Recently a policy, DMED, is proposed and proved to achieve the asymptotic
bound for the model that each reward distribution is supported in a known
bounded interval, e.g. [0,1]. However, the derived regret bound is described in
an asymptotic form and the performance in finite time has been unknown. We
inspect this policy and derive a finite-time regret bound by refining large
deviation probabilities to a simple finite form.

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.

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.

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.

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 new algorithm to find local maximum and minimum of a holonomic
function and apply it for the Fisher-Bingham integral on the sphere $S^n$,
which is used in the directional statistics. The method utilizes the theory and
algorithms of holonomic systems.

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.

We give an expository review of applications of computational algebraic
statistics to design and analysis of fractional factorial experiments based on
our recent works. For the purpose of design, the techniques of Gr\"obner bases
and indicator functions allow us to treat fractional factorial designs without
distinction between regular designs and non-regular designs. For the purpose of
analysis of data from fractional factorial designs, the techniques of Markov
bases allow us to handle discrete observations.

We propose procedures for testing whether stock price processes are
martingales based on limit order type betting strategies. We first show that
the null hypothesis of martingale property of a stock price process can be
tested based on the capital process of a betting strategy. In particular with
high frequency Markov type strategies we find that martingale null hypotheses
are rejected for many stock price processes.

We propose minimum empirical divergence (MED) policy for the multiarmed
bandit problem. We prove asymptotic optimality of the proposed policy for the
case of finite support models. In our setting, Burnetas and Katehakis has
already proposed an asymptotically optimal policy. For choosing an arm our
policy uses a criterion which is dual to the quantity used in Burnetas and
Katehakis. Our criterion is easily computed by a convex optimization technique
and has an advantage in practical implementation.

In this paper we propose an investing strategy based on neural network models
combined with ideas from game-theoretic probability of Shafer and Vovk. Our
proposed strategy uses parameter values of a neural network with the best
performance until the previous round (trading day) for deciding the investment
in the current round. We compare performance of our proposed strategy with
various strategies including a strategy based on supervised neural network
models and show that our procedure is competitive with other strategies.

We propose a sequential optimizing betting strategy in the multi-dimensional
bounded forecasting game in the framework of game-theoretic probability of
Shafer and Vovk (2001). By studying the asymptotic behavior of its capital
process, we prove a generalization of the strong law of large numbers, where
the convergence rate of the sample mean vector depends on the growth rate of
the quadratic variation process. The growth rate of the quadratic variation
process may be slower than the number of rounds or may even be zero.

We discuss connecting tables with zero-one entries by a subset of a Markov
basis. In this paper, as a Markov basis we consider the Graver basis, which
corresponds to the unique minimal Markov basis for the Lawrence lifting of the
original configuration. Since the Graver basis tends to be large, it is of
interest to clarify conditions such that a subset of the Graver basis, in
particular a minimal Markov basis itself, connects tables with zero-one
entries. We give some theoretical results on the connectivity of tables with
zero-one entries.