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.
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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
Received on Fri Aug 02 2002 - 10:14:28 PDT