47 research outputs found
On Generalized Computable Universal Priors and their Convergence
Solomonoff unified Occam's razor and Epicurus' principle of multiple
explanations to one elegant, formal, universal theory of inductive inference,
which initiated the field of algorithmic information theory. His central result
is that the posterior of the universal semimeasure M converges rapidly to the
true sequence generating posterior mu, if the latter is computable. Hence, M is
eligible as a universal predictor in case of unknown mu. The first part of the
paper investigates the existence and convergence of computable universal
(semi)measures for a hierarchy of computability classes: recursive, estimable,
enumerable, and approximable. For instance, M is known to be enumerable, but
not estimable, and to dominate all enumerable semimeasures. We present proofs
for discrete and continuous semimeasures. The second part investigates more
closely the types of convergence, possibly implied by universality: in
difference and in ratio, with probability 1, in mean sum, and for Martin-Loef
random sequences. We introduce a generalized concept of randomness for
individual sequences and use it to exhibit difficulties regarding these issues.
In particular, we show that convergence fails (holds) on generalized-random
sequences in gappy (dense) Bernoulli classes.Comment: 22 page
Sequential Predictions based on Algorithmic Complexity
This paper studies sequence prediction based on the monotone Kolmogorov
complexity Km=-log m, i.e. based on universal deterministic/one-part MDL. m is
extremely close to Solomonoff's universal prior M, the latter being an
excellent predictor in deterministic as well as probabilistic environments,
where performance is measured in terms of convergence of posteriors or losses.
Despite this closeness to M, it is difficult to assess the prediction quality
of m, since little is known about the closeness of their posteriors, which are
the important quantities for prediction. We show that for deterministic
computable environments, the "posterior" and losses of m converge, but rapid
convergence could only be shown on-sequence; the off-sequence convergence can
be slow. In probabilistic environments, neither the posterior nor the losses
converge, in general.Comment: 26 pages, LaTe
On Martin-Löf convergence of Solomonoff’s mixture
We study the convergence of Solomonoff’s universal mixture on individual Martin-Löf random sequences. A new result is presented extending the work of Hutter and Muchnik (2004) by showing that there does not exist a universal mixture that converges on all Martin-Löf random sequences
Algorithmic Complexity Bounds on Future Prediction Errors
We bound the future loss when predicting any (computably) stochastic sequence
online. Solomonoff finitely bounded the total deviation of his universal
predictor from the true distribution by the algorithmic complexity of
. Here we assume we are at a time and already observed .
We bound the future prediction performance on by a new
variant of algorithmic complexity of given , plus the complexity of the
randomness deficiency of . The new complexity is monotone in its condition
in the sense that this complexity can only decrease if the condition is
prolonged. We also briefly discuss potential generalizations to Bayesian model
classes and to classification problems.Comment: 21 page
On Universal Prediction and Bayesian Confirmation
The Bayesian framework is a well-studied and successful framework for
inductive reasoning, which includes hypothesis testing and confirmation,
parameter estimation, sequence prediction, classification, and regression. But
standard statistical guidelines for choosing the model class and prior are not
always available or fail, in particular in complex situations. Solomonoff
completed the Bayesian framework by providing a rigorous, unique, formal, and
universal choice for the model class and the prior. We discuss in breadth how
and in which sense universal (non-i.i.d.) sequence prediction solves various
(philosophical) problems of traditional Bayesian sequence prediction. We show
that Solomonoff's model possesses many desirable properties: Strong total and
weak instantaneous bounds, and in contrast to most classical continuous prior
densities has no zero p(oste)rior problem, i.e. can confirm universal
hypotheses, is reparametrization and regrouping invariant, and avoids the
old-evidence and updating problem. It even performs well (actually better) in
non-computable environments.Comment: 24 page
Towards a Universal Theory of Artificial Intelligence based on Algorithmic Probability and Sequential Decision Theory
Decision theory formally solves the problem of rational agents in uncertain
worlds if the true environmental probability distribution is known.
Solomonoff's theory of universal induction formally solves the problem of
sequence prediction for unknown distribution. We unify both theories and give
strong arguments that the resulting universal AIXI model behaves optimal in any
computable environment. The major drawback of the AIXI model is that it is
uncomputable. To overcome this problem, we construct a modified algorithm
AIXI^tl, which is still superior to any other time t and space l bounded agent.
The computation time of AIXI^tl is of the order t x 2^l.Comment: 8 two-column pages, latex2e, 1 figure, submitted to ijca
Uniform test of algorithmic randomness over a general space
The algorithmic theory of randomness is well developed when the underlying
space is the set of finite or infinite sequences and the underlying probability
distribution is the uniform distribution or a computable distribution. These
restrictions seem artificial. Some progress has been made to extend the theory
to arbitrary Bernoulli distributions (by Martin-Loef), and to arbitrary
distributions (by Levin). We recall the main ideas and problems of Levin's
theory, and report further progress in the same framework.
- We allow non-compact spaces (like the space of continuous functions,
underlying the Brownian motion).
- The uniform test (deficiency of randomness) d_P(x) (depending both on the
outcome x and the measure P should be defined in a general and natural way.
- We see which of the old results survive: existence of universal tests,
conservation of randomness, expression of tests in terms of description
complexity, existence of a universal measure, expression of mutual information
as "deficiency of independence.
- The negative of the new randomness test is shown to be a generalization of
complexity in continuous spaces; we show that the addition theorem survives.
The paper's main contribution is introducing an appropriate framework for
studying these questions and related ones (like statistics for a general family
of distributions).Comment: 40 pages. Journal reference and a slight correction in the proof of
Theorem 7 adde