Solomonoff’s optimal but noncomputable method for inductive inference assumes that observation sequences x are drawn from an recursive prior distribution μ(x). Instead of using the unknown μ(x) he predicts using the celebrated universal enumerable prior M(x) which for all x exceeds any recursive μ(x), save for a constant factor independent of x. The simplicity measure M(x) naturally implements “Occam’s razor” and is closely related to the Kolmogorov complexity of x. However, M assigns high probability to certain data x that are extremely hard to compute. This does not match our intuitive notion of simplicity. Here we suggest a more plausible measure derived from the fastest way of computing data. In absence of contrarian evidence, we assume that the physical world is generated by a computational process, and that any possibly infinite sequence of observations is therefore computable in the limit (this assumption is more radical and stronger than Solomonoff’s). Then we replace M by the novel Speed Prior S, under which the cumulative a priori probability of all data whose computation through an optimal algorithm requires more than O(n) resources is 1/n. We show that the Speed Prior allows for deriving a computable strategy for optimal prediction of future y, given p ast x. Then we consider the case that the data actually stem from a nonoptimal, unknown computational process, and use Hutter’s recent results to derive excellent expected loss bounds for S-based inductive inference. We conclude with several nontraditional predictions concerning the future of our universe.
|Title of host publication
|Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)
|Number of pages
|Published - Jan 1 2002