gibbs sampling for bayesian non conjugate and hierarchical models by using auxiliary variables summary we demonstrate the use of auxiliary or latent variables for sampling non standard densities which arise in the context of the bayesian analysis of non conjugate and hierarchical models by using a gibbs sampler their strategic use can result in a gibbs sampler having easily sampled full conditionals we propose such a procedure to simplify or speed up the markov chain monte carlo algorithm the strength of this approach lies in its generality and its ease of implementation the aim of the paper therefore is to provide an alternative sampling algorithm to rejection based methods and other sampling approaches such as the metropolis hastings algorithm

correlated random measures abstractwe develop correlated random measures random measures where the atom weights can exhibit a flexible pattern of dependence and use them to develop powerful hierarchical bayesian nonparametric models hierarchical bayesian nonparametric models are usually built from completely random measures a poisson process based construction in which the atom weights are independent completely random measures imply strong independence assumptions in the corresponding hierarchical model and these assumptions are often misplaced in real world settings correlated random measures address this limitation they model correlation within the measure by using a gaussian process in concert with the poisson process with correlated random measures for example we can develop a latent feature model for which we can infer both the properties of the latent features and their dependency pattern we develop several other examples as well we study a correlated random measure model of pairwise count data we derive an eff

gibbs sampling methods for stick breaking priors a rich and flexible class of random probability measures which we call stick breaking priors can be constructed using a sequence of independent beta random variables examples of random measures that have this characterization include the dirichlet process its two parameter extension the two parameter poisson dirichlet process finite dimensional dirichlet priors and beta two parameter processes the rich nature of stick breaking priors offers bayesians a useful class of priors for nonparametric problems while the similar construction used in each prior can be exploited to develop a general computational procedure for fitting them in this article we present two general types of gibbs samplers that can be used to fit posteriors of bayesian hierarchical models based on stick breaking priors the first type of gibbs sampler referred to as a polya urn gibbs sampler is a generalized version of a widely used gibbs sampling method currently employed for dirichlet process computing this method applies t

the profile of bubbling solutions of a class of fourth order geometric equations on 4 manifolds abstract we study a class of fourth order geometric equations defined on a 4 dimensional compact riemannian manifold which includes the q curvature equation we obtain sharp estimates on the difference near the blow up points between a bubbling sequence of solutions and the standard bubble

stability of compacton solutions the stability of the recently discovered compacton solutions is studied by means of both linear stability analysis as well as lyapunov stability criteria from the results obtained it follows that unlike solitons all the allowed compacton solutions are stable as the stability condition is satisfied for arbitrary values of the nonlinearity parameter the results are shown to be true even for the higher order nonlinear dispersion equations for compactons some new conservation laws for the higher order nonlinear dispersion equations are also presented

combinatorial clustering and the beta negative binomial process we develop a bayesian nonparametric approach to a general family of latent class problems in which individuals can belong simultaneously to multiple classes and where each class can be exhibited multiple times by an individual we introduce a combinatorial stochastic process known as the negative binomial process rm nbp as an infinite dimensional prior appropriate for such problems we show that the rm nbp is conjugate to the beta process and we characterize the posterior distribution under the beta negative binomial process rm bnbp and hierarchical models based on the rm bnbp the rm hbnbp we study the asymptotic properties of the rm bnbp and develop a three parameter extension of the rm bnbp that exhibits power law behavior we derive mcmc algorithms for posterior inference under the rm hbnbp and we present experiments using these algorithms in the domains of image segmentation object recognition and document analysis

nonparametric bayes estimators based on beta processes in models for life history data the article studies the problem of finding bayes estimators for cumulative hazard rates and related quantities w r t prior distributions that correspond to cumulative hazard rate processes with nonnegative independent increments a particular class of prior processes termed beta processes is introduced and is shown to constitute a conjugate class an important bonus of the approach based on cumulative hazards is that more complicated models for life history data than the simple life table situation can be treated for example time inhomogeneous markov chains we find posterior distributions and derive bayes estimators in such models and present a semiparametric bayesian analysis of the cox regression model

controlling the reinforcement in bayesian non parametric mixture models summary the paper deals with the problem of determining the number of components in a mixture model we take a bayesian non parametric approach and adopt a hierarchical model with a suitable non parametric prior for the latent structure a commonly used model for such a problem is the mixture of dirichlet process model here we replace the dirichlet process with a more general non parametric prior obtained from a generalized gamma process the basic feature of this model is that it yields a partition structure for the latent variables which is of gibbs type this relates to the well known exchangeable product partition models if compared with the usual mixture of dirichlet process model the advantage of the generalization that we are examining relies on the availability of an additional parameter belonging to the interval 0 1 it is shown that such a parameter greatly influences the clustering behaviour of the model a value of that is close to 1 generates a large number of clusters most of which are of small size then a reinforcement mechanism which is driven by acts on the mass allocation by penalizing clusters of small size and favouring those few groups containing a large number of elements these features turn out to be very useful in the context of mixture modelling since it is difficult to specify a priori the reinforcement rate it is reasonable to specify a prior for hence the strength of the reinforcement mechanism is controlled by the data

compactness of solutions to some geometric fourth order equations we prove compactness of solutions to some fourth order equations with exponential nonlinearities on four manifolds the proof is based on a refined bubbling analysis for which the main estimates are given in integral form our result is used in a subsequent paper to find critical points via minimax arguments of some geometric functional which give rise to conformal metrics of constant q curvature as a byproduct of our method we also obtain compactness of such metrics

bias corrected maximum likelihood estimator of the negative binomial dispersion parameter nous decrivons un estimateur du maximum de vraisemblance avec correction du premier ordre pour le biais sur le parametre de dispersion d une distribution binomiale negative cet estimateur est compare en terme debiais et d efficacite avec l estimateur du maximum de vraisemblance etudie par piegorsch 1990 biometrics 46 863 867 les estimateurs de moment et du maximum de quasi vraisemblance etendue de clark et perry 1989 biometrics 45 309 316 et un estimateur de double quasi vraisemblance etendue l estimateur du maximum de vraisemblance avec correction du biais a des proprietes de biais et d efficacite superieures dans la plupart des cas pour faciliter la comparaison nous donnons les resultats pour le modele binomial negatif a deux parametres cependant un exemple impliquant la regression binomiale negative est donne

hierarchical dirichlet processes we consider problems involving groups of data where each observation within a group is a draw from a mixture model and where it is desirable to share mixture components between groups we assume that the number of mixture components is unknown a priori and is to be inferred from the data in this setting it is natural to consider sets of dirichlet processes one for each group where the well known clustering property of the dirichlet process provides a nonparametric prior for the number of mixture components within each group given our desire to tie the mixture models in the various groups we consider a hierarchical model specifically one in which the base measure for the child dirichlet processes is itself distributed according to a dirichlet process such a base measure being discrete the child dirichlet processes necessarily share atoms thus as desired the mixture models in the different groups necessarily share mixture components we discuss representations of hierarchical dirichlet processes

markov chain monte carlo in approximate dirichlet and beta two parameter process hierarchical models summary we present some easy to construct random probability measures which approximate the dirichlet process and an extension which we will call the beta two parameter process the nature of these constructions makes it simple to implement markov chain monte carlo algorithms for fitting nonparametric hierarchical models and mixtures of nonparametric hierarchical models for the dirichlet process we consider a truncation approximation as well as a weak limit approximation based on a mixture of dirichlet processes the same type of truncation approximation can also be applied to the beta two parameter process both methods lead to posteriors which can be fitted using markov chain monte carlo algorithms that take advantage of blocked coordinate updates these algorithms promote rapid mixing of the markov chain and can be readily applied to normal mean mixture models and to density estimation problems we prefer the truncation approximations since a simple device for monitoring the adequacy of the approximation can be easily computed from the output of the gibbs sampler furthermore for the dirichlet process the truncation approximation offers an exponentially higher degree of accuracy over the weak limit approximation for the same computational effort we also find that a certain beta two parameter process may be suitable for finite mixture modelling because the distinct number of sampled values from this process tends to match closely the number of components of the underlying mixture distribution

poisson gamma random field models for spatial statistics summary doubly stochastic bayesian hierarchical models are introduced to account for uncertainty and spatial variation in the underlying intensity measure for point process models inhomogeneous gamma process random fields and more generally markov random fields with infinitely divisible distributions are used to construct positively autocorrelated intensity measures for spatial poisson point processes these in turn are used to model the number and location of individual events a data augmentation scheme and markov chain monte carlo numerical methods are employed to generate samples from bayesian posterior and predictive distributions the methods are developed in both continuous and discrete settings and are applied to a problem in forest ecology

an estimate on the blowing up solutions of a fourth order equation abstract in this short paper i prove a harnack type inequality of the blowing up solutions for a class of fourth order equations with exponential growth on a compact four manifold the main method i use is the moving plane method

nonparametric bayesian dictionary learning for analysis of noisy and incomplete images nonparametric bayesian methods are considered for recovery of imagery based upon compressive incomplete and or noisy measurements a truncated beta bernoulli process is employed to infer an appropriate dictionary for the data under test and also for image recovery in the context of compressive sensing significant improvements in image recovery are manifested using learned dictionaries relative to using standard orthonormal image expansions the compressive measurement projections are also optimized for the learned dictionary additionally we consider simpler incomplete measurements defined by measuring a subset of image pixels uniformly selected at random spatial interrelationships within imagery are exploited through use of the dirichlet and probit stick breaking processes several example results are presented with comparisons to other methods in the literature

existence of conformal metrics with constant q curvature given a compact four dimensional manifold we prove existence of conformal metrics with constant q curvature under generic assumptions the problem amounts to solving a fourth order nonlinear elliptic equation with variational structure since the corresponding euler functional is in general unbounded from above and from below we employ topological methods and min max schemes jointly with the compactness result of 35

a sup inf inequality for some nonlinear elliptic equations involving exponential nonlinearities let u be a solution of the equation u v x eu in a domain r2 where 0 a v b and v is lipschitz continuous we prove that sup u can be controlled in terms of inf u more precisely supku inf u c a b k v l for any compact subset k this extends an earlier result of shafrir who obtained a similar conclusion when v 1

maximum likelihood estimation for the negative binomial dispersion parameter a 2 this yf y k for k 0 u 0 anscombe 1950 here u e y and k 1 a also var y u a2 k a aa2 this distribution has numerous applications in the biological sciences see e g breslow 1984 margolin kim and risko 1989 and the references in clark and perry 1989 anscombe 1950 observed that strictly speaking the maximum likelihood ml estimator of k k does not have a distribution since there exists a finite probability of observing a data set from which k may not be calculated this occurs when the sample mean exceeds the sample variance a more useful parameterization involves a 1 k giving

extremal metrics of zeta function determinants on 4 manifolds in conformal geometry the sobolev inequality at a critical exponent has received much attention in particular the determination of the best constants has played a crucial role in the yamabe problem in dimension two the analogous problem deals with the moser trudinger inequality on a compact riemann surface m2 there exists a constant c c m so that f e47w2 c m if f ivw12 1 and f w 0 the connection of this inequality with geometry comes through the zeta functional determinant of the laplacian as defined by ray singer for a riemannian metric g let 0 a1 a2 be the spectrum

negative binomial process count and mixture modeling the seemingly disjoint problems of count and mixture modeling are united under the negative binomial nb process a gamma process is employed to model the rate measure of a poisson process whose normalization provides a random probability measure for mixture modeling and whose marginalization leads to an nb process for count modeling a draw from the nb process consists of a poisson distributed finite number of distinct atoms each of which is associated with a logarithmic distributed number of data samples we reveal relationships between various count and mixture modeling distributions and construct a poisson logarithmic bivariate distribution that connects the nb and chinese restaurant table distributions fundamental properties of the models are developed and we derive efficient bayesian inference it is shown that with augmentation and normalization the nb process and gamma nb process can be reduced to the dirichlet process and hierarchical dirichlet process respectively these relationships highlight theoretical structural and computational advantages of the nb process a variety of nb processes including the beta geometric beta nb marked beta nb marked gamma nb and zero inflated nb processes with distinct sharing mechanisms are also constructed these models are applied to topic modeling with connections made to existing algorithms under poisson factor analysis example results show the importance of inferring both the nb dispersion and probability parameters

bayesian hierarchical models with conjugate full conditional distributions for dependent data from the natural exponential family we introduce a bayesian approach for analyzing possibly high dimensional dependent data that are distributed according to a member from the natural exponential family of distributions this probl

frequency of frequencies distributions and size dependent exchangeable random partitions abstractmotivated by the fundamental problem of modeling the frequency of frequencies fof distribution this paper introduces the concept of a cluster structure to define a probability function that governs the joint distribution of a random count and its exchangeable random partitions a cluster structure naturally arising from a completely random measure mixed poisson process allows the probability distribution of the random partitions of a subset of a population to be dependent on the population size a distinct and motivated feature that makes it more flexible than a partition structure this allows it to model an entire fof distribution whose structural properties change as the population size varies a fof vector can be simulated by drawing an infinite number of poisson random variables or by a stick breaking construction with a finite random number of steps a generalized negative binomial process model is proposed to generate a cluster structure where in the prior the number of clusters is f

the two parameter poisson dirichlet distribution derived from a stable subordinator the two parameter poisson dirichlet distribution denoted mathsf pd alpha theta is a probability distribution on the set of decreasing positive sequences with sum 1 the usual poisson dirichlet distribution with a single parameter theta introduced by kingman is mathsf pd 0 theta known properties of mathsf pd 0 theta including the markov chain description due to vershik shmidt and ignatov are generalized to the two parameter case the size biased random permutation of mathsf pd alpha theta is a simple residual allocation model proposed by engen in the context of species diversity and rediscovered by perman and the authors in the study of excursions of brownian motion and bessel processes for 0 alpha 1 mathsf pd alpha 0 is the asymptotic distribution of ranked lengths of excursions of a markov chain away from a state whose recurrence time distribution is in the domain of attraction of a stable law of index alpha formulae in this case trace back to work of darling lamperti and wendel in the 1950s and 1960s the distribution of ranked lengths of excursions of a one dimensional brownian motion is mathsf pd 1 2 0 and the corresponding distribution for a brownian bredge is mathsf pd 1 2 1 2 the mathsf pd alpha 0 and mathsf pd alpha alpha distributions admit a similar interpretation in terms of the ranked lengths of excursions of a semistable markov process whose zero set is the range of a stable subordinator of index alpha

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

multistage estimation compared with fixed sample size estimation of the negative binomial parameter k the negative binomial distribution often fits biological data well since the mean is of primary interest the parameterization in terms of the mean and a parameter k has become almost standard given a precise estimate of k sequential methods of estimating the mean are available however precise estimation of k has been elusive in this paper estimators of k by the method of moments and by maximum likelihood estimation based on samples of size 50 and 100 are investigated by using monte carlo methods although the maximum likelihood estimators tend to have smaller mean square errors than the method of moments estimators neither can be considered good a multistage approach to the estimation of k is presented and monte carlo results indicate that this procedure gives a mean square error less than that of comparable fixed sample size estimators two truncation rules for the multistage procedure are also given and compared

retrospective markov chain monte carlo methods for dirichlet process hierarchical models inference for dirichlet process hierarchical models is typically performed using markov chain monte carlo methods which can be roughly categorized into marginal and conditional methods the former integrate out analytically the infinite dimensional component of the hierarchical model and sample from the marginal distribution of the remaining variables using the gibbs sampler conditional methods impute the dirichlet process and update it as a component of the gibbs sampler since this requires imputation of an infinite dimensional process implementation of the conditional method has relied on finite approximations in this paper we show how to avoid such approximations by designing two novel markov chain monte carlo algorithms which sample from the exact posterior distribution of quantities of interest the approximations are avoided by the new technique of retrospective sampling we also show how the algorithms can obtain samples from functionals of the dirichlet process the marginal and the conditional methods are compared and a careful simulation study is included which involves a non conjugate model different datasets and prior specifications