# The Speed Prior: A New Simplicity Measure Yielding Near-Optimal Computable Predictions

## Abstract

Solomonoff’s optimal but *non*computable 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 past *x*. Then we consider the case that the data actually stem from a *non*optimal, 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.

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