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Generalized Conformal Prediction

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Algorithmic Learning in a Random World

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Abstract

In the previous chapters we assumed that the data are generated from an exchangeable probability measure. In this chapter we generalize the method of conformal prediction to cover arbitrary statistical models that belong to the class of, as we call them, online compression models. Interesting online compression models include, e.g., partial exchangeability models, Gaussian models, and causal networks.

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References

  1. Asarin, E.A.: Some properties of Kolmogorov δ-random finite sequences. Theory Probab. Appl. 32, 507–508 (1987)

    Article  MATH  Google Scholar 

  2. Asarin, E.A.: On some properties of finite objects random in the algorithmic sense. Soviet Math. Doklady 36, 109–112 (1988). The Russian original published in 1987

    Google Scholar 

  3. Baker, G.A.: The probability that the mean of a second sample will differ from the mean of a first sample by less than a certain multiple of the standard deviation of the first sample. Ann. Math. Stat. 6, 197–201 (1935)

    Article  MATH  Google Scholar 

  4. Bernardo, J.M., Smith, A.F.M.: Bayesian Theory. Wiley, Chichester (1994)

    Book  MATH  Google Scholar 

  5. Bourbaki, N.: Elements of the History of Mathematics. Springer, Berlin (1994)

    MATH  Google Scholar 

  6. Cox, D.R., Hinkley, D.V.: Theoretical Statistics. Chapman and Hall, London (1974)

    Book  MATH  Google Scholar 

  7. Cramér, H.: Mathematical Methods of Statistics. Princeton University Press, Princeton (1946)

    MATH  Google Scholar 

  8. Dawid, A.P.: Intersubjective statistical models. In: Koch, G.S., Spizzichino, F. (eds.) Exchangeability in Probability and Statistics, pp. 217–232. North-Holland, Amsterdam (1982)

    Google Scholar 

  9. de Finetti, B.: Sur la condition d’équivalence partielle. In: Actualités Scientifiques et Industrielles, vol. 739. Hermann, Paris (1938). An English translation, under the title “On the condition of partial exchangeability”, is included in [175], pp. 193–206

  10. Draper, N.R., Smith, H.: Applied Regression Analysis, 3rd edn. Wiley, New York (1998)

    Book  MATH  Google Scholar 

  11. Dudley, R.M.: Real Analysis and Probability. Cambridge University Press, Cambridge (2002). Original edition published in 1989 by Wadsworth

    Google Scholar 

  12. Fisher, R.A.: The goodness of fit of regression formulae and the distribution of regression coefficients. J. R. Stat. Soc. 85, 597–612 (1922)

    Article  Google Scholar 

  13. Fisher, R.A.: Applications of “Student’s” distribution. Metron 5, 90–104 (1925)

    MATH  Google Scholar 

  14. Fisher, R.A.: Statistical Methods and Scientific Inference, 3rd edn. Hafner, New York (1973). Included in [112]. First edition: 1956

  15. Fisher, R.A.: Statistical Methods for Research Workers, 14th (revised and enlarged) edn. Hafner, New York (1973). Included in [112]. First edition: 1925

  16. Freedman, D.A.: Invariants under mixing which generalise de Finetti’s theorem. Ann. Math. Stat. 33, 916–923 (1962)

    Article  MATH  Google Scholar 

  17. Freedman, D.A.: Invariants under mixing which generalise de Finetti’s theorem: continuous time parameter. Ann. Math. Stat. 34, 1194–1216 (1963)

    Article  MATH  Google Scholar 

  18. Hastie, T., Montanari, A., Rosset, S., Tibshirani, R.: Surprises in high-dimensional ridgeless least squares interpolation. Ann. Stat. 50, 949–986 (2022)

    Article  MATH  Google Scholar 

  19. Kingman, J.F.C.: On random sequences with spherical symmetry. Biometrika 59, 492–493 (1972)

    Article  MATH  Google Scholar 

  20. Kingman, J.F.C.: Uses of exchangeability. Ann. Probab. 6, 183–197 (1978)

    Article  MATH  Google Scholar 

  21. Kolmogorov, A.N.: Sur une formule limite de M. A. Khintchine. Comptes rendus des Séances de l’Académie des Sciences 186, 824–825 (1928)

    Google Scholar 

  22. Kolmogorov, A.N.: Über das Gesetz des iterierten Logarithmus. Math. Ann. 101, 126–135 (1929)

    Article  MATH  Google Scholar 

  23. Kolmogorov, A.N.: Grundbegriffe der Wahrscheinlichkeitsrechnung. Springer, Berlin (1933). Published in English as Foundations of the Theory of Probability (Chelsea, New York). First edition, 1950; second edition, 1956

    Google Scholar 

  24. Kolmogorov, A.N.: On tables of random numbers. Sankhya Ind. J. Stat. A 25, 369–376 (1963)

    MATH  Google Scholar 

  25. Kolmogorov, A.N.: Three approaches to the quantitative definition of information. Problems Inf. Transm. 1, 1–7 (1965)

    Google Scholar 

  26. Kolmogorov, A.N.: Logical basis for information theory and probability theory. IEEE Trans. Inf. Theory IT-14, 662–664 (1968)

    Article  MATH  Google Scholar 

  27. Kolmogorov, A.N.: Combinatorial foundations of information theory and the calculus of probabilities. Russian Math. Surv. 38, 29–40 (1983). This article was written in 1970 in connection with Kolmogorov’s talk at the International Mathematical Congress in Nice

    Google Scholar 

  28. Lauritzen, S.L.: Statistical Models as Extremal Families. Aalborg University Press, Aalborg (1982)

    MATH  Google Scholar 

  29. Lauritzen, S.L.: Extremal Families and Systems of Sufficient Statistics. Lecture Notes in Statistics, vol. 49. Springer, New York (1988)

    Google Scholar 

  30. Martin-Löf, P.: The definition of random sequences. Inf. Control 9, 602–619 (1966)

    Article  MATH  Google Scholar 

  31. Martin-Löf, P.: Repetitive structures (with discussion). In: Barndorff-Nielsen, O.E., Blæsild, P., Schou, G. (eds.) Proceedings of Conference on Foundational Questions in Statistical Inference, pp. 271–294. Aarhus, Denmark (1974)

    Google Scholar 

  32. Montgomery, D.C., Peck, E.A., Vining, G.G.: Introduction to Linear Regression Analysis, 6th edn. Wiley, Hoboken (2021)

    MATH  Google Scholar 

  33. Ryabko, D.: Relaxing i.i.d. assumption in online pattern recognition. Tech. Rep. CS-TR-03-11, Department of Computer Science, Royal Holloway, University of London (2003)

    Google Scholar 

  34. Sampson, A.R.: A tale of two regressions. J. Am. Stat. Assoc. 69, 682–689 (1974)

    Article  MATH  Google Scholar 

  35. Schervish, M.J.: Theory of Statistics. Springer, New York (1995)

    Book  MATH  Google Scholar 

  36. Seal, H.L.: Studies in the history of probability and statistics. XV: The historical development of the Gauss linear model. Biometrika 54, 1–24 (1967)

    MATH  Google Scholar 

  37. Shiryaev, A.N.: Probability-1, 3rd edn. Springer, New York (2016)

    Book  MATH  Google Scholar 

  38. Shiryaev, A.N.: Probability-2, 3rd edn. Springer, New York (2019)

    Book  MATH  Google Scholar 

  39. Smith, A.F.M.: On random sequences with centred spherical symmetry. J. R. Stat. Soc. B 43, 208–209 (1981)

    MATH  Google Scholar 

  40. Stuart, A., Ord, K.J., Arnold, S.: Kendall’s Advanced Theory of Statistics, Vol. 2a: Classical Inference and the Linear Model, 6th edn. Arnold, London (1999)

    Google Scholar 

  41. Tibshirani, R.J., Barber, R.F., Candès, E.J., Ramdas, A.: Conformal prediction under covariate shift. In: Advances in Neural Information Processing Systems, vol. 32, pp. 2530–2540. Curran Associates, Red Hook (2019)

    Google Scholar 

  42. Vovk, V.: On the concept of the Bernoulli property. Russian Math. Surv. 41, 247–248 (1986). Russian original: . Another English translation with proofs: arXiv report 1612.08859

  43. Vovk, V.: Kolmogorov’s complexity conception of probability. In: Hendricks, V.F., Pedersen, S.A., Jørgensen, K.F. (eds.) Probability Theory: Philosophy, Recent History and Relations to Science, pp. 51–69. Kluwer, Dordrecht (2001)

    Chapter  Google Scholar 

  44. Vovk, V.: Well-calibrated predictions from on-line compression models. Theor. Comput. Sci. 364, 10–26 (2006). ALT 2003 Special Issue

    Google Scholar 

  45. Vovk, V., Shafer, G.: Kolmogorov’s contributions to the foundations of probability. Problems Inf. Trans. 39, 21–31 (2003)

    Article  MATH  Google Scholar 

  46. Vovk, V., Gammerman, A., Shafer, G.: Algorithmic Learning in a Random World. Springer, New York (2005). This is the first edition of this book

    Google Scholar 

  47. Vovk, V., Nouretdinov, I., Gammerman, A.: On-line predictive linear regression. Ann. Stat. 37, 1566–1590 (2009). See also arXiv:math/0511522 [math.ST] (November 2011)

  48. Wilks, S.S.: Determination of sample sizes for setting tolerance limits. Ann. Math. Stat. 12, 91–96 (1941)

    Article  MATH  Google Scholar 

  49. Williams, D.: Probability with Martingales. Cambridge University Press, Cambridge (1991)

    Book  MATH  Google Scholar 

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Authors and Affiliations

  1. Royal Holloway, University of London, London, UK

    Vladimir Vovk & Alexander Gammerman

  2. Rutgers University, Newark, NJ, USA

    Glenn Shafer

Authors
  1. Vladimir Vovk
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  2. Alexander Gammerman
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  3. Glenn Shafer
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Vovk, V., Gammerman, A., Shafer, G. (2022). Generalized Conformal Prediction. In: Algorithmic Learning in a Random World. Springer, Cham. https://doi.org/10.1007/978-3-031-06649-8_11

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