219 research outputs found

    Universality, optimality, and randomness deficiency

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    A Martin-Löf test UU is universal if it captures all non-Martin-Löf random sequences, and it is optimal if for every ML-test VV there is a c∈ωc∈ω such that ∀n(Vn+c⊆Un)∀n(Vn+c⊆Un). We study the computational differences between universal and optimal ML-tests as well as the effects that these differences have on both the notion of layerwise computability and the Weihrauch degree of LAYLAY, the function that produces a bound for a given Martin-Löf random sequence's randomness deficiency. We prove several robustness and idempotence results concerning the Weihrauch degree of LAYLAY, and we show that layerwise computability is more restrictive than Weihrauch reducibility to LAYLAY. Along similar lines we also study the principle RDRD, a variant of LAYLAY outputting the precise randomness deficiency of sequences instead of only an upper bound as LAYLAY

    Algorithmic statistics: forty years later

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    Algorithmic statistics has two different (and almost orthogonal) motivations. From the philosophical point of view, it tries to formalize how the statistics works and why some statistical models are better than others. After this notion of a "good model" is introduced, a natural question arises: it is possible that for some piece of data there is no good model? If yes, how often these bad ("non-stochastic") data appear "in real life"? Another, more technical motivation comes from algorithmic information theory. In this theory a notion of complexity of a finite object (=amount of information in this object) is introduced; it assigns to every object some number, called its algorithmic complexity (or Kolmogorov complexity). Algorithmic statistic provides a more fine-grained classification: for each finite object some curve is defined that characterizes its behavior. It turns out that several different definitions give (approximately) the same curve. In this survey we try to provide an exposition of the main results in the field (including full proofs for the most important ones), as well as some historical comments. We assume that the reader is familiar with the main notions of algorithmic information (Kolmogorov complexity) theory.Comment: Missing proofs adde

    Algorithmic Complexity Bounds on Future Prediction Errors

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    We bound the future loss when predicting any (computably) stochastic sequence online. Solomonoff finitely bounded the total deviation of his universal predictor MM from the true distribution mumu by the algorithmic complexity of mumu. Here we assume we are at a time t>1t>1 and already observed x=x1...xtx=x_1...x_t. We bound the future prediction performance on xt+1xt+2...x_{t+1}x_{t+2}... by a new variant of algorithmic complexity of mumu given xx, plus the complexity of the randomness deficiency of xx. The new complexity is monotone in its condition in the sense that this complexity can only decrease if the condition is prolonged. We also briefly discuss potential generalizations to Bayesian model classes and to classification problems.Comment: 21 page

    Algorithmic randomness and layerwise computability

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    International audienceIn this article we present the framework of layerwise computability. We explain the origin of this notion, its main features and properties, and we illustrate it with several concrete examples: decomposition of measures, random closed sets, Brownian motion

    Minimum Description Length Induction, Bayesianism, and Kolmogorov Complexity

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    The relationship between the Bayesian approach and the minimum description length approach is established. We sharpen and clarify the general modeling principles MDL and MML, abstracted as the ideal MDL principle and defined from Bayes's rule by means of Kolmogorov complexity. The basic condition under which the ideal principle should be applied is encapsulated as the Fundamental Inequality, which in broad terms states that the principle is valid when the data are random, relative to every contemplated hypothesis and also these hypotheses are random relative to the (universal) prior. Basically, the ideal principle states that the prior probability associated with the hypothesis should be given by the algorithmic universal probability, and the sum of the log universal probability of the model plus the log of the probability of the data given the model should be minimized. If we restrict the model class to the finite sets then application of the ideal principle turns into Kolmogorov's minimal sufficient statistic. In general we show that data compression is almost always the best strategy, both in hypothesis identification and prediction.Comment: 35 pages, Latex. Submitted IEEE Trans. Inform. Theor

    Weihrauch-completeness for layerwise computability

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    We introduce the notion of being Weihrauch-complete for layerwise computability and provide several natural examples related to complex oscillations, the law of the iterated logarithm and Birkhoff's theorem. We also consider hitting time operators, which share the Weihrauch degree of the former examples but fail to be layerwise computable

    Computable Measure Theory and Algorithmic Randomness

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    International audienceWe provide a survey of recent results in computable measure and probability theory, from both the perspectives of computable analysis and algorithmic randomness, and discuss the relations between them
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