Prediction and Dimension

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 polynomialtime 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 show that deterministic predictability also has this property if X and Y are computably presentable. We show that 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 below by the segmented self-information of the predictability of X, and that these bounds are tight. 1 This author’s work was done while a senior research scientist at NEC Research Institute. 2 This author’s research was supported in part by National Science Foundation Grants 9988483 and 0344187. Much of the work was done while this author was on sabbatical at NEC Research Institute. Preprint submitted to Elsevier Science 3 October 2003 Fig. 1.1. Prediction-dimension diagrams for k = 2, 3, 4.