474 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
Computational limits to nonparametric estimation for ergodic processes
A new negative result for nonparametric estimation of binary ergodic
processes is shown. I The problem of estimation of distribution with any degree
of accuracy is studied. Then it is shown that for any countable class of
estimators there is a zero-entropy binary ergodic process that is inconsistent
with the class of estimators. Our result is different from other negative
results for universal forecasting scheme of ergodic processes.Comment: submitted to IEEE trans I
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
Algorithmic randomness and stochastic selection function
We show algorithmic randomness versions of the two classical theorems on
subsequences of normal numbers. One is Kamae-Weiss theorem (Kamae 1973) on
normal numbers, which characterize the selection function that preserves normal
numbers. Another one is the Steinhaus (1922) theorem on normal numbers, which
characterize the normality from their subsequences. In van Lambalgen (1987), an
algorithmic analogy to Kamae-Weiss theorem is conjectured in terms of
algorithmic randomness and complexity. In this paper we consider two types of
algorithmic random sequence; one is ML-random sequences and the other one is
the set of sequences that have maximal complexity rate. Then we show
algorithmic randomness versions of corresponding theorems to the above
classical results.Comment: submitted to CCR2012 special issue. arXiv admin note: text overlap
with arXiv:1106.315
Some explicit formulae for the distributions of words (Probability Symposium)
Parts of the paper have been presented in [23, 24]
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