Power and limitations of conformal martingales

This paper, accompanying my poster at ISIPTA 2019 (5 July 2019), poses the problem of investigating the power and limitations of conformal martingales as a means of detecting deviations from randomness. It starts from a brief review of conformal change detection, including CUSUM and Shiryaev-Roberts versions, and establishing simple validity results limiting the frequency of false alarms. It then gives several preliminary results in the direction of efficiency and discusses connections between randomness, exchangeability, and conformal martingales.

[1]  D. McDonald.  A cusum procedure based on sequential ranks , 1990 .

[2]  R. Mises.  Grundlagen der Wahrscheinlichkeitsrechnung , 1919 .

[3]  G. Lorden.  PROCEDURES FOR REACTING TO A CHANGE IN DISTRIBUTION , 1971 .

[4]  Vladimir Vovk,et al.  Conformal Prediction for Reliable Machine Learning: Theory, Adaptations and Applications , 2014 .

[5]  Harry Wechsler,et al.  A Martingale Framework for Detecting Changes in Data Streams by Testing Exchangeability , 2010, IEEE Transactions on Pattern Analysis and Machine Intelligence.

[6]  A. Shiryayev.  On Tables of Random Numbers , 1993 .

[7]  V. Vovk.  On the concept of the Bernoulli property , 1986 .

[8]  Per Martin-Löf,et al.  The Definition of Random Sequences , 1966, Inf. Control..

[9]  A. Shiryaev.  On Optimum Methods in Quickest Detection Problems , 1963 .

[10]  E. S. Page.  CONTINUOUS INSPECTION SCHEMES , 1954 .

[11]  A. Kolmogorov,et al.  ALGORITHMS AND RANDOMNESS , 1988 .

[12]  A. N. Kolmogorov,et al.  Foundations of the theory of probability , 1960 .

[13]  Stephen E. Fienberg,et al.  Testing Statistical Hypotheses , 2005 .

[14]  Marie Schmidt,et al.  Nonparametrics Statistical Methods Based On Ranks , 2016 .

[15]  Helly.  Grundbegriffe der Wahrscheinlichkeitsrechnung , 1936 .

[16]  S. W. Roberts.  A Comparison of Some Control Chart Procedures , 1966 .

[17]  M. Pollak.  Average Run Lengths of an Optimal Method of Detecting a Change in Distribution. , 1987 .

[18]  A. Kolmogorov.  Three approaches to the quantitative definition of information , 1968 .

[19]  Alexander Gammerman,et al.  Inductive Conformal Martingales for Change-Point Detection , 2017, COPA.

[20]  Alexander Gammerman,et al.  Plug-in martingales for testing exchangeability on-line , 2012, ICML.

[21]  A. N. Kolmogorov.  Combinatorial foundations of information theory and the calculus of probabilities , 1983 .

[22]  Andrei N. Kolmogorov,et al.  Logical basis for information theory and probability theory , 1968, IEEE Trans. Inf. Theory.

[23]  W. Gasarch,et al.  The Book Review Column 1 Coverage Untyped Systems Simple Types Recursive Types Higher-order Systems General Impression 3 Organization, and Contents of the Book , 2022 .

[24]  H. Robbins.  A Remark on Stirling’s Formula , 1955 .

[25]  Wenyu Du,et al.  On robustness of the Shiryaev–Roberts change-point detection procedure under parameter misspecification in the post-change distribution , 2017, Commun. Stat. Simul. Comput..