Exact prediction and universal coding for trees

Abstract

-No closed formula for the optimal predictor (estimator) exists for the model of order preserving distributions on a finite tree. We present an algorithm of low complexity for a large class of trees.Let T = (T, ≤) be a finite rooted tree and consider the model P of all distributions over the nodes of T which respect the ordering (P ∈ P means P (a) ≤ P (b) whenever a ≤ bhence the root has minimal, the leaves maximal probability). For the unique optimal predictor P * ∈ P, sup P ∈P D(P P * ) is minimal among all distributions in P. It is our goal to determine P * exactly. Presently, results in this direction (exact rather than numerical or asymptotic results) are somewhat sporadic. The starting point of exact results of the type here considered is Ryabko [1]. The author, jointly with Peter Harremoës, has pointed to exact results for Bernoulli sources, cf. [3]. In [2] numerical methods are indicated, but based on the same theoretical reasoning as here (and with some associated exact results, not stated there). As a distant goal we mention the possibility to base asymptotically optimal prediction for, say Bernoulli sources on exact prediction results.Our concern here is a case-study: No side information, models defined via a tree structure on the basic alphabet, taken finite. Even so, the problem is complicated and to further simplify, we limit the discussion: For a sequence k = (k1, . . . , kn) of natural numbers, T = T [k] denotes the rooted tree with k1 branches emerging from the root, with k2 branches emerging from each node in level 1 etc. until kn nodes emerging from each node in level n − 1. Trees of this type have uniform branching and k is the branching pattern. . For each of them, the optimal predictor is indicated by showing the weights assigned to each node. The last example is a bit surprising in that the root and its immediate successor are assigned the same weight, hence the same probability. The marked nodes show the spectrum σ(T ) of the trees. σ(T ) consists of the active anchors, nodes a for which the uniform distribution over {b|b ≥ a} contributes to the optimal predictor (i.e., the uniform distribution has positive weight in the

Computation of Universal Objects for Distributions Over Co-Trees

“…It is due to Ryabko who developed a closed formula for the universal predictor, cf., [4]. For the larger subclass of co-trees with uniform branching, an algorithm was announced in Topsøe [5] but the details were never published.…”

“…The transitive nature of the lemma is best revealed by generalizing the result. 5 With reference to notions from Section IV, a blocking set for with maximal bracket is set-theoretically maximal (minimal) if it is not a proper subset (superset) of some other blocking set for with maximal bracket. theoretically largest such set and the set-theoretically smallest such set.…”