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TR02-011 | 14th October 2001 00:00
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#### The nonprobabilistic approach to learning the best prediction.

**Abstract:**
The problem of predicting a sequence $x_1, x_2,.... $ where each $x_i$ belongs

to a finite alphabet $A$ is considered. Each letter $x_{t+1}$ is predicted

using information on the word $x_1, x_2, ...., x_t $ only. We use the game

theoretical interpretation which can be traced to Laplace where there exists a gambler who

tries to estimate probabilities for the letter $x_{t+1}$ in order to maximize

his capital . The optimal method of prediction is described for the case when

the sequence $x_1, x_2,.... $ is generated by a stationary and ergodic source. It turns out

that the optimal method is based only on estimations of conditional probabilities.

In particular, it means that if we work in the framework of the ergodic and

stationary source model, we cannot consider pattern recognition and other complex

and interesting tools, because even the optimal method does not need them.

That is why we suggest a so-called nonprobabilistic

approach which is not based on the stationary and ergodic source model and

show that complex algorithms of prediction can be considered in the framework

of this approach.

The new approach is to consider a set of all infinite

sequences (over a given finite alphabet) and estimate the size of sets of

predictable sequences with the help of the Hausdorff dimension.

This approach enables us first, to show that there exist large sets of well

predictable sequences which have zero measure for each stationary and ergodic

measure. (In fact, it means that such sets are invisible in the framework of

the ergodic and stationary source model and shows the necessity of the new approach.)

Second, it is shown that there exist quite large sets of such sequences

that can be predicted well by complex algorithms which use not only estimations of

conditional probabilities.