arXiv subjects

Hayato Takahashi

Publications and source records attributed to Hayato Takahashi.

14 recordsLinked to original sources

arXiv2025

Universal parameterized family of distributions of runs

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 \(μ\)-overlapping probabilities for \(μ\geq 0\) 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.

math.PR↗

arXiv2023

Bayesian definition of random sequences with respect to conditional probabilities

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.

cs.IT↗

arXiv2022

The explicit formulae for the distributions of nonoverlapping words and its applications to statistical tests for pseudo random numbers

The distributions of the number of occurrences of words (the distributions of words for short) play key roles in information theory, statistics, probability theory, ergodic theory, computer science, and DNA analysis. Bassino et al. 2010 and Regnier et al. 1998 showed generating functions of the distributions of words for all sample sizes. Robin et al. 1999 presented generating functions of the distributions for the return time of words and demonstrated a recurrence formula for these distributions. These generating functions are rational functions; except for simple cases, it is difficult to expand them into power series. In this paper, we study finite-dimensional generating functions of the distributions of nonoverlapping words for each fixed sample size and demonstrate the explicit formulae for the distributions of words for the Bernoulli models. Our results are generalized to nonoverlapping partial words. We study statistical tests that depend on the number of occurrences of words and the number of block-wise occurrences of words, respectively. We demonstrate that the power of the test that depends on the number of occurrences of words is significantly large compared to the other one. Finally, we apply our results to statistical tests for pseudo random numbers.

cs.IT↗

arXiv2019

The distributions of sliding block patterns in finite samples and the inclusion-exclusion principles for partially ordered sets

In this paper we show the distributions of sliding block patterns for Bernoulli processes with finite alphabet, which is not based on the induction on sample size. We show a new inclusion-exclusion formula in multivariate generating function form on partially ordered sets, and show a simpler expression of generating functions of the number of pattern occurrences in finite samples. We show higher moments of the sliding block patterns and power of tests based on sliding block patterns.

cs.IT↗

arXiv2017

Recent progress on conditional randomness

In this article, recent progress on ML-randomness with respect to conditional probabilities is reviewed. In particular a new result of conditional randomness with respect to mutually singular probabilities are shown, which is a generalization of Hanssen's result (2010) for Bernoulli processes.

math.LO↗

arXiv2016

Conditional probabilities and van Lambalgen theorem revisited

The definition of conditional probability in case of continuous distributions was an important step in the development of mathematical theory of probabilities. How can we define this notion in algorithmic probability theory? In this survey we discuss the developments in this direction trying to explain what are the difficulties and what can be done to avoid them. Almost all the results discussed in this paper have been published (and we provide the references), but we tried to put them into perspective and to explain the proofs in a more intuitive way. We assume that the reader is familiar with basic notions of measure theory and algorithmic randomness.

math.LO↗

arXiv2015

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.

cs.IT↗

arXiv2015

Generalization of van Lambalgen's theorem and blind randomness for conditional probabilities

Generalization of the Lambalgen's theorem is studied with the notion of Hippocratic (blind) randomness without assuming computability of conditional probabilities. In [Bauwence 2014], a counter-example for the generalization of Lambalgen's theorem is shown when the conditional probability is not computable. In this paper, it is shown that (i) finiteness of martingale for blind randomness, (ii) classification of two blind randomness by likelihood ratio test, (iii) sufficient conditions for the generalization of the Lambalgen's theorem, and (iv) an example that satisfies the Lambalgen's theorem but the conditional probabilities are not computable for all random parameters.

math.LO↗

arXiv2014

Universal test for Hippocratic randomness

Hippocratic randomness is defined in a similar way to Martin-Lof randomness, however it does not assume computability of the probability and the existence of universal test is not assured. We introduce the notion of approximation of probability and show the existence of the universal test (Levin-Schnorr theorem) for Hippocratic randomness when the logarithm of the probability is approximated within additive constant.

cs.IT↗

arXiv2012

Algorithmic analogies to kamae-Weiss theorem on normal numbers

In this paper we study subsequences of random numbers. In Kamae (1973), selection functions that depend only on coordinates are studied, and their necessary and sufficient condition for the selected sequences to be normal numbers is given. In van Lambalgen (1987), an algorithmic analogy to the theorem is conjectured in terms of algorithmic randomness and Kolmogorov complexity. In this paper, we show different algorithmic analogies to the theorem.

cs.IT↗

arXiv2011

Some limits to nonparametric estimation for ergodic processes

A new negative result for nonparametric distribution 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. We also introduce a related result by B. Weiss.

cs.IT↗

arXiv2011

Weak randomness and Kamae's theorem on normal numbers

A function from sequences to their subsequences is called selection function. A selection function is called admissible (with respect to normal numbers) if for all normal numbers, their subsequences obtained by the selection function are normal numbers. In Kamae (1973) selection functions that are not depend on sequences (depend only on coordinates) are studied, and their necessary and sufficient condition for admissibility is given. In this paper we introduce a notion of weak randomness and study an algorithmic analogy to the Kamae's theorem.

cs.IT↗

arXiv2011

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.

cs.IT↗

arXiv2010

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.

cs.IT↗