Physics is an inductive science. You try to find a law that matches the data, and hope that it generalizes on unseen data. When asked about the nature of their inductive business, most physicists refer to Popper's ideas from the 1930s (falsifiability etc), and sometimes to Occam's razor: simple explanations are better explanations. Most physicists, however, are not even aware of the fact that there is a formal basis for their inductive science, provided by the field of computational learning theory (COLT), in particular, the theory of universal induction. The contents of the following COLT paper may be old news to some on this list. ------------------------------------------------------------------ The Speed Prior: a new simplicity measure yielding near-optimal computable predictions (Juergen Schmidhuber, IDSIA)

In J. Kivinen and R. H. Sloan, eds, Proc. 15th Annual Conf. on Computational Learning Theory (COLT), 216-228, Springer, 2002; based on section 6 of http://arXiv.org/abs/quant-ph/0011122 (2000) http://www.idsia.ch/~juergen/speedprior.html ftp://ftp.idsia.ch/pub/juergen/colt.ps Solomonoff's optimal but noncomputable method for inductive inference assumes that observation sequences x are drawn from an recursive prior distribution mu(x). Instead of using the unknown mu(x) he predicts using the celebrated universal enumerable prior M(x) which for all x exceeds any recursive mu(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. Assuming our own universe is sampled from S, we predict: it won't get many times older than it is now; large scale quantum computation won't work well; beta decay is not random but due to some fast pseudo-random generator which we should try to discover. Juergen Schmidhuber http://www.idsia.ch/~juergen