72,794 research outputs found
Bayesian definition of random sequences with respect to conditional probabilities
We study Martin-L\"{o}f random (ML-random) points on computable probability
measures on sample and parameter spaces (Bayes models). We consider four
variants of conditional random sequences with respect to the conditional
distributions: two of them are defined by ML-randomness on Bayes models and the
others are defined by blind tests for conditional distributions. We consider a
weak criterion for conditional ML-randomness and show that only variants of
ML-randomness on Bayes models satisfy the criterion. We show that these four
variants of conditional randomness are identical when the conditional
probability measure is computable and the posterior distribution converges
weakly to almost all parameters. We compare ML-randomness on Bayes models with
randomness for uniformly computable parametric models. It is known that two
computable probability measures are orthogonal if and only if their ML-random
sets are disjoint. We extend these results for uniformly computable parametric
models. Finally, we present an algorithmic solution to a classical problem in
Bayes statistics, i.e.~the posterior distributions converge weakly to almost
all parameters if and only if the posterior distributions converge weakly to
all ML-random parameters.Comment: revised versio
Universal Coding and Prediction on Martin-L\"of Random Points
We perform an effectivization of classical results concerning universal
coding and prediction for stationary ergodic processes over an arbitrary finite
alphabet. That is, we lift the well-known almost sure statements to statements
about Martin-L\"of random sequences. Most of this work is quite mechanical but,
by the way, we complete a result of Ryabko from 2008 by showing that each
universal probability measure in the sense of universal coding induces a
universal predictor in the prequential sense. Surprisingly, the effectivization
of this implication holds true provided the universal measure does not ascribe
too low conditional probabilities to individual symbols. As an example, we show
that the Prediction by Partial Matching (PPM) measure satisfies this
requirement. In the almost sure setting, the requirement is superfluous.Comment: 12 page
Limits of permutation sequences
A permutation sequence is said to be convergent if the density of occurrences
of every fixed permutation in the elements of the sequence converges. We prove
that such a convergent sequence has a natural limit object, namely a Lebesgue
measurable function with the additional properties that,
for every fixed , the restriction is a cumulative
distribution function and, for every , the restriction
satisfies a "mass" condition. This limit process is well-behaved:
every function in the class of limit objects is a limit of some permutation
sequence, and two of these functions are limits of the same sequence if and
only if they are equal almost everywhere. An ingredient in the proofs is a new
model of random permutations, which generalizes previous models and might be
interesting for its own sake.Comment: accepted for publication in the Journal of Combinatorial Theory,
Series B. arXiv admin note: text overlap with arXiv:1106.166
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