Stochastic prediction procedure

In probability theory and statistics, a stochastic prediction procedure is based on a Bernoulli space and may be used to make predictions under specific conditions. In contrast to a prediction obtained in traditional science, predictions obtained by means of a stochastic predíction procedure meet a given reliability requirement and are optimal with respect to accuracy. A prediction procedure refers to a random variable X and predicts future events for X depending on the initial conditions or more precisely said on what is known about the initial conditions.

Mathematical formulation

A stochastic prediction procedure is a mathematical function denoted $A_{X}^{(\beta )}$ defined on sets representing the possible initial conditions and having images that are the predictions. The function $A_{X}^{(\beta )}$ is derived meeting the following two requirements:

• The stochastic prediction procedure $A_{X}^{(\beta )}$ shall yield predictions which will occur with a probability of at least $\beta$ where $\beta$ is called the reliability level of $A_{X}^{(\beta )}$.
• The stochastic prediction procedure $A_{X}^{(\beta )}$ shall yield predictions with optimal accuracy, where accuracy is defined by the size of the prediction.

The mathematical task consists of deriving a function $A_{X}^{(\beta )}$ which meets the above formulated two requirements.

Stochastic predictions

The predictions made by means of $A_{X}^{(\beta )}$ refer to the random variable X. The random variable X has a certain range of variability denoted by ${\mathfrak {X}}$ implying that any meaningful prediction is a subset of ${\mathfrak {X}}$. The future development is uncertain and uncertainty is generated by ignorance and randomness. In order to meet the reliability requirement specified by the reliability level $\beta$ it is necessary to consider both. The two sources of uncertainty are explicitly contained in the corresponding Bernoulli Space.

The derivation of a stochastic prediction procedure is therefore based on a Bernoulli Space which represents a stochastic model of the change from past to future taking into account the characteristic human ignorance about the initial conditions and the inherent randomness of the evolution of universe. Thus, it becomes possible to derive predictions procedures which yield reliable predictions with optimal accuracy. Note that because of randomness it is in principle impossible to predict the indeterminate future outcome and at the same time meet a specified reliability requirement. This is impossible, even in the rare case that the initial conditions are known.

References

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