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A diffusion taking value in probability measures on a graph with a vertex set $V$, $\sum_{i\in V}x_i\delta_i$, is studied. The masses on each vertices satisfy the stochastic differential equation of the form $dx_i=\sum_{j\in N(i)}\sqrt{x_ix_j}dB_{ij}$ on the simplex, where $\{B_{ij}\}$ are independent standard Brownian motions with skew symmetry and $N(i)$ is the neighbour of the vertex $i$. A dual Markov chain on integer partitions to the Markov semigroup associated with the diffusion is used to show that the support of an extremal stationary state of the adjoint semigroup is an independent set of the graph. We also investigate the diffusion with a linear drift, which gives a killing of the dual Markov chain on a finite integer lattice. The Markov chain is used to study the unique stationary state of the diffusion, which generalizes the Dirichlet distribution. Two applications of the diffusions are discussed: analysis of an algorithm to find an independent set of a graph, and a Bayesian graph selection based on computation of probability of a sample by using coupling from the past., Comment: 30 pages, 3 figures

We show that Pfaffians or contiguity relations of hypergeometric functions of several variables give a direct sampling algorithm from toric models in statistics, which is a Markov chain on a lattice generated by a matrix $A$. A correspondence among graphical models and $A$-hypergeometric system is discussed and we give a sum formula of special values of $A$-hypergeometric polynomials. Some hypergeometric series which are interesting in view of statistics are presented., Comment: 40 pages, 2 figures

The Ewens sampling formula is a distribution related to the random partition of a positive integer. In this study, we investigate the issue of non-existence solutions in parameter estimation under the distribution. As a result, the first and second moments matching estimators to the uniformly minimum variance unbiased estimator are derived using the Ewens sampling formula in asymptotic sense. A Monte Carlo simulation study is performed to evaluate the efficiency of the resulting estimators., Comment: 25 pages, 3 figures

A procedure for asymptotic bias reduction of maximum likelihood estimates of generic estimands is developed. The estimator is realized as a plug-in estimator, where the parameter maximizes the penalized likelihood with a penalty function that satisfies a quasi-linear partial differential equation of the first order. The integration of the partial differential equation with the aid of differential geometry is discussed. Applications to generalized linear models, linear mixed-effects models, and a location-scale family are presented., Comment: 49 pages, 4 figures

In this paper, we present a consistent procedure to assess the significance of gravitational wave events observed by laser interferometric gravitational wave detectors based on the background distribution of detection statistic. We propose a non-parametric method to estimate $p$-value. Based on the estimated $p$-values, we propose a new procedure to assess the significance of a particular event with $q$-value which is the minimum false discovery rate that can be attained when calling the event significant. The $q$-value gives us a criterion on the significance of events which is different from $P_{\rm astro}$ which is used in the LIGO-Virgo analysis and in other analysis. The proposed procedure is applied to the 1-OGC and 2-OGC catalogs [2][3]. For most of the events which were claimed significant in [2] and [3], we also obtain the same results. However, there are differences in the significance for several marginal events. Since the proposed procedure does not require any assumptions on signal and noise, it is very simple and straightforward. The procedure is also applicable to other searches for gravitational waves whose background distribution of detection statistic is difficult to know., Comment: Accepted by PTEP, 18 pages, 8 figure

Consider two forms of sampling from a population: (i) drawing $s$ samples of $n$ elements with replacement and (ii) drawing a single sample of $ns$ elements. In this paper, under the setting where the descending order population frequency follows the Poisson--Dirichlet distribution with parameter $\theta$, we report that the magnitude relation of the Fisher information, which sample partitions converted from samples (i) and (ii) possess, can change depending on the parameters, $n$, $s$, and $\theta$. Roughly speaking, if $\theta$ is small relative to $n$ and $s$, the Fisher information of (i) is larger than that of (ii); on the contrary, if $\theta$ is large relative to $n$ and $s$, the Fisher information of (ii) is larger than that of (i). The result represents one aspect of random distributions., Comment: 20 pages

A distribution whose normalization constant is an A-hypergeometric polynomial is called an A-hypergeometric distribution. Such a distribution is in turn a generalization of the generalized hypergeometric distribution on the contingency tables with fixed marginal sums. In this paper, we will see that an A-hypergeometric distribution with a homogeneous matrix of two rows, especially, that associated with the rational normal curve, appears in inferences involving exchangeable partition structures. An exact sampling algorithm is presented for the general (any number of rows) A-hypergeometric distributions. Then, the maximum likelihood estimation of the A-hypergeometric distribution associated with the rational normal curve, which is an algebraic exponential family, is discussed. The information geometry of the Newton polytope is useful for analyzing the full and the curved exponential family. Algebraic methods are provided for evaluating the A-hypergeometric polynomials., Comment: 36 pages, 2 figures

The star-shaped $\Lambda$-coalescent and corresponding $\Lambda$-Fleming-Viot process where the $\Lambda$ measure has a single atom at unity are studied in this paper. The transition functions and stationary distribution of the $\Lambda$-Fleming-Viot process are derived in a two-type model with mutation. The distribution of the number of non-mutant lines back in time in the star-shaped $\Lambda$-coalescent is found. Extensions are made to a model with $d$ types, either with parent independent mutation or general Markov mutation, and an infinitely-many-types model when $d\to \infty$. An eigenfunction expansion for the transition functions is found which has polynomial right eigenfunctions and left eigenfunctions described by hyperfunctions. A further star-shaped model with general frequency dependent change is considered and the stationary distribution in the Fleming-Viot process derived. This model includes a star-shaped $\Lambda$-Fleming-Viot process with mutation and selection. In a general $\Lambda$-coalescent explicit formulae for the transition functions and stationary distribution when there is mutation are unknown, however in this paper explicit formulae are derived in the star-shaped coalescent.

Gibbs-type exchangeable random partitions, which is a class of multiplicative measures on the set of positive integer partitions, appear in various contexts, including Bayesian statistics, random combinatorial structures, and stochastic models of diversity in various phenomena. Some distributional results on ordered sizes in the Gibbs partition are established by introducing associated partial Bell polynomials and analysis of the generating functions. The combinatorial approach is applied to derive explicit results on asymptotic behavior of the extreme sizes in the Gibbs partition. Especially, Ewens-Pitman partition, which is the sample from the Poisson-Dirichlet process and has been discussed from rather model-specific viewpoints, and a random partition which was recently introduced by Gnedin, are discussed in the details. As by-products, some formulas for the associated partial Bell polynomials are presented., Comment: 34 pages, 1 figure

Approximate Bayesian computation (ABC) is a likelihood-free approach for Bayesian inferences based on a rejection algorithm method that applies a tolerance of dissimilarity between summary statistics from observed and simulated data. Although several improvements to the algorithm have been proposed, none of these improvements avoid the following two sources of approximation: 1) lack of sufficient statistics: sampling is not from the true posterior density given data but from an approximate posterior density given summary statistics; and 2) non-zero tolerance: sampling from the posterior density given summary statistics is achieved only in the limit of zero tolerance. The first source of approximation can be improved by adding a summary statistic, but an increase in the number of summary statistics could introduce additional variance caused by the low acceptance rate. Consequently, many researchers have attempted to develop techniques to choose informative summary statistics. The present study evaluated the utility of a kernel-based ABC method (Fukumizu et al. 2010, arXiv:1009.5736 and 2011, NIPS 24: 1549-1557) for complex problems that demand many summary statistics. Specifically, kernel ABC was applied to population genetic inference. We demonstrate that, in contrast to conventional ABCs, kernel ABC can incorporate a large number of summary statistics while maintaining high performance of the inference., Comment: 31 pages, 5 figures

Known results on the moments of the distribution generated by the two-locus Wright-Fisher diffusion model and a duality between the diffusion process and the ancestral process with recombination are briefly summarized. A numerical methods for computing moments by a Markov chain Monte Carlo and a method to compute closed-form expressions of the moments are presented. By using the duality argument properties of the ancestral recombination graph are studied in terms of the moments., Comment: 25 pages, no figures

Gene conversion is a mechanism by which a double-strand break in a DNA molecule is repaired using a homologous DNA molecule as a template. As a result, one gene is 'copied and pasted' onto the other gene. It was recently reported that the direction of gene conversion appears to be biased towards G and C nucleotides. In this paper a stochastic model of the dynamics of the bias in gene conversion is developed for a finite population of members in a multigene family. The dual process is the biased voter model, which generates an ancestral random graph for a given sample. An importance-sampling algorithm for computing the likelihood of the sample is also given., Comment: 29 pages, 2 figures

The ancestral selection graph in population genetics was introduced by KroneNeuhauser (1997) as an analogue of the coalescent genealogy of a sample of genes from a neutrally evolving population. The number of particles in this graph, followed backwards in time, is a birth and death process with quadratic death and linear birth rates. In this paper an explicit form of the probability distribution of the number of particles is obtained by using the density of the allele frequency in the corresponding diffusion model obtained by Kimura (1955). It is shown that the process of fixation of the allele in the diffusion model corresponds to convergence of the ancestral process to its stationary measure. The time to fixation of the allele conditional on fixation is studied in terms of the ancestral process., Comment: 36 pages, 5 figures; minor correction, figures added

By using BPS closed string, the entropy is calculated of the extremal five dimensional black hole consisting of Dirichlet onebranes, Dirichlet fivebranes and Kaluza-Klein momentum in the flat background approximation. In our formulations we consider two kinds of BPS closed strings with or without a winding number. In the former case heavy excitation modes of closed strings are used to derive the entropy. In the latter case we have no oscillator modes and consider collective motion of such massless closed strings. The entropy is given by the number of the ways how we divide the Kaluza-Klein momentum among the massless closed strings. In both cases the black hole entropy is the same as the Bekenstein-Hawking entropy. We argue that the collective modes of closed strings without winding is equivalent to a single closed string with winding. We propose that the two closed string pictures are connected with the open string pictures by the modular transformation., Comment: 21pages, no figures, explanations are improved

When a particle moves around a Kerr black hole, it radiates gravitational waves.Some of these waves are absorbed by the black hole. We calculate such absorption of gravitational waves induced by a particle of mass mu in a circular orbit on an equatorial plane around a Kerr black hole of mass M. We assume that the velocity of the particle v is much smaller than the speed of light c and calculate the energy absorption rate analytically. We adopt an analytic technique for the Teukolsky equation developed by Mano, Suzuki and Takasugi. We obtain the energy absorption rate to O((v/c)^8) compared to the lowest order. We find that the black hole absorption occurs at O((v/c)^5) beyond the Newtonian-quadrapole luminosity at infinity in the case when the black hole is rotating, which is O((v/c)^3) lower than the non-rotating case. Using the energy absorption rate, we investigate its effects on the orbital evolution of coalescing compact binaries., Comment: 22 pages, ptptex, no figures

The analytical solutions reported in our previous paper are given as series of hypergeometric or Coulomb wave functions. By using them, we can get the Teukolsky functions analytically in a desired accuracy. For the computation, the deep understanding of their properties is necessary. We summarize the main result: The relative normalization between the solutions with a spin weight s and -s is given analytically by using the Teukolsky-Starobinsky (T-S) identities. By examining the asymptotic behaviors of our solution and combined with the T-S identities and the Wronskian, we found nontrivial identities between the sums of coefficients of the series. These identities will serve to make various expression in simpler forms and also become a powerful tool to test the accuracy of the computation. As an application, we investigated the absorption rate and the evaporation rate of black hole and obtain interesting analytic results., Comment: 28 pages

Analytic solutions of the Regge-Wheeler equation are presented in the form of series of hypergeometric functions and Coulomb wave functions which have different regions of convergence. Relations between these solutions are established. The series solutions are given as the Post-Minkowskian expansion with respect to a parameter $\epsilon \equiv 2M\omega$, $M$ being the mass of black hole. This expansion corresponds to the post-Newtonian expansion when they are applied to the gravitational radiation from a particle in circular orbit around a black hole. These solutions can also be useful for numerical computations., Comment: 22 pages

Analytic solutions of the Teukolsky equation in Kerr geometries are presented in the form of series of hypergeometric functions and Coulomb wave functions. Relations between these solutions are established. The solutions provide a very powerful method not only for examining the general properties of solutions and physical quantities when they are applied to, but also for numerical computations. The solutions are given in the expansion of a small parameter $\epsilon \equiv 2M\omega$, $M$ being the mass of black hole, which corresponds to Post-Minkowski expansion by $G$ and to post-Newtonian expansion when they are applied to the gravitational radiation from a particle in circular orbit around a black hole. It is expected that these solutions will become a powerful weapon to construct the theoretical template towards LIGO and VIRGO projects., Comment: 24 pages, minor modifications