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## Über dieses Buch

"Decision Systems and Non-stochastic Randomness" is the first systematic presentation and mathematical formalization (including existence theorems) of the statistical regularities of non-stochastic randomness. The results presented in this book extend the capabilities of probability theory by providing mathematical techniques that allow for the description of uncertain events that do not fit standard stochastic models. The book demonstrates how non-stochastic regularities can be incorporated into decision theory and information theory, offering an alternative to the subjective probability approach to uncertainty and the unified approach to the measurement of information. This book is intended for statisticians, mathematicians, engineers, economists or other researchers interested in non-stochastic modeling and decision theory.

## Inhaltsverzeichnis

### Chapter 1. Introduction

There are some questions that never age. It is answers that grow old. We begin our book with such a question. “Four kings—Hanoverians—George I, George II, George III, and George IV all died on the same day of the week, Saturday. The probability that we are dealing here with a random event is extremely low. Can we make a conclusion that Saturday is a day of ill omen for this dynasty? And if so, then what is the value of statistical methods of hypothesis verification?”
V. I. Ivanenko

### Chapter 2. Decision Systems

Decision theory emerged from the requirements of diverse fields of human activity such as medicine, gambling, politics, warfare, economics and finance, and engineering. Perhaps this is the reason for the terminological diversity that sometimes impedes not only mutual understanding between specialists in different fields but also the development of decision theory itself. In this sense, control theory has been more fortunate, for its terminology turned out to be common to many spheres of its application.
V. I. Ivanenko

### Chapter 3. Indifferent Uncertainty

In the control of technological processes of mass production, one often finds the following decision situation. In every realization of a technological operation, it is necessary to choose the value of some operational parameter, for example, the time t of a treatment, knowing the following information: (1) The “ideal” value t* of the parameter t—that is, the value that we would choose if we knew all necessary variables or inputs—is contained within certain limits t 1t* ≤ t 2. (2) If t deviates from t* by no more than a given value ∆t > 0, the article is good; otherwise it is bad or wasted. (3) The technological operation is carried out many times; therefore it is expedient to estimate it by the average quantity of wasted articles over infinitely many realizations.
V. I. Ivanenko

### Chapter 4. Nonstochastic Randomness

The results of Chapter 3 allow us to look in a new way at the problem of decision-making under uncertainty as a whole. Let us begin by noticing that in research in this field there is one at first sight terminological, but in fact rather fundamental, confusion. Some authors, following [46], mention only two types of behavior of the cause–effect mechanism that generates consequences: (1) so-called complete uncertainty, or just uncertainty, when nothing is known about the mechanism and (2) so-called risk, when consequences are random with a given probability distribution. Others, such as [80] and [79], consider this second variant as one of the main types of uncertainty, and call it correspondingly stochastic uncertainty or a first information situation.
V. I. Ivanenko

### Chapter 5. General Decision Problems

The regularities on Θ provide a mathematical tool that can be used for the description of mass events depending on a parameter and estimated on average. It is interesting, however, that this conclusion can be obtained regardless of the results of Chapter 4.
V. I. Ivanenko

### Chapter 6. Experiment in Decision Problems

So far, we have considered the case in which we know beforehand the set Θ of values of the unknown parameter θ and the function L, that is, the scheme Z = (Θ, U, L). It follows from Theorem 5.2 that for a univalent assignment of the criterion L Z * , it is sufficient to know as well the statistical regularity P on Θ describing the supposed behavior of θ ∈ Θ. In this way, the pair S = (Z, P) becomes a complete description of the decision problem T = {S, L Z * } (Remark 5.1).
V. I. Ivanenko

### Chapter 7. Informativity of Experiment in Bayesian Decision Problems

In this chapter we shall study the question of interconnection between the concept of informativity of experiment and Shannon’s concept “quantity of information,” contained in this experiment. Here the class of decision problems and experiments is significantly narrowed with respect to previous chapters. Therefore it is convenient to change slightly the notation and terminology.
V. I. Ivanenko

### Chapter 8. Reducibility of Experiments in Multistep Decision Problems

The notion of informativity of experiment (Chapters 6, 7) is useful for the extension of some stochastic decision problems to random-in-a-broad-sense decision problems. One such extension that concerns a multistep decision problem is considered here. A multistep decision problem arises when the decision-maker has to make sequential decisions in the same situation [14]. If an experiment is included in this system, then sequential accumulation of information—decrease of uncertainty—becomes feasible [18]. Indeed, let n be the number of stages in the multistep problem. Let the decision-maker before making the (k + 1)th decision, k = 0, 1, …, n, perform an experiment (observation).
V. I. Ivanenko

### Chapter 9. Concluding Remarks

Decision-making under uncertainty accompanies all human activity, and so it is difficult to locate the beginnings of research in this field. At any rate, the consideration of problems in which nothing was known about the parameter but the set of its possible values—total uncertainty problems—can be found already in the work of Johann Bernoulli, Laplace, and Bayes alongside problems in which the unknown parameter was random with a given distribution. However, the “principle of insufficient foundation” (Bayes postulate) suggested therein, which in the case of total uncertainty recommended the consideration of all values of the unknown parameter as equiprobable, eventually proved to be logically inconsistent, and in 1854 it was subjected to serious criticism by J. Bull [29]. Attempts to save this principle by a formalization of conditions under which it does not lead to contradictions (see, for example, [10]) were undertaken later, but the overwhelming majority of researchers preferred to eschew the principle entirely, and as a result, quite a number of approaches to problems with uncertainty appeared.
V. I. Ivanenko

### Backmatter

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