Bernoulli space

The Bernoulli space^[1] is a mathematical model for the transition from past to future in due consideration of uncertainty of future developments. The Bernoulli space is at the core of Bernoulli stochastics and represents the basis for reliable and accurate predictions and measurements.

The Bernoulli space does not assume a "ideal world" as, for example, physics which is based on the belief in truth and causality. In contrast, the Bernoulli space admits human ignorance and cosmological randomness which both generate uncertainty. The Bernoulli Space as a mathematical model of change can therefore be regarded as the means for obtaining reliable and accurate predictions.

Uncertainty

Before the model can be introduced, the term uncertainty^[2] must be uniquely explained since in everyday speech it can have several inconsistent meanings. In everyday speech uncertainty may refer to determinate, but unknown facts, or to the indeterminate future that cannot be known as it does not exist so far.

The difference between facts and future events is obvious, a fact is specified and if quantified has a fixed value, while a future event is indeterminate and therefore may or may not occur. A fact might be unknown, but nevertheless it is fixed. A future event cannot be known or unknown because it is subject to randomness and may occur or not occur.

For the sake of a unique interpretation, uncertainty is explained here as the inability to predict a future development exactly. This inability is due to ignorance about the past and randomness about the future. Ignorance and randomness generate uncertainty and for describing uncertainty quantitatively both sources have to be taken into account.

Ignorance

A fact is given by its (true) value. Ignorance means that the true value is unknown. Actually, only very rarely the true values of considered attributes are known. In general they are unknown. What is known are values which are not the true value. By the knowledge about "what is not" it is practically always possible to specify a set of values which cover the true but unknown one. This set reflects the knowledge about "what is no" or the ignorance about "what is". The larger the set is, the larger is the ignorance. In the rare case that the true value is known, the corresponding set is given by a singleton.

Randomness

Randomness refers to a future event and reflects its propensity to occur. The propensity can be small or large depending on the given situation. More than 300 years ago, the Swiss theologian and mathematician Jacob Bernoulli succeeded to quantify randomness of a future event. He explained the occurrence propensity of a future event by the degree of certainty of its occurrence and called this degree probability.^[3] If the occurrence of a given event is impossible, there is not the slightest propensity and the degree of certainty, i.e., the probability is 0. On the other hand if the occurrence of a given event is certain, then the propensity reaches its maximum and the degree of certainty adopts the value 1.

By fixing the zero and the unit of randomness, Jacob Bernoulli succeeded in quantifying it. Unfortunately his achievements were neither noticed nor understood by contemporary scientists and fell into oblivion after his premature death in 1705 when his masterpiece Ars conjectandi was still not completed. Finally, 8 years after Jacob had died, it was published in 1713 by Jacob's nephew Nikolaus, but it was too late to have a decisive influence on the development of science. The concept of probability which was introduced by Jacob Bernoulli as a quantification of randomness, degenerated to an ambiguous concept with many inconsistent interpretations and, consequently, the International System of Units^[4] (SI) does not contain a unit for randomness. Because of the shortcoming, it is not possible to express randomness adequately. For example, the probability that an interaction of a given kind will take place between a nucleus and an incident neutron is measured by an effective cross-sectional area also called cross section, i.e., in square centimeter which makes not much sense.

During the last two decades, however, the original interpretation was revived by the mathematician Elart von Collani and used to develop a new branch of science dealing with uncertainty and named Bernoulli Stochastics in memoriam of Jakob Bernoulli.

Future and past

Uncertainty as interpreted here refers to the future development which is always more or less strongly limited by its relation to the past. If this relation would be causal, then the future would be a mere transformation of the past or in other words the past would uniquely determine the future. In such a case there is no randomness, but only ignorance.

Until recently, science assumed that there is no randomness and the evolution of the universe follows strict causal laws. Only during the last century when looking closer at the elementary particles randomness was acknowledged for the world of particles, but still not for the macroworld. The consequence of this neglect is the fact that mankind has not developed so far powerful methods to handle successfully risk and uncertainty and thus solve the related problems.

In contrast to the prevailing belief in science, the Bernoulli Space is not based on a deterministic transition from known past to future implying that the relation cannot be represented by a function. Instead the unknown past as well as the random future must be described reflecting the generated uncertainty.

Random variable X

The future outcome of the considered process is indeterminate, i.e., each repetition of the process yields in general a different outcome. This is due to randomness and therefore the variable which represents the future outcomes of the process is called a random variable generally denoted X.

A random variable X will adopt one value of the set of possible values. This set is called range of variability of X. If the process is repeated then it is in general observed that some elements of the range of variability occur more frequently and other less frequently. This is due to different probabilities of the corresponding events. To obtain a complete model about the future development of X it is necessary to know besides its range of variability its probability distribution denoted $P_X$ which assigns to each future event of X its probability, i.e., its degree of certainty to occur.

Deterministic variable D

Those aspects of the past which have an influence on the future development with respect to X form the determinate initial conditions, which are represented by a variable, say D. Since value of D is fixed, D is called a deterministic variable.

Generally, the actual value of the deterministic variable D is not known. However, in any case a bounded set can be specified which contains the actual value of D.

The Bernoulli space as a model of uncertainty

A Bernoulli space is a model for the change from past to future. It therefore refers to a pair of variables $(X,D)$ where the random variable X represents the aspect of interest in the future, while the deterministic variable D represents the initial conditions, i.e., those aspects of the past which are relevant for X. The Bernoulli space of $(X,D)$ is denoted $\mathcal{B}_{X,D}$ and consists of three components $\mathcal{B}_{X,D} = (\mathcal{D}, \mathcal{X},\mathcal{P})$.

1. The first component is a bounded set of real numbers which is called ignorance space denoted $\mathcal{D}$ and which contains all those values of the deterministic variable D which cannot be excluded from being the true value of D according to the available knowledge. If the ignorance space is a singleton, i.e., $\mathcal{D}= \{d\}$ then the actual value of D is known, in all other cases the actual value is unknown. Each element of the ignorance space $\mathcal{D}$ might be the actual value and therefore each value must be equally taken into account in order to exclude that subjective opinion falsify the objective approach.

1. The second component is a function called variability function and denoted $\mathcal{X}$. The variability function assigns to each level of knowledge (= subset of the ignorance space) the corresponding range of variability of the random variable X. It follows that each image of the variability function consists of those values of X which might occur in the future, under the condition that the actual value of D is an element of its argument, i.e., the considered subset of the ignorance space. The variability function yields the extent of the variability of the random variable X and often whenever safety is involved only the largest and/or the smallest value of the range of variability of X are of interest. Replacing in such cases the bounded range of variability by an unbounded set, for example, the set of real numbers, may yield dangerous results.

1. The third component is called random structure function denoted $\mathcal{P}$ and assigns to each level of knowledge (= subset of the ignorance space) a probability distribution over the corresponding image of the variability function. The probability distribution assigns to each event give by a subset of the range of variability, the corresponding probability, i.e., the degree of certainty of its occurrence.

References

1. Elart von Collani (ed.), Defining the Science Stochastics, Heldermann Verlag, Lemgo, 2004.
2. Elart von Collani, Defining and Modeling Uncertainty, Journal of Uncertain Systems Vol.2, No.3, 202–211, 2008, .
3. "Probability, indeed, is degree of certainty, and differs from the latter as a part differs from the whole." in: Jacob Bernoulli, The Art of Conjecturing, translated by Edith Dudley Sykka, 2006, Johns Hopkins University Press, Baltimore, p. 315.
4. The International System of Units (SI), 8th edition, 2006, .

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