The Blackwell Prediction Algorithm for Infinite 0-1 Sequences, and a Generalization

Abstract

Let x1,x2,... be a (not necessarily random) infinite 0–1 sequence. We wish to sequentially predict the sequence. This means that, for each n ≥ 1, we will guess the value of x _n+1, basing our guess on knowledge of x1,x2,..., x _n. Of interest are algorithms which predict well for all 0–1 sequences. An example is the Blackwell algorithm discussed in Sect. 1. In Sect. 2 we introduce a generalization of Blackwell’s algorithm to the case of three categories. This three-category algorithm will be explained using a ge-ometric model (the so-called prediction prism), and it will be shown to be a natural generalization of Blackwell’s two-category algorithm.

The Blackwell algorithm has interesting properties. It predicts arbituary 0–1 sequences as well or better than independent, identically distributed Bernoulli variables, for which it is optimal. Such Bernoulli variables are consequently the hardest to predict. Similar results hold for the three-category generalization of Blackwell’s algorithm.

We thank T. Sellke for translating and helping to revise this paper.