Prediction and dimension

Abstract

Given a set X of sequences over a finite alphabet, we investigate the following three quantities. (i) The feasible predictability of X is the highest success ratio that a polynomial-time randomized predictor can achieve on all sequences in X. (ii) The deterministic feasible predictability of X is the highest success ratio that a polynomial-time deterministic predictor can achieve on all sequences in X. (iii) The feasible dimension of X is the polynomial-time effectivization of the classical Hausdorff dimension ("fractal dimension") of X. Predictability is known to be stable in the sense that the feasible predictability of X∪Y is always the minimum of the feasible predictabilities of X and Y. We showt hat deterministic predictability also has this property if X and Y are computably presentable. We showtha t deterministic predictability coincides with predictability on singleton sets. Our main theorem states that the feasible dimension of X is bounded above by the maximum entropy of the predictability of X and bounded belowb y the segmented self-information of the predictability of X, and that these bounds are tight.