Title: Bayesian definition of random sequences with respect to conditional probabilities.
Author: Hayato Takahashi
Abstract: We study Martin-Löf random (ML-random) points on computable probability measures
on sample and parameter spaces(Bayes models). We consider variants of conditional randomness
defined by ML-randomness on Bayes models and those of conditional blind randomness.
We show that variants of conditional blind randomness are ill-defined from the Bayes statistical point of view.
We prove that if the sets of random sequences of uniformly computable parametric models are pairwise disjoint
then there is a consistent estimator for the model.
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.
Title: Computational limits to nonparametric estimation for ergodic processes.
Author: Hayato Takahashi
Abstract: A new negative result for nonparametric estimation of binary ergodic processes is shown.
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.
Title: Universal parameterized family of distributions of runs
Author: Hayato Takahashi
Abstract: We present explicit formulae for parameterized families of probabilities of the number of nonoverlapping words
and increasing nonoverlapping words
in independent and identically distributed (i.i.d.) finite valued random variables, respectively.
Then we provide an explicit formula for a parameterized family of probabilities of the number of runs, which
generalizes \(\mu\)-overlapping probabilities for \(\mu\geq 0\) in i.i.d.~binary valued random variables.
We also demonstrate exact probabilities of the number of runs whose size are exactly given numbers (Mood 1940).
The number of arithmetic operations required to compute our formula for generalized probabilities of runs
is linear order of sample size for fixed number of parameters and range.
To analyse these number of arithmetic operations for unbounded number of parameters,
we show an asymptotic formula for
the number of integer partitions that are less than or equal to given number as a special case of Meinardus's theorem.