An operational characterization of the notion of probability by algorithmic randomness and its applications Kohtaro Tadaki Department of Computer Science, College of Engineering Chubu University Nagoya, Japan CCR 2015, June 24th, 2015, Institute of Computer Science, Heidelberg University 1
Abstract The notion of probability plays an important role in almost all areas of sci- ence and technology. In modern mathematics, however, probability theory means nothing other than measure theory, and the operational characteri- zation of the notion of probability does not seem to be established yet. In this talk, based on the toolkit of algorithmic randomness we present an operational characterization of the notion of probability. We use the notion of Martin-L¨ of randomness with respect to Bernoulli mea- sure to present the operational characterization. As the first step of the research of this line, in this talk we consider the case of finite probability space, i.e., the case where the sample space of the underlying probability space is finite, for simplicity. We give a natural operational characterization of the notion of conditional probability, and show how to represent the notion of the independence of random variables/events by the operational characterization. Finally, we mention some of the applications of our formalism to the general areas of science and technology. 2
Historical Background 3
Historical Background At the beginning of the past century, there was a comprehensive attempt to provide an operational characterization for the notion of probability. Namely, von Mises developed a mathematical theory of repetitive events which was aimed at reformulating the theory of probability and statistics based on an operational characterization of the notion of probability. In the attempt, he introduced the notion of collective as a mathematical idealization of a long sequence of outcomes of experiments or observations repeated under a set of invariable conditions, such as the repeated tossing of a coin or of a pair of dice. The collective plays a role as an operational characterization of the notion of probability, and is an infinite sequence of sample points of a probability space. In 1939, however, Ville revealed the defect of the notion of collective from the aspect of randomness. In addition, the collective has an intrinsic defect that it cannot exclude the possibility that an event with probability zero may occur. 4
Historical Background In 1966, Martin-L¨ of introduced the definition of random sequences, which is called Martin-L¨ of randomness nowadays, and plays a central role in the recent development of algorithmic randomness. At the same time, he introduced the notion of Martin-L¨ of randomness with respect to Bernoulli measure. He then pointed out that this notion over- comes the defect of collective, and this can be regarded precisely as the collective which von Mises wanted to define. However, Martin-L¨ of himself did not develop probability theory based on Martin-L¨ of random sequence with respect to Bernoulli measure. The aim of this talk is to develop an operational characterization of the notion of probability based on Martin-L¨ of random sequence with respect to Bernoulli measure, according to von Mises’s idea for reformulating proba- bility theory based on the collective. 5
Probability Space 6
Finite Probability Space We give an operational characterization of the notion of probability for a finite probability space. Definition A finite probability space is a mapping P : Ω → [0 , 1] which satisfies the following: (i) The domain of definition Ω is a non-empty finite set. (ii) � a ∈ Ω P ( a ) = 1. Here, Ω is called the sample space , and elements in Ω are called sample points or elementary events . A subset of Ω is called an event . For each event A , P ( A ) is defined by � P ( A ) := P ( a ) , a ∈ A and is called the probability of A . Note that most probability spaces appearing in engineering are finite. 7
Algorithmic Randomness 8
Bernoulli measure Then Ω ∗ denotes the set of all finite Let Ω be a non-empty finite set. strings over Ω, and Ω ∞ denotes the set of all infinite sequences over Ω. Let P : Ω → [0 , 1] be a finite probability space. Bernoulli measure λ P on Ω ∞ has the following property: For every σ ∈ Ω ∗ , P ( a ) N a ( σ ) , � [ σ ] ≺ � � λ P = a ∈ Ω where [ σ ] ≺ denotes the set of all infinite sequences over Ω which have σ as a prefix, and N a ( σ ) denotes the number of the occurrences of the element a in a finite string σ over Ω. 9
Martin-L¨ of randomness with respect to Bernoulli measure Definition [Martin-L¨ of 1966] Let P : Ω → [0 , 1] be a finite probability space. (i) A Martin-L¨ of P -test over Ω is a uniformly recursively enumerable se- quence { G n } n ∈ N ⊂ Ω ∗ such that for every n ∈ N , [ G n ] ≺ � ≤ 2 − n , � λ P where [ G n ] ≺ := { α ∈ Ω ∞ | Some prefix of α is in G n } . (ii) α ∈ Ω ∞ is called Martin-L¨ of P -random if for every Martin-L¨ of P -test { G n } n ∈ N over Ω, ∞ [ G n ] ≺ . � α / ∈ n =0 Remark In this talk, a finite probability space P is not required to be com- putable at all (except for the results related to van Lambalgen’s Theorem). Thus, Bernoulli measure λ P is not necessarily computable. 10
An Operational Characterization of the Notion of Probability: Ensemble 11
Ensemble We propose that a Martin-L¨ of P -random sequence of elementary events gives an operational characterization of the notion of probability. Since this notion plays a central role in our formalism, we call it ensemble , in particu- lar, instead of collective for distinction. The name “ensemble” comes from physics. Definition [Ensemble] Let P : Ω → [0 , 1] be a finite probability space. A Martin-L¨ of P -random sequence is called an ensemble for the finite probability space P . Consider an infinite sequence α ∈ Ω ∞ of outcomes which is being generated by infinitely repeated trials described by the finite probability space P . The operational characterization of the notion of probability for the finite prob- ability space P is thought to be completed if the property which the infinite sequence α has to satisfy is determined. We thus propose the following thesis. Thesis Let P : Ω → [0 , 1] be a finite probability space. An infinite se- quence of outcomes in Ω which is being generated by infinitely repeated trials described by the finite probability space P is an ensemble for P . 12
We check the validity of the thesis in what follows. 13
What is “probability” ? 14
“Necessary Conditions” for the Notion of Probability to Satisfy Consider an infinite sequence α ∈ Ω ∞ of outcomes which is being generated by infinitely repeated trials described by a finite probability space P . Accord- ing to our intuitive understanding on the notion of probability, the necessary conditions which the notion of probability ought to satisfy seem as follows: • The law of large numbers holds for α . • An event with probability zero never occurs in α . • α must be closed under a computable shuffling. • α must be closed under the selection by a computable se- lection function. • · · · · · · · · · · · · 15
The law of large numbers holds for ensembles Theorem [The law of large numbers] Let P : Ω → [0 , 1] be a finite probability space. For every α ∈ Ω ∞ , if α is an ensemble for P , then (i) [Martin-L¨ of 1966] the law of large numbers holds for α , that is, for every a ∈ Ω, # of a in α ↾ n lim = P ( a ) . n →∞ n (ii) Actually, there exists a single Martin-L¨ of P -test over Ω such that, for every α ∈ Ω ∞ , if α passes the test then the law of large numbers holds for α . This theorem holds even if the finite probability space P is not computable. 16
An event with probability zero never occurs in ensembles Consider the finite probability space P : { a, b } → [0 , 1] such that P ( a ) = 0 and P ( b ) = 1. Consider the infinite sequence α = b, a, b, b, b, b, b, b, b, b, b, b, . . . . . . . Since # of a in α ↾ n lim = 0 = P ( a ) , n →∞ n the law of large numbers certainly holds for α . However, the event a with probability zero has occurred in α once. This contradicts our intuition, in particular, contradicts the notion of probability in quantum mechanics. Thus, the law of large numbers is insufficient to characterizes the notion of probability, and the notion of probability is more than the law of large numbers. Theorem [Martin-L¨ of 1966] Let P : Ω → [0 , 1] be a finite probability space, and let a ∈ Ω. Suppose that P ( a ) = 0. Then, for every α ∈ Ω ∞ , if α is an ensemble for P , then α does not contain a at all. 17
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