Computability of Probability Distributions and Characteristic Functions

Mori, Takakazu; Tsujii, Yoshiki; Yasugi, Mariko

doi:10.2168/lmcs-9(3:9)2013

Cited by 9 publications

(7 citation statements)

References 7 publications

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“…The random variable X has a discrete probability distribution if A is at most countable. If we denote by P X ({x}) the probability of the event {X = x} = {ω ∈ | X(ω) = x}, then the discrete probability distribution of X is completely defined by the numbers P X ({x}) [23, p. 159]; see also [24,29]).…”

Section: A Glimpse Of Probability Theorymentioning

confidence: 99%

A probabilistic anytime algorithm for the halting problem

Calude

Dumitrescu

2018

COM

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The Halting Problem, the most (in)famous undecidable problem, has important applications in theoretical and applied computer science and beyond, hence the interest in its approximate solutions.Experimental results reported on various models of computation suggest that halting programs are not uniformly distributedrunning times play an important role. A reason is that a program which eventually stops but does not halt "quickly", stops at a time which is algorithmically compressible.In this paper we work with running times to define a class of computable probability distributions on the set of halting programs in order to construct an anytime algorithm for the Halting problem with a probabilistic evaluation of the error of the decision.

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Section: A Glimpse Of Probability Theorymentioning

confidence: 99%

A probabilistic anytime algorithm for the halting problem

Calude

Dumitrescu

2018

COM

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“…Many of these papers follow the type-2 effectivity approach [Wei00, BHW08] or the domain theory approach [AJ94]. Most of these papers have been concerned with computable representations of measures or probability distributions [Wei99, Mül99, WW06, SS06, Sch07, Eda09, HR09d, MTY13,Col]. While most of these representations are equivalent, the generality of the underlying spaces vary.…”

Section: Randomness and Constructive Mathematicsmentioning

confidence: 99%

Computable randomness and betting for computable probability spaces

Rute

2016

Mathematical Logic Qtrly

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Abstract. Unlike Martin-Löf randomness and Schnorr randomness, computable randomness has not been defined, except for a few ad hoc cases, outside of Cantor space. This paper offers such a definition (actually, several equivalent definitions), and further, provides a general method for abstracting "bit-wise" definitions of randomness from Cantor space to arbitrary computable probability spaces. This same method is also applied to give machine characterizations of computable and Schnorr randomness for computable probability spaces, extending the previously known results. The paper contains a new type of randomness-endomorphism randomness-which the author hopes will shed light on the open question of whether Kolmogorov-Loveland randomness is equivalent to Martin-Löf randomness. The last section contains ideas for future research.

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“…If {µ n } n∈N is a sequence of probability measures, then {µ n } n∈N converges vaguely if and only if it converges weakly (see [4]). This fact is used in [8] to define an effective notion of convergence for probability measures on M(R). However, this definition is not a suitable effective analogue to classical vague convergence for non-probability measures on M(R).…”

Section: Introductionmentioning

confidence: 99%

Effective weak and vague convergence of measures on the real line

Rojas¹

2021

Preprint

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We expand our effective framework for weak convergence of measures on the real line by showing that effective convergence in the Prokhorov metric is equivalent to effective weak convergence. In addition, we establish a framework for the study of the effective theory of vague convergence of measures. We introduce a uniform notion and a non-uniform notion of vague convergence, and we show that both these notions are equivalent. However, limits under effective vague convergence may not be computable even when they are finite. We give an example of a finite incomputable effective vague limit measure, and we provide a necessary and sufficient condition so that effective vague convergence produces a computable limit. Finally, we determine a sufficient condition for which effective weak and vague convergence of measures coincide. As a corollary, we obtain an effective version of the equivalence between classical weak and vague convergence of sequences of probability measures.

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Computability of Probability Distributions and Characteristic Functions

Cited by 9 publications

References 7 publications

A probabilistic anytime algorithm for the halting problem

A probabilistic anytime algorithm for the halting problem

Computable randomness and betting for computable probability spaces

Effective weak and vague convergence of measures on the real line

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