arXiv subjects
Publications and source records attributed to Akimichi Takemura.
Markov basis for statistical model of contingency tables gives a useful tool for performing the conditional test of the model via Markov chain Monte Carlo method. In this paper we derive explicit forms of Markov bases for change point models and block diagonal effect models, which are typical block-wise effect models of two-way contingency tables, and perform conditional tests with some real data sets.
In this paper we give an explicit and algorithmic description of Graver basis for the toric ideal associated with a simple undirected graph and apply the basis for testing the beta model of random graphs by Markov chain Monte Carlo method.
We give a unified treatment of the convergence of random series and the rate of convergence of strong law of large numbers in the framework of game-theoretic probability of Shafer and Vovk (2001). We consider games with the quadratic hedge as well as more general weaker hedges. The latter corresponds to existence of an absolute moment of order smaller than two in the measure-theoretic framework. We prove some precise relations between the convergence of centered random series and the convergence of the series of prices of the hedges. When interpreted in measure-theoretic framework, these results characterize convergence of a martingale in terms of convergence of the series of conditional absolute moments. In order to prove these results we derive some fundamental results on deterministic strategies of Reality, who is a player in a protocol of game-theoretic probability. It is of particular interest, since Reality's strategies do not have any counterparts in measure-theoretic framework, ant yet they can be used to prove results, which can be interpreted in measure-theoretic framework.
Let B be a real hyperplane arrangement which is stable under the action of a Coxeter group W. Then B acts naturally on the set of chambers of B. We assume that B is disjoint from the Coxeter arrangement A=A(W) of W. In this paper, we show that the W-orbits of the set of chambers of B are in one-to-one correspondence with the chambers of C=A\cup B which are contained in an arbitrarily fixed chamber of A. From this fact, we find that the number of W-orbits of the set of chambers of B is given by the number of chambers of C divided by the order of W. We will also study the set of chambers of C which are contained in a chamber b of B. We prove that the cardinality of this set is equal to the order of the isotropy subgroup W_b of b. We illustrate these results with some examples, and solve an open problem in Kamiya, Takemura and Terao [Ranking patterns of unfolding models of codimension one, Adv. in Appl. Math. (2010)] by using our results.
In this article we give a computational study of combinatorics of the discriminantal arrangements. The discriminantal arrangements are parametrized by two positive integers n and k such that n>k. The intersection lattice of the discriminantal arrangement with the parameter (n,k) is isomorphic to the intersection lattice of the hyperplane arrangement generated by n generic points in the d-dimensional vector space where d=n-k-1. The combinatorics of the discriminantal arrangements is very hard, except for the special cases of the Boolean arrangements (k=0) and the braid arrangements (k=1). We review some results on the intersection lattices of the arrangements generated by generic points and use them to obtain some computational results on the characteristic polynomials of the discriminantal arrangements.
In this paper we give a review of the method of imsets introduced by Studeny (2005) from a geometric point of view. Elementary imsets span a polyhedral cone and its dual cone is the cone of supermodular functions. We review basic facts on the structure of these cones. Then we derive some new results on the following topics: i) extreme rays of the cone of standardized supermodular functions, ii) faces of the cones, iii) small relations among elementary imsets, and iv) some computational results on Markov basis for the toric ideal defined by elementary imsets.
Arrangement theory plays an essential role in the study of the unfolding model used in many fields. This paper describes how arrangement theory can be usefully employed in solving the problems of counting (i) the number of admissible rankings in an unfolding model and (ii) the number of ranking patterns generated by unfolding models. The paper is mostly expository but also contains some new results such as simple upper and lower bounds for the number of ranking patterns in the unidimensional case.
The methodology of Markov basis initiated by Diaconis and Sturmfels(1998) stimulated active research on Markov bases for more than ten years. It also motivated improvements of algorithms for Grobner basis computation for toric ideals, such as those implemented in 4ti2. However at present explicit forms of Markov bases are known only for some relatively simple models, such as the decomposable models of contingency tables. Furthermore general algorithms for Markov bases computation often fail to produce Markov bases even for moderate-sized models in a practical amount of time. Hence so far we could not perform exact tests based on Markov basis methodology for many important practical problems. In this article we propose to use lattice bases for performing exact tests, in the case where Markov bases are not known. Computation of lattice bases is much easier than that of Markov bases. With many examples we show that the approach with lattice bases is practical. We also check that its performance is comparable to Markov bases for the problems where Markov bases are known.
We give an exposition and numerical studies of upper hedging prices in multinomial models from the viewpoint of linear programming and the game-theoretic probability of Shafer and Vovk. We also show that, as the number of rounds goes to infinity, the upper hedging price of a European option converges to the solution of the Black-Scholes-Barenblatt equation.
For statistical analysis of multiway contingency tables we propose modeling interaction terms in each maximal compact component of a hierarchical model. By this approach we can search for parsimonious models with smaller degrees of freedom than the usual hierarchical model, while preserving conditional independence structures in the hierarchical model. We discuss estimation and exacts tests of the proposed model and illustrate the advantage of the proposed modeling with some data sets.
We prove a game-theoretic version of Levy's zero-one law, and deduce several corollaries from it, including non-stochastic versions of Kolmogorov's zero-one law, the ergodicity of Bernoulli shifts, and a zero-one law for dependent trials. Our secondary goal is to explore the basic definitions of game-theoretic probability theory, with Levy's zero-one law serving a useful role.
We study properties of semi-elementary imsets and elementary imsets introduced by Studeny (2005). The rules of the semi-graphoid axiom (decomposition, weak union and contraction) for conditional independence statements can be translated into a simple identity among three semi-elementary imsets. By recursively applying the identity, any semi-elementary imset can be written as a sum of elementary imsets, which we call a representation of the semi-elementary imset. A semi-elementary imset has many representations. We study properties of the set of possible representations of a semi-elementary imset and prove that all representations are connected by relations among four elementary imsets.
In the multiarmed bandit problem a gambler chooses an arm of a slot machine to pull considering a tradeoff between exploration and exploitation. We study the stochastic bandit problem where each arm has a reward distribution supported in a known bounded interval, e.g. [0,1]. For this model, policies which take into account the empirical variances (i.e. second moments) of the arms are known to perform effectively. In this paper, we generalize this idea and we propose a policy which exploits the first d empirical moments for arbitrary d fixed in advance. The asymptotic upper bound of the regret of the policy approaches the theoretical bound by Burnetas and Katehakis as d increases. By choosing appropriate d, the proposed policy realizes a tradeoff between the computational complexity and the expected regret.
We give an exponential lower bound for the Graver complexity of the incidence matrix of a complete bipartite graph of arbitrary size. Our result is a generalization of the result by Berstein and Onn (2009) for 3xr complete bipartite graphs, r \ge 3.
We present a geometrical method for analyzing sequential estimating procedures. It is based on the design principle of the second-order efficient sequential estimation provided in Okamoto, Amari and Takeuchi (1991). By introducing a dual conformal curvature quantity, we clarify the conditions for the covariance minimization of sequential estimators. These conditions are further elabolated for the multidimensional curved exponential family. The theoretical results are then numerically examined by using typical statistical models, von Mises-Fisher and hyperboloid models.
An admissible estimator of the eigenvalues of the variance-covariance matrix is given for multivariate normal distributions with respect to the scale-invariant squared error loss.
We consider hyperplane arrangements generated by generic points and study their intersection lattices. These arrangements are known to be equivalent to discriminantal arrangements. We show a fundamental structure of the intersection lattices by decomposing the poset ideals as direct products of smaller lattices corresponding to smaller dimensions. Based on this decomposition we compute the Möbius functions of the lattices and the characteristic polynomials of the arrangements up to dimension six.
We derive a Markov basis consisting of moves of degree at most three for two-state toric homogeneous Markov chain model of arbitrary length without parameters for initial states. Our basis consists of moves of degree three and degree one, which alter the initial frequencies, in addition to moves of degree two and degree one for toric homogeneous Markov chain model with parameters for initial states.