A game-theoretic analysis of updating sets of probabilities

Peter D. Grünwald, Joseph Y. Halpern
Proceedings of the 24th Conference on Uncertainty in Artificial Intelligence, PMLR R6:240-247, 2008.

Abstract

We consider how an agent should update her uncertainty when it is represented by a set P of probability distributions and the agent observes that a random variable X takes on value x, given that the agent makes decisions using the minimax criterion, perhaps the best-studied and most commonly-used criterion in the literature. We adopt a game-theoretic framework, where the agent plays against a bookie, who chooses some distribution from P. We consider two reasonable games that differ in what the bookie knows when he makes his choice. Anomalies that have been observed before, like time inconsistency, can be understood as arising because different games are being played, against bookies with different information. We characterize the important special cases in which the optimal decision rules according to the minimax criterion amount to either conditioning or simply ignoring the information. Finally, we consider the relationship between conditioning and calibration when uncertainty is described by sets of probabilities.

Cite this Paper

BibTeX
@InProceedings{pmlr-vR6-grunwald08a, title = {A game-theoretic analysis of updating sets of probabilities}, author = {Gr\"{u}nwald, Peter D. and Halpern, Joseph Y.}, booktitle = {Proceedings of the 24th Conference on Uncertainty in Artificial Intelligence}, pages = {240--247}, year = {2008}, editor = {McAllester, David A. and Myllymäki, Petri}, volume = {R6}, series = {Proceedings of Machine Learning Research}, month = {09--12 Jul}, publisher = {PMLR}, pdf = {https://raw.githubusercontent.com/mlresearch/r6/main/assets/grunwald08a/grunwald08a.pdf}, url = {https://proceedings.mlr.press/r6/grunwald08a.html}, abstract = {We consider how an agent should update her uncertainty when it is represented by a set P of probability distributions and the agent observes that a random variable X takes on value x, given that the agent makes decisions using the minimax criterion, perhaps the best-studied and most commonly-used criterion in the literature. We adopt a game-theoretic framework, where the agent plays against a bookie, who chooses some distribution from P. We consider two reasonable games that differ in what the bookie knows when he makes his choice. Anomalies that have been observed before, like time inconsistency, can be understood as arising because different games are being played, against bookies with different information. We characterize the important special cases in which the optimal decision rules according to the minimax criterion amount to either conditioning or simply ignoring the information. Finally, we consider the relationship between conditioning and calibration when uncertainty is described by sets of probabilities.}, note = {Reissued by PMLR on 09 October 2024.} }
Endnote
%0 Conference Paper %T A game-theoretic analysis of updating sets of probabilities %A Peter D. Grünwald %A Joseph Y. Halpern %B Proceedings of the 24th Conference on Uncertainty in Artificial Intelligence %C Proceedings of Machine Learning Research %D 2008 %E David A. McAllester %E Petri Myllymäki %F pmlr-vR6-grunwald08a %I PMLR %P 240--247 %U https://proceedings.mlr.press/r6/grunwald08a.html %V R6 %X We consider how an agent should update her uncertainty when it is represented by a set P of probability distributions and the agent observes that a random variable X takes on value x, given that the agent makes decisions using the minimax criterion, perhaps the best-studied and most commonly-used criterion in the literature. We adopt a game-theoretic framework, where the agent plays against a bookie, who chooses some distribution from P. We consider two reasonable games that differ in what the bookie knows when he makes his choice. Anomalies that have been observed before, like time inconsistency, can be understood as arising because different games are being played, against bookies with different information. We characterize the important special cases in which the optimal decision rules according to the minimax criterion amount to either conditioning or simply ignoring the information. Finally, we consider the relationship between conditioning and calibration when uncertainty is described by sets of probabilities. %Z Reissued by PMLR on 09 October 2024.
APA
Grünwald, P.D. & Halpern, J.Y.. (2008). A game-theoretic analysis of updating sets of probabilities. Proceedings of the 24th Conference on Uncertainty in Artificial Intelligence, in Proceedings of Machine Learning Research R6:240-247 Available from https://proceedings.mlr.press/r6/grunwald08a.html. Reissued by PMLR on 09 October 2024.

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