probability inequalities for sums of bounded random variables. J. Am. Stat. Assoc., 58(301):13–30, 1963.
  9. Time-uniform, nonparametric, nonasymptotic confidence sequences. Ann. Statist., 49(2):1055–1080, 2021.
  10. Parameter-free online convex optimization with sub-exponential noise. In Conf. Learn. Theory, pages 1802–1823. PMLR, 2019.
  11. The performance of universal encoding. IEEE Trans. Inf. Theory, 27(2):199–207, 1981.
  12. Tze Leung Lai. On confidence sequences. Ann. Statist., 4(2):265–280, 1976.
  13. Li, Li, and Dai’s Contribution to the Discussion of “Estimating Means of Bounded Random Variables by Betting” by Waudby-Smith and Ramdas. J. R. Stat. Soc. Series B Stat. Methodol., page qkad111, October 2023. ISSN 1369-7412. doi: 10.1093/jrsssb/qkad111.
  14. Martingale methods for sequential estimation of convex functionals and divergences. IEEE Trans. Inf. Theory, 69(7):4641–4658, 2023. doi: 10.1109/TIT.2023.3250099.
  15. Concentration inequalities for the empirical distribution of discrete distributions: beyond the method of types. Inf. Inference, 9(4):813–850, 2020.
  16. Tight concentrations and confidence sequences from the regret of universal portfolio. IEEE Trans. Inf. Theory, 70(1):436–455, 2024. doi: 10.1109/TIT.2023.3330187. arXiv:2110.14099.
  17. Admissible anytime-valid sequential inference must rely on nonnegative martingales. arXiv preprint arXiv:2009.03167, September 2020.
  18. On confidence sequences for bounded random processes via universal gambling strategies. arXiv preprint arXiv:2207.12382, 2022.
  19. On the near-optimality of betting confidence sets for bounded means. arXiv preprint arXiv:2310.01547, 2023.
  20. Simultaneous confidence intervals and sample size determination for multinomial proportions. J. Am. Statist. Assoc., 90(429):366–369, 1995.
  21. Open problem: Fast and optimal online portfolio selection. In Conf. Learn. Theory, pages 3864–3869. PMLR, 2020.
  22. Jean Ville. Etude critique de la notion de collectif. Bull. Amer. Math. Soc, 45(11):824, 1939.
  23. Confidence sequences for sampling without replacement. In H. Larochelle, M. Ranzato, R. Hadsell, M.F. Balcan, and H. Lin, editors, Adv. Neural Inf. Proc. Syst., volume 33, pages 20204–20214. Curran Associates, Inc., 2020a.
  24. Estimating means of bounded random variables by betting. arXiv preprint arXiv:2010.09686, 2020b.
  25. RiLACS: Risk Limiting Audits via Confidence Sequences. E-Vote-ID 2021, page 130, 2021.
  26. Time-uniform self-normalized concentration for vector-valued processes. arXiv preprint arXiv:2310.09100, 2023.
Citations (2)
View on Semantic Scholar

Summary

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

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

Tweets