1,949 research outputs found
Algorithmic randomness and monotone complexity on product space
We study algorithmic randomness and monotone complexity on product of the set
of infinite binary sequences. We explore the following problems: monotone
complexity on product space, Lambalgen's theorem for correlated probability,
classification of random sets by likelihood ratio tests, decomposition of
complexity and independence, Bayesian statistics for individual random
sequences. Formerly Lambalgen's theorem for correlated probability is shown
under a uniform computability assumption in [H. Takahashi Inform. Comp. 2008].
In this paper we show the theorem without the assumption
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
Effective Capacity and Randomness of Closed Sets
We investigate the connection between measure and capacity for the space of
nonempty closed subsets of {0,1}*. For any computable measure, a computable
capacity T may be defined by letting T(Q) be the measure of the family of
closed sets which have nonempty intersection with Q. We prove an effective
version of Choquet's capacity theorem by showing that every computable capacity
may be obtained from a computable measure in this way. We establish conditions
that characterize when the capacity of a random closed set equals zero or is
>0. We construct for certain measures an effectively closed set with positive
capacity and with Lebesgue measure zero
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
Algorithmic Randomness and Capacity of Closed Sets
We investigate the connection between measure, capacity and algorithmic
randomness for the space of closed sets. For any computable measure m, a
computable capacity T may be defined by letting T(Q) be the measure of the
family of closed sets K which have nonempty intersection with Q. We prove an
effective version of Choquet's capacity theorem by showing that every
computable capacity may be obtained from a computable measure in this way. We
establish conditions on the measure m that characterize when the capacity of an
m-random closed set equals zero. This includes new results in classical
probability theory as well as results for algorithmic randomness. For certain
computable measures, we construct effectively closed sets with positive
capacity and with Lebesgue measure zero. We show that for computable measures,
a real q is upper semi-computable if and only if there is an effectively closed
set with capacity q
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