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On unstable and unoptimal prediction
Author(s) -
Kalociński Dariusz,
Steifer Tomasz
Publication year - 2019
Publication title -
mathematical logic quarterly
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.473
H-Index - 28
eISSN - 1521-3870
pISSN - 0942-5616
DOI - 10.1002/malq.201800085
Subject(s) - randomness , mathematics , sequence (biology) , limit (mathematics) , phenomenon , context (archaeology) , function (biology) , mean squared prediction error , binary number , limit of a function , discrete mathematics , statistics , arithmetic , mathematical analysis , epistemology , paleontology , philosophy , genetics , evolutionary biology , biology
We consider the notion of prediction functions (or predictors) studied before in the context of randomness and stochasticity by Ko, and later by Ambos‐Spies and others. Predictor is a total computable function which tries to predict bits of some infinite binary sequence. The prediction error is defined as the limit of the number of incorrect answers divided by the number of answers given so far. We discuss indefiniteness of prediction errors for weak 1‐generics and show that this phenomenon affects certain c.e. sequences as well. On the other hand, a notion of optimal predictor is considered. It is shown that there is a sequence for which increasingly better predictors exist but for which no predictor is optimal.