Stochastic prediction procedure

Mathematical formulation

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

The mathematical task consists of deriving a function A X ( β ) which meets the above formulated two requirements.

Stochastic predictions

The predictions made by means of A X ( β ) refer to the random variable X. The random variable X has a certain range of variability denoted by X implying that any meaningful prediction is a subset of 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 β 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.

Predictions in science

The characteristic issues of a stochastic prediction procedures shall be illustrated by a comparison with a prediction made and published by the University Corporation for Atmospheric Research (UCAR) which is described in as follows:

The involved scientists of the University Corporation for Atmospheric Research (UCAR) reported on March 6, 2006, about their new prediction method:

The first striking difference refers to the use of the word "accuracy". In case of a stochastic prediction procedure it refers to the size of the predicted event. The accuracy is largest, if the predicted event is a singleton and thus has the size 1. The larger the size is, the worse the accuracy becomes. If the size of the predicted event is too large, then the prediction is useless.

In the case of the UCAR predictions, an accuracy of more than 98% is claimed for the comparison with the past eight solar cycles. The meaning of this statement for the future cycles remains unclear. It resembles in some sense a statement about the reliability, i.e., the probability that the predicted event will actually occur. But this interpretation is wrong, since the statement refers to the past and not to the future. Maybe, it constitutes a relative frequency, however, in this case the corresponding event is unclear. It is always possible to develop a model that matches the past, but this does not mean that the probability of predicted future events can be assessed. Moreover, it appears that the UCAR predictions are given by singletons. If this is really the case then the accuracy would be highest, however the reliability would be equal to zero or at least close to zero. This is by the way the usual case in physics, which yields predictions with highest accuracy which, however, almost never occur.