estimation of regression models for the mean of repeated outcomes under nonignorable nonmonotone nonresponse we propose a new class of models for making inference about the mean of a vector of repeated outcomes when the outcome vector is incompletely observed in some study units and missingness is nonmonotone each model in our class is indexed by a set of unidentified selection bias functions which quantify the residual association of the outcome at each occasion t and the probability that this outcome is missing after adjusting for variables observed prior to time t and for the past nonresponse pattern in particular selection bias functions equal to zero encode the investigator s a priori belief that nonresponse of the next outcome does not depend on that outcome after adjusting for the observed past we call this assumption sequential explainability since each model in our class is nonparametric it fits the data perfectly well as such our models are ideal for conducting sensitivity analyses aimed at evaluating the impact that different degrees of departure from sequential explainability have on inference about the marginal means of interest although the marginal means are identified under each of our models their estimation is not feasible in practice because it requires the auxiliary estimation of conditional expectations and probabilities given high dimensional variables we henceforth discuss the estimation of the marginal means under each model in our class assuming additionally that at each occasion either one of the following two models holds a parametric model for the conditional probability of nonresponse given current outcomes and past recorded data or a parametric model for the conditional mean of the outcome on the nonrespondents given the past recorded data we call the resulting procedure 2 t multiply robust as it protects at each of the t time points against misspecification of one of these two working models although not against simultaneous misspecification of both we extend our proposed class of models and estimators to incorporate data configurations which include baseline covariates and a parametric model for the conditional mean of the vector of repeated outcomes given the baseline covariates

elliptic solutions to difference non linear equations and related many body problems we study algebro geometric finite gap and elliptic solutions of fully discretized kp or 2d toda equations in bilinear form they are hirota s difference equation for functions starting from a given algebraic curve we express the function and the baker akhiezer function in terms of the riemann theta function we show that the elliptic solutions when the function is an elliptic polynomial form a subclass of the general algebro geometric solutions we construct the algebraic curves of the elliptic solutions the evolution of zeros of the elliptic solutions is governed by the discrete time generalization of the ruijsenaars schneider many body system the zeros obey equations which have the form of nested bethe ansatz equations known from integrable quantum field theories we discuss the lax representation and the action angle type variables for the many body system we also discuss elliptic solutions to discrete analogues of kdv sine gordon and 1d toda equations and describe the loci of the zeros

nonlinear quantum dynamical system based on the kadomtsev petviashvili ii equation the structure of soliton solutions of classical integrable nonlinear evolution equations which can be solved through the hirota transformation suggests a new way for the construction of nonlinear quantum dynamical systems that are based on the classical equations in the new approach the classical soliton solution is mapped into an operator u which is a nonlinear functional of the particle number operators over a fock space of quantum particles u obeys the evolution equation the classical soliton solutions are the eigenvalues of u in multi particle states in the fock space the construction easily allows for the incorporation of particle interactions which generate soliton effects that do not have a classical analog in this paper this new approach is applied to the case of the kadomtsev petviashvili ii equation the nonlinear quantum dynamical system describes a set of m 2s 1 particles with intrinsic spin s which interact in clusters of 1 n m 1 particles

the semiparametric bernstein von mises theorem in a smooth semiparametric estimation problem the marginal posterior for the parameter of interest is expected to be asymptotically normal and satisfy frequentist criteria of optimality if the model is endowed with a suitable prior it is shown that under certain straightforward and interpretable conditions the assertion of le cam s acclaimed but strictly parametric bernstein von mises theorem univ california publ statist 1 1953 277 329 holds in the semiparametric situation as well as a consequence bayesian point estimators achieve efficiency for example in the sense of hajek s convolution theorem z wahrsch verw gebiete 14 1970 323 330 the model is required to satisfy differentiability and metric entropy conditions while the nuisance prior must assign nonzero mass to certain kullback leibler neighborhoods ghosal ghosh and van der vaart ann statist 28 2000 500 531 in addition the marginal posterior is required to converge at parametric rate which appears to be the most stringent condition in examples the results are applied to estimation of the linear coefficient in partial linear regression with a gaussian prior on a smoothness class for the nuisance

constructing solutions to the ultradiscrete painleve equations we investigate the nature of particular solutions to the ultradiscrete painleve equations we start by analysing the autonomous limit and show that the equations possess an explicit invariant which leads naturally to the ultradiscrete analogue of elliptic functions for the ultradiscrete painleve equations ii and iii we present special solutions reminiscent of the casorati determinant ones which exist in the continuous and discrete cases finally we analyse the discrete painleve equation i and show how it contains both the continuous and the ultradiscrete ones as particular limits

bayesian conditional tensor factorizations for high dimensional classification abstractin many application areas data are collected on a categorical response and high dimensional categorical predictors with the goals being to build a parsimonious model for classification while doing inferences on the important predictors in settings such as genomics there can be complex interactions among the predictors by using a carefully structured tucker factorization we define a model that can characterize any conditional probability while facilitating variable selection and modeling of higher order interactions following a bayesian approach we propose a markov chain monte carlo algorithm for posterior computation accommodating uncertainty in the predictors to be included under near low rank assumptions the posterior distribution for the conditional probability is shown to achieve close to the parametric rate of contraction even in ultra high dimensional settings the methods are illustrated using simulation examples and biomedical applications supplementary materials for this artic

posterior consistency of dirichlet mixtures in density estimation a dirichlet mixture of normal densities is a useful choice for a prior distribution on densities in the problem of bayesian density estimation in the recent years efficient markov chain monte carlo method for the computation of the posterior distribution has been developed the method has been applied to data arising from different fields of interest the important issue of consistency was however left open in this paper we settle this issue in affirmative

box and ball system as a realization of ultradiscrete nonautonomous kp equation cellular automata which are realized by dynamics of several kinds of balls in an infinite array of boxes are investigated they show soliton patterns even in the case when each box has arbitrary capacity the analytical expression for the soliton patterns are obtained using ultradiscretization of the nonautonomous discrete kp equation

bayesian nonparametric analysis of longitudinal studies in the presence of informative missingness summaryin longitudinal clinical trials one often encounters missingness that is thought to be nonignorable such missingness introduces identifiability issues resulting in causal effects being nonparametrically unidentified it is then prudent to conduct a sensitivity analysis to assess how much of the inference is being driven by untestable assumptions needed to identify the effects of interest we introduce a bayesian nonparametric framework for conducting inference in the presence of nonignorable nonmonotone missingness this framework focuses on the specification of an auxiliary working prior on the space of complete data generating mechanisms this prior induces a prior on the observed data generating mechanism which is then used in conjunction with an identifying restriction to conduct inference advantages of this approach include a flexible modelling framework access to simple computational methods strong theoretical support straightforward sensitivity analysis and applicability to nonmonotone missingness

multiple addition theorem for discrete and continuous nonlinear problems the addition relation for the riemann theta functions and for its limits which leads to the appearance of exponential functions in soliton type equations is discussed the form of addition property presented resolves itself to the factorization of the n tuple product of the shifted functions and it seems to be useful for analysis of soliton type continuous and discrete processes in the n 1 space time a close relation with the natural generalization of bi and tri linear operators into multiple linear operators concludes the paper

ultradiscretization of elliptic functions and its applications to integrable systems it is shown that there exist three kinds of ultradiscrete analogues of jacobi s elliptic functions in this process the asymptotic behaviour of the poles and the zeros of the functions plays a crucial role using the ultradiscrete analogues and an addition formula exact solutions to the ultradiscrete kp equation are constructed and their relation to the ultradiscrete qrt system is discussed

simplex factor models for multivariate unordered categorical data gaussian latent factor models are routinely used for modeling of dependence in continuous binary and ordered categorical data for unordered categorical variables gaussian latent factor models lead to challenging computation and complex modeling structures as an alternative we propose a novel class of simplex factor models in the single factor case the model treats the different categorical outcomes as independent with unknown marginals the model can characterize flexible dependence structures parsimoniously with few factors and as factors are added any multivariate categorical data distribution can be accurately approximated using a bayesian approach for computation and inferences a markov chain monte carlo mcmc algorithm is proposed that scales well with increasing dimension with the number of factors treated as unknown we develop an efficient proposal for updating the base probability vector in hierarchical dirichlet models theoretical properties are described and we evaluate the appr

the bayesian bootstrap the bayesian bootstrap is the bayesian analogue of the bootstrap instead of simulating the sampling distribution of a statistic estimating a parameter the bayesian bootstrap simulates the posterior distribution of the parameter operationally and inferentially the methods are quite similar because both methods of drawing inferences are based on somewhat peculiar model assumptions and the resulting inferences are generally sensitive to these assumptions neither method should be applied without some consideration of the reasonableness of these model assumptions in this sense neither method is a true bootstrap procedure yielding inferences unaided by external assumptions

periodic multiwave solutions to the toda type cellular automaton an ultradiscretization of the riemann theta function is proposed the ultradiscretization satisfies an addition formula which is an ultradiscrete analogue of an addition formula for the riemann theta function using the addition formula periodic multiwave solutions to the toda type cellular automaton are obtained

pattern mixture models for multivariate incomplete data consider a random sample on variables x1 xv with some values of xv missing selection models specify the distribution of x1 xv over respondents and nonrespondents to xv and the conditional distribution that xv is missing given x1 xv in contrast pattern mixture models specify the conditional distribution of x 1 xv given that xv is observed or missing respectively and the marginal distribution of the binary indicator for whether or not xv is missing for multivariate data with a general pattern of missing values the literature has tended to adopt the selection modeling approach see for example little and rubin here pattern mixture models are proposed for this more general problem pattern mixture models are chronically underidentified in particular for the case of univariate nonresponse mentioned above there are no data on the distribution of xv given x1 xv 1 in the stratum with xv missing thus the models require restrictions or prior information to identify the paramet

a tropical analogue of fay s trisecant identity and the ultra discrete periodic toda lattice we introduce a tropical analogue of fay s trisecant identity for a special family of hyperelliptic tropical curves we apply it to obtain the general solution of the ultra discrete toda lattice with periodic boundary conditions in terms of the tropical riemann s theta function

adaptive bayesian multivariate density estimation with dirichlet mixtures summary we show that rate adaptive multivariate density estimation can be performed using bayesian methods based on dirichlet mixtures of normal kernels with a prior distribution on the kernel s covariance matrix parameter we derive sufficient conditions on the prior specification that guarantee convergence to a true density at a rate that is minimax optimal for the smoothness class to which the true density belongs no prior knowledge of smoothness is assumed the sufficient conditions are shown to hold for the dirichlet location mixture of normals prior with a gaussian base measure and an inverse wishart prior on the covariance matrix parameter locally h older

bayesian density estimation and inference using mixtures abstract we describe and illustrate bayesian inference in models for density estimation using mixtures of dirichlet processes these models provide natural settings for density estimation and are exemplified by special cases where data are modeled as a sample from mixtures of normal distributions efficient simulation methods are used to approximate various prior posterior and predictive distributions this allows for direct inference on a variety of practical issues including problems of local versus global smoothing uncertainty about density estimates assessment of modality and the inference on the numbers of components also convergence results are established for a general class of normal mixture models

the space of interactions in neural network models the typical fraction of the space of interactions between each pair of n ising spins which solve the problem of storing a given set of p random patterns as n bit spin configurations is considered the volume is calculated explicitly as a function of the storage ratio alpha p n of the value kappa 0 of the product of the spin and the magnetic field at each site and of the magnetisation m here m may vary between 0 no correlation and 1 completely correlated the capacity increases with the correlation between patterns from alpha 2 for correlated patterns with kappa 0 and tends to infinity as m tends to 1 the calculations use a saddle point method and the order parameters at the saddle point are assumed to be replica symmetric this solution is shown to be locally stable a local iterative learning algorithm for updating the interactions is given which will converge to a solution of given kappa provided such solutions exist

bayesian nonparametric longitudinal data analysis abstractpractical bayesian nonparametric methods have been developed across a wide variety of contexts here we develop a novel statistical model that generalizes standard mixed models for longitudinal data that include flexible mean functions as well as combined compound symmetry cs and autoregressive ar covariance structures ar structure is often specified through the use of a gaussian process gp with covariance functions that allow longitudinal data to be more correlated if they are observed closer in time than if they are observed farther apart we allow for ar structure by considering a broader class of models that incorporates a dirichlet process mixture dpm over the covariance parameters of the gp we are able to take advantage of modern bayesian statistical methods in making full predictive inferences and about characteristics of longitudinal profiles and their differences across covariate combinations we also take advantage of the generality of our model which provides for estimation

pattern mixture models with proper time dependence summary recently pattern mixture modelling has become a popular tool for modelling incomplete longitudinal data such models are under identified in the sense that for any dropout pattern the data provide no direct information on the distribution of the unobserved outcomes given the observed ones one simple way of overcoming this problem ordinary extrapolation of sufficiently simple pattern specific models often produces rather unlikely descriptions several authors consider identifying restrictions instead molenberghs et al 1998 have constructed identifying restrictions corresponding to missing at random in this paper the family of restrictions where drop out does not depend on future unobserved observations is identified the ideas are illustrated using a clinical study of alzheimer patients

partial and latent ignorability in missing data problems summary when an assumption of missing at random is untenable it becomes necessary to model missingdata indicators which carry information about the parameters of the complete data population within a given application however researchers may believe that some aspects of missingness are ignorable but others are not we argue that there are two different ways to formalize the notion that only part of the missingness is ignorable these approaches correspond to assumptions that we call partially missing at random and latently missing at random we explain these concepts and apply them in a latent class analysis of survey questions with item nonresponse

integrable time discretisation of the ruijsenaars schneider model an exactly integrable symplectic correspondence is derived which in a continuum limit leads to the equations of motion of the relativistic generalization of the calogero moser system that was introduced for the first time by ruijsenaars and schneider for the discrete time model the equations of motion take the form of bethe ansatz equations for the inhomogeneous spin 1 2 xyz heisenberg magnet we present a lax pair the sympletic structure and prove the involutivity of the invariants exact solutions are investigated in the rational and hyperbolic trigonometric limits of the system that is given in terms of elliptic functions these solutions are connected with discrete soliton equations the results obtained allow us to consider the bethe ansatz equations as ones giving an integrable symplectic correspondence mixing the parameters of the quantum integrable system and the parameters of the corresponding bethe wavefunction

box and ball system with a carrier and ultradiscrete modified kdv equation a new soliton cellular automaton is proposed it is defined by an array of an infinite number of boxes a finite number of balls and a carrier of balls moreover it reduces to a discrete equation obtained from the discrete modified korteweg de vries equation through a limit an algebraic expression of soliton solutions is also proposed

pattern mixture and selection models for analysing longitudinal data with monotone missing patterns summary we examine three pattern mixture models for making inference about parameters of the distribution of an outcome of interest y that is to be measured at the end of a longitudinal study when this outcome is missing in some subjects we show that these pattern mixture models also have an interpretation as selection models because these models make unverifiable assumptions we recommend that inference about the distribution of y be repeated under a range of plausible assumptions we argue that of the three models considered only one admits a parameterization that facilitates the examination of departures from the assumption of sequential ignorability the three models are nonparametric in the sense that they do not impose restrictions on the class of observed data distributions owing to the curse of dimensionality the assumptions that are encoded in these models are sufficient for identification but not for inference we describe additional flexible and easily interpretable assumptions under which it is possible to construct estimators that are well behaved with moderate sample sizes these assumptions define semiparametric models for the distribution of the observed data we describe a class of estimators which up to asymptotic equivalence comprise all the consistent and asymptotically normal estimators of the parameters of interest under the postulated semiparametric models we illustrate our methods with the analysis of data from a randomized clinical trial of contracepting women

perceptron like learning in time summating neural networks this paper investigates the ability of a single layer time summating neural network to associate and store temporal sequences in particular the associative learning of temporal sequences is reformulated as an equivalent classification task involving static patterns this leads to a generalization of the perceptron learning rule and convergence theorem to the case of temporal sequences using geometrical arguments based on linear separability it is shown how a time summating network can handle temporal features such as ordering and coarticulation effects such an ability is a consequence of the fact that the time summating network develops an activity trace consisting of a decaying sum of all previous inputs to the network on the other hand such an activity trace may also lead to an accumulation of errors in the presence of noisy inputs this motivates a modification of the perceptron learning rule involving the introduction of a stability parameter that guarantees a certain level of robustness to noise the performance of the network in the presence of random input sequences is then analysed using statistical mechanical techniques finally it is shown how with small modifications the time summating network can be trained to store and recall complex sequences

temporal sequence storage capacity of time summating neural networks the maximum capacity for storage of temporal sequences in a time summating neural network is analysed within a statistical mechanical framework each neuron of such a network maintains an activity trace consisting of a decaying sum of all previous inputs to that neuron the maximum storage capacity is expressed as a function of the temporal correlations of the local fields of the neurons and evaluated using a perturbation expansion in powers of the decay rate

tetrahedra octahedra and cubo octahedra integrable geometry of multi ratios geometric and algebraic aspects of multi ratios m2n are investigated in detail connections with menelaus theorem clifford configurations and maxwell s reciprocal quadrangles are utilized to associate the multi ratios m4 m8 and m8 with tetrahedra octahedra and cubo octahedra respectively integrable maps defined on face centred fcc lattices and irregular lattices composed of the face centres of simple cubic lattices are constructed and related to the discrete kp and bkp equations and the integrable discrete darboux system governing conjugate lattices an interpretation in terms of integrable irregular lattices of slopes on the plane is also given