Computable Measure Theory and Algorithmic Randomness

Hoyrup, Mathieu; Rute, Jason

doi:10.1007/978-3-030-59234-9_7

Cited by 3 publications

(6 citation statements)

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“…That is: each U k can be effectively approximated as a union of basic subsets of 2 ; and this approximation is uniform in k, in the sense that a single machine can handle each of the U k . 34 Let be a computable probability measure on 2 . Then an effective -null set is a subset T of 2 that can be written in the form T = U k , for some uniformly effective family of open sets, {U k }, satisfying (U k ) ≤ 2 -k .…”

Section: Learning and Bernoulli Measuresmentioning

confidence: 99%

“…Choosing j large enough to ensure that 1/c m ≤ 2 j , we have 33 Much of the argument of this section was suggested by Christopher Porter (in private communication). 34 And if we equip 2 with the product topology (as in footnote 6), then each U k is an open subset of 2 . 35 Commentary: such T are -null sets that can be effectively specified in a certain sense.…”

Section: Learning and Bernoulli Measuresmentioning

confidence: 99%

“… 63 For more on this prior, see, e.g., [21, pp. 119–121], [65, 740 f. and 750], and [34, examples 7.2.4, 7.2.27, and 7.4.14]. …”

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confidence: 99%

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Unprincipled

Belot

2023

The Review of Symbolic Logic

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It is widely thought that chance should be understood in reductionist terms: claims about chance should be understood as claims that certain patterns of events are instantiated. There are many possible reductionist theories of chance, differing as to which possible pattern of events they take to be chance-making. It is also widely taken to be a norm of rationality that credence should defer to chance: special cases aside, rationality requires that one’s credence function, when conditionalized on the chance-making facts, should coincide with the objective chance function. It is a shortcoming of a theory of chance if it implies that this norm of rationality is unsatisfiable. The primary goal of this paper is to show, on the basis of considerations concerning computability and inductive learning, that this shortcoming is more common than one would have hoped.

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Section: Learning and Bernoulli Measuresmentioning

confidence: 99%

Section: Learning and Bernoulli Measuresmentioning

confidence: 99%

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Unprincipled

Belot

2023

The Review of Symbolic Logic

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“…This matches Theorem 5 in reducing a seemingly different setting for randomness to the case of binary sequences. See [48] for a general perspective on this phenomenon. Further to Definition 1 above, taken from [44], in what follows I also adopt [Definition 4.2] in [44] as a definition (or characterizing) of computability (which in this approach presupposes continuity): Definition 6. if (X, B) and (X , B ) are effective topological spaces, then f : X → X is computable iff for each U ∈ B the inverse image f −1 (U ) ⊂ X is open and computable.…”

Section: From 'For P-almost Every X' To 'For All P-random X'mentioning

confidence: 99%

“…, where [σ] = σA ω , is quite natural (here O(X) is the topology of X). If P is computable as defined in clause 6, P is a computable point in the effective space of all probability measures on X [46][47][48], where a point x in an effective topological space is deemed computable if {x} = ∩ n V n for some computable sequence V n .…”

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confidence: 99%

Typical = Random

Landsman

2023

Axioms

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This expository paper advocates an approach to physics in which “typicality” is identified with a suitable form of algorithmic randomness. To this end various theorems from mathematics and physics are reviewed. Their original versions state that some property Φ(x) holds for P-almost all x∈X, where P is a probability measure on some space X. Their more refined (and typically more recent) formulations show that Φ(x) holds for all P-random x∈X. The computational notion of P-randomness used here generalizes the one introduced by Martin-Löf in 1966 in a way now standard in algorithmic randomness. Examples come from probability theory, analysis, dynamical systems/ergodic theory, statistical mechanics, and quantum mechanics (especially hidden variable theories). An underlying philosophical theme, inherited from von Mises and Kolmogorov, is the interplay between probability and randomness, especially: which comes first?

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Computer Science for Continuous Data

Brauße

Collins²,

Ziegler³

2022

Computer Algebra in Scientific Computing

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Computable Measure Theory and Algorithmic Randomness

Abstract: We 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.

Cited by 3 publications

References 65 publications

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Typical = Random

Computer Science for Continuous Data

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