Towards a strictly frequentist theory of imprecise probability

Christian Fröhlich, Rabanus Derr, Robert C. Williamson
Proceedings of the Thirteenth International Symposium on Imprecise Probability: Theories and Applications, PMLR 215:230-240, 2023.

Abstract

Strict frequentism defines probability as the limiting relative frequency in an infinite sequence. What if the limit does not exist? We present a broader theory, which is applicable also to statistical phenomena that exhibit diverging relative frequencies. In doing so, we develop a close connection with imprecise probability: the cluster points of relative frequencies yield a coherent upper prevision. We show that a natural frequentist definition of conditional probability recovers the generalized Bayes rule. We prove constructively that, for a finite set of elementary events, there exists a sequence for which the cluster points of relative frequencies coincide with a prespecified set, thereby providing strictly frequentist semantics for coherent upper previsions.

Cite this Paper

BibTeX
@InProceedings{pmlr-v215-frohlich23a, title = {Towards a strictly frequentist theory of imprecise probability}, author = {Fr\"ohlich, Christian and Derr, Rabanus and Williamson, Robert C.}, booktitle = {Proceedings of the Thirteenth International Symposium on Imprecise Probability: Theories and Applications}, pages = {230--240}, year = {2023}, editor = {Miranda, Enrique and Montes, Ignacio and Quaeghebeur, Erik and Vantaggi, Barbara}, volume = {215}, series = {Proceedings of Machine Learning Research}, month = {11--14 Jul}, publisher = {PMLR}, pdf = {https://proceedings.mlr.press/v215/frohlich23a/frohlich23a.pdf}, url = {https://proceedings.mlr.press/v215/frohlich23a.html}, abstract = {Strict frequentism defines probability as the limiting relative frequency in an infinite sequence. What if the limit does not exist? We present a broader theory, which is applicable also to statistical phenomena that exhibit diverging relative frequencies. In doing so, we develop a close connection with imprecise probability: the cluster points of relative frequencies yield a coherent upper prevision. We show that a natural frequentist definition of conditional probability recovers the generalized Bayes rule. We prove constructively that, for a finite set of elementary events, there exists a sequence for which the cluster points of relative frequencies coincide with a prespecified set, thereby providing strictly frequentist semantics for coherent upper previsions.} }
Endnote
%0 Conference Paper %T Towards a strictly frequentist theory of imprecise probability %A Christian Fröhlich %A Rabanus Derr %A Robert C. Williamson %B Proceedings of the Thirteenth International Symposium on Imprecise Probability: Theories and Applications %C Proceedings of Machine Learning Research %D 2023 %E Enrique Miranda %E Ignacio Montes %E Erik Quaeghebeur %E Barbara Vantaggi %F pmlr-v215-frohlich23a %I PMLR %P 230--240 %U https://proceedings.mlr.press/v215/frohlich23a.html %V 215 %X Strict frequentism defines probability as the limiting relative frequency in an infinite sequence. What if the limit does not exist? We present a broader theory, which is applicable also to statistical phenomena that exhibit diverging relative frequencies. In doing so, we develop a close connection with imprecise probability: the cluster points of relative frequencies yield a coherent upper prevision. We show that a natural frequentist definition of conditional probability recovers the generalized Bayes rule. We prove constructively that, for a finite set of elementary events, there exists a sequence for which the cluster points of relative frequencies coincide with a prespecified set, thereby providing strictly frequentist semantics for coherent upper previsions.
APA
Fröhlich, C., Derr, R. & Williamson, R.C.. (2023). Towards a strictly frequentist theory of imprecise probability. Proceedings of the Thirteenth International Symposium on Imprecise Probability: Theories and Applications, in Proceedings of Machine Learning Research 215:230-240 Available from https://proceedings.mlr.press/v215/frohlich23a.html.

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