The semimeasure property of algorithmic probability - "Feature" or "bug"?

Abstract

An unknown process is generating a sequence of symbols, drawn from an alphabet,. Given an initial segment of the sequence, how can one predict the next symbol? Ray Solomonoff's theory of inductive reasoning rests on the idea that a useful estimate of a sequence's true probability of being outputted by the unknown process is provided by its algorithmic probability (its probability of being outputted by a species of probabilistic Turing machine). However algorithmic probability is a "semimeasure": i.e., the sum, over all, of the conditional algorithmic probabilities of the next symbol being x, may be less than 1. Solomonoff thought that algorithmic probability must be normalized, to eradicate this semimeasure property, before it can yield acceptable probability estimates. This paper argues, to the contrary, that the semimeasure property contributes substantially, in its own right, to the power of an algorithmic-probability-based theory of induction, and that normalization is unnecessary. © 2013 Springer-Verlag Berlin Heidelberg.