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Algorithmic identification of probabilities is hard

Abstract : Reading more and more bits from an infinite binary sequence that is random for a Bernoulli measure with parameter p, we can get better and better approximations of p using the strong law of large numbers. In this paper, we study a similar situation from the viewpoint of inductive inference. Assume that p is a computable real, and we have to eventually guess the program that computes p. We show that this cannot be done computably, and extend this result to more general computable distributions. We also provide a weak positive result showing that looking at a sequence X generated according to some computable probability measure, we can guess a sequence of algorithms that, starting from some point, compute a measure that makes X Martin-Löf random.
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https://hal-lirmm.ccsd.cnrs.fr/lirmm-01803441
Contributor : Alexander Shen Connect in order to contact the contributor
Submitted on : Wednesday, May 30, 2018 - 3:28:39 PM
Last modification on : Wednesday, October 27, 2021 - 10:20:48 AM

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Laurent Bienvenu, Santiago Figueira, Benoit Monin, Alexander Shen. Algorithmic identification of probabilities is hard. Journal of Computer and System Sciences, Elsevier, 2018, 95, pp.98-108. ⟨10.1016/j.jcss.2018.01.002⟩. ⟨lirmm-01803441⟩

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