Abstract
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 past 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.
This paper is based on section 6 of TR IDSIA-20-00, Version 2.0: http://www.idsia.ch/~juergen/toesv2/ http://arXiv.org/abs/quant-ph/0011122 (public physics archive)
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Schmidhuber, J. (2002). The Speed Prior: A New Simplicity Measure Yielding Near-Optimal Computable Predictions. In: Kivinen, J., Sloan, R.H. (eds) Computational Learning Theory. COLT 2002. Lecture Notes in Computer Science(), vol 2375. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-45435-7_15
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DOI: https://doi.org/10.1007/3-540-45435-7_15
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