nonparametric bayes conditional distribution modeling with variable selection this article considers a methodology for flexibly characterizing the relationship between a response and multiple predictors goals are 1 to estimate the conditional response distribution addressing the distributional changes across the predictor space and 2 to identify important predictors for the response distribution change both within local regions and globally we first introduce the probit stick breaking process psbp as a prior for an uncountable collection of predictor dependent random distributions and propose a psbp mixture psbpm of normal regressions for modeling the conditional distributions a global variable selection structure is incorporated to discard unimportant predictors while allowing estimation of posterior inclusion probabilities local variable selection is conducted relying on the conditional distribution estimates at different predictor points an efficient stochastic search sampling algorithm is proposed for posterior computation the methods are illustrated through simulation and applied to an epidemiologic study

a time series ddp for functional proteomics profiles using a new type of array technology the reverse phase protein array rppa we measure time course protein expression for a set of selected markers that are known to co regulate biological functions in a pathway structure to accommodate the complex dependent nature of the data including temporal correlation and pathway dependence for the protein markers we propose a mixed effects model with temporal and protein specific components we develop a sequence of random probability measures rpm to account for the dependence in time of the protein expression measurements marginally for each rpm we assume a dirichlet process dp model the dependence is introduced by defining multivariate beta distributions for the unnormalized weights of the stick breaking representation we also acknowledge the pathway dependence among proteins via a conditionally autoregressive car model applying our model to the rppa data we reveal a pathway dependent functional profile for the set of proteins as well as marginal expression profiles over time for individual markers

inference for mixtures of finite polya tree models mixtures of polya tree models provide a flexible alternative when a parametric model may only hold approximately i provide computational strategies for obtaining full semiparametric inference for mixtures of finite polya tree models given a standard parameterization including models that would be troublesome to fit using dirichlet process mixtures recommendations are put forth on choosing the level of a finite polya tree and model comparison is discussed several examples demonstrate the utility of finite polya tree modeling including data fit to generalized linear mixed models and several survival models

estimating normal means with a dirichlet process prior abstract in this article the dirichlet process prior is used to provide a nonparametric bayesian estimate of a vector of normal means in the past there have been computational difficulties with this model this article solves the computational difficulties by developing a gibbs sampler algorithm the estimator developed in this article is then compared to parametric empirical bayes estimators peb and nonparametric empirical bayes estimators npeb in a monte carlo study the monte carlo study demonstrates that in some conditions the peb is better than the npeb and in other conditions the npeb is better than the peb the monte carlo study also shows that the estimator developed in this article produces estimates that are about as good as the peb when the peb is better and produces estimates that are as good as the npeb estimator when that method is better

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

functional clustering by bayesian wavelet methods summary we propose a nonparametric bayes wavelet model for clustering of functional data the wavelet based methodology is aimed at the resolution of generic global and local features during clustering and is suitable for clustering high dimensional data based on the dirichlet process the nonparametric bayes model extends the scope of traditional bayes wavelet methods to functional clustering and allows the elicitation of prior belief about the regularity of the functions and the number of clusters by suitably mixing the dirichlet processes posterior inference is carried out by gibbs sampling with conjugate priors which makes the computation straightforward we use simulated as well as real data sets to illustrate the suitability of the approach over other alternatives

marginal likelihood from the metropolis hastings output this article provides a framework for estimating the marginal likelihood for the purpose of bayesian model comparisons the approach extends and completes the method presented in chib 1995 by overcoming the problems associated with the presence of intractable full conditional densities the proposed method is developed in the context of mcmc chains produced by the metropolis hastings algorithm whose building blocks are used both for sampling and marginal likelihood estimation thus economizing on prerun tuning effort and programming experiments involving the logit model for binary data hierarchical random effects model for clustered gaussian data poisson regression model for clustered count data and the multivariate probit model for correlated binary data are used to illustrate the performance and implementation of the method these examples demonstrate that the method is practical and widely applicable

maass cusp forms abstract it is shown that under certain standard assumptions such as extended riemann hypotheses the scattering matrix s for generic sl 2 r is unexpectedly of order 2 this leads to the conjecture that the generic cofinite has very few maass cusp forms

illustration of bayesian inference in normal data models using gibbs sampling abstract the use of the gibbs sampler as a method for calculating bayesian marginal posterior and predictive densities is reviewed and illustrated with a range of normal data models including variance components unordered and ordered means hierarchical growth curves and missing data in a crossover trial in all cases the approach is straightforward to specify distributionally and to implement computationally with output readily adapted for required inference summaries

an estimation method for the semiparametric mixed effects model summary a semiparametric mixed effects regression model is proposed for the analysis of clustered or longitudinal data with continuous ordinal or binary outcome the common assumption of gaussian random effects is relaxed by using a predictive recursion method newton and zhang 1999 to provide a nonparametric smooth density estimate a new strategy is introduced to accelerate the algorithm parameter estimates are obtained by maximizing the marginal profile likelihood by powell s conjugate direction search method monte carlo results are presented to show that the method can improve the mean squared error of the fixed effects estimators when the random effects distribution is not gaussian the usefulness of visualizing the random effects density itself is illustrated in the analysis of data from the wisconsin sleep survey the proposed estimation procedure is computationally feasible for quite large data sets

disappearance of cusp forms in special families in his 1954 g6ttingen lectures atle selberg sb posed the question of the asymptotic density of the poles of the eisenstein series for r a cofinite non cocompact subgroup of psl 2 r the question is known to be equivalent to the asymptotic count of the number of eigenvalues for the laplacebeltrami operator d acting in l2 h r where h is the upper halfplane with hyperbolic metric in general the spectral decomposition of d consists of the continuous band 1 4 oo with multiplicity the number of cusps of h r and a discrete component possibly in 1 4 0o spanned by eigenfunctions w1i wp2 see hj1 kb lxp for the eigenvalues aj dpj ajpj o consider the spectral counting function n r ij v n 0 see dips ps2 cor 2 3 luo lu recently established the unconditional result his argument does not involve the lindel6f hypothesis deshouillers et

bayesian semiparametric analysis of structural equation models with mixed continuous and unordered categorical variables recently structural equation models sems have been applied for analyzing interrelationships among observed and latent variables in biological and medical research latent variables in these models are typically assumed to have a normal distribution this article considers a bayesian semparametric sem with covariates and mixed continuous and unordered categorical variables in which the explanatory latent variables in the structural equation are modeled via an appropriate truncated dirichlet process with a stick breaking procedure results obtained from a simulation study and an analysis of a real medical data set are presented to illustrate the methodology

bayesian and conditional frequentist testing of a parametric model versus nonparametric alternatives testing the fit of data to a parametric model can be done by embedding the parametric model in a nonparametric alternative and computing the bayes factor of the parametric model to the nonparametric alternative doing so by specifying the nonparametric alternative via a polya tree process is particularly attractive from both theoretical and methodological perspectives among the benefits is a degree of computational simplicity that even allows for robustness analyses to be implemented default nonsubjective versions of this analysis are developed herein in the sense that recommended choices are provided for the many features of the polya tree process that need to be specified considerable discussion of these features is also provided to assist those who might be interested in subjective choices a variety of examples involving location scale models are studied finally it is shown that the resulting procedure can be viewed as a conditional frequentist test resulting in data dependent reported err

additive cubic spline regression with dirichlet process mixture errors abstract the goal of this article is to develop a flexible bayesian analysis of regression models for continuous and categorical outcomes in the models we study covariate or regression effects are modeled additively by cubic splines and the error distribution that of the latent outcomes in the case of categorical data is modeled as a dirichlet process mixture we employ a relatively unexplored but attractive basis in which the spline coefficients are the unknown function ordinates at the knots we exploit this feature to develop a proper prior distribution on the coefficients that involves the first and second differences of the ordinates quantities about which one may have prior knowledge we also discuss the problem of comparing models with different numbers of knots or different error distributions through marginal likelihoods and bayes factors which are computed within the framework of chib 1995 as extended to dpm models by basu and chib 2003 the techniques are illustrated with simulated and real data

semiparametric bayesian inference for stochastic frontier models in this paper we propose a semiparametric bayesian framework for the analysis of stochastic frontiers and efficiency measurement the distribution of inefficiencies is modelled nonparametrically through a dirichlet process prior we suggest prior distributions and implement a bayesian analysis through an efficient markov chain monte carlo sampler which allows us to deal with practically relevant sample sizes we also consider the case where the efficiency distribution varies with firm characteristics the methodology is applied to a cost frontier estimated from a panel data set on 382 u s hospitals

a semiparametric bayesian model for randomised block designs summary a model is proposed for a bayesian semiparametric analysis of randomised block experiments the model is a hierarchical model in which a dirichlet process is inserted at the middle stage for the distribution of the block effects this model allows an arbitrary distribution of block effects and it results in effective estimates of treatment contrasts block effects and the distribution of block effects an effective computational strategy is presented for describing the posterior distribution

clustering and variable selection in the presence of mixed variable types and missing data we consider the problem of model based clustering in the presence of many correlated mixed continuous and discrete variables some of which may have missing values discrete variables are treated with a latent continuous variable approach and the dirichlet process is used to construct a mixture model with an unknown number of components variable selection is also performed to identify the variables that are most influential for determining cluster membership the work is motivated by the need to cluster patients thought to potentially have autism spectrum disorder on the basis of many cognitive and or behavioral test scores there are a modest number of patients 486 in the data set along with many 55 test score variables many of which are discrete valued and or missing the goal of the work is to 1 cluster these patients into similar groups to help identify those with similar clinical presentation and 2 identify a sparse subset of tests that inform the clusters in order to eliminate unnecessary testing the proposed approach compares very favorably with other methods via simulation of problems of this type the results of the autism spectrum disorder analysis suggested 3 clusters to be most likely while only 4 test scores had high 0 5 posterior probability of being informative this will result in much more efficient and informative testing the need to cluster observations on the basis of many correlated continuous discrete variables with missing values is a common problem in the health sciences as well as in many other disciplines

nonparametric bayes testing of changes in a response distribution with an ordinal predictor in certain biomedical studies one may anticipate changes in the shape of a response distribution across the levels of an ordinal predictor for instance in toxicology studies skewness and modality might change as dose increases to address this issue we propose a bayesian nonparametric method for testing for distribution changes across an ordinal predictor using a dynamic mixture of dirichlet processes we allow the response distribution to change flexibly at each level of the predictor in addition by assigning mixture priors to the hyperparameters we can obtain posterior probabilities of no effect of the predictor and identify the lowest dose level for which there is an appreciable change in distribution the method also provides a natural framework for performing tests across multiple outcomes we apply our method to data from a genotoxicity experiment

marginal likelihood and bayes factors for dirichlet process mixture models we present a method for comparing semiparametric bayesian models constructed under the dirichlet process mixture dpm framework with alternative semiparameteric or parameteric bayesian models a distinctive feature of the method is that it can be applied to semiparametric models containing covariates and hierarchical prior structures and is apparently the first method of its kind formally the method is based on the marginal likelihood estimation approach of chib 1995 and requires estimation of the likelihood and posterior ordinates of the dpm model at a single high density point an interesting computation is involved in the estimation of the likelihood ordinate which is devised via collapsed sequential importance sampling extensive experiments with synthetic and real data involving semiparametric binary data regression models and hierarchical longitudinal mixed effects models are used to illustrate the implementation performance and applicability of the method

bayesian nonparametric inference on stochastic ordering we consider bayesian inference about collections of unknown distributions subject to a partial stochastic ordering to address problems in testing of equalities between groups and estimation of group specific distributions we propose classes of restricted dependent dirichlet process priors these priors have full support in the space of stochastically ordered distributions and can be used for collections of unknown mixture distributions to obtain a flexible class of mixture models theoretical properties are discussed efficient methods are developed for posterior computation using markov chain monte carlo simulation and the methods are illustrated using data from a study of dna damage and repair

sequential imputations and bayesian missing data problems abstract for missing data problems tanner and wong have described a data augmentation procedure that approximates the actual posterior distribution of the parameter vector by a mixture of complete data posteriors their method of constructing the complete data sets is closely related to the gibbs sampler both required iterations and similar to the em algorithm convergence can be slow we introduce in this article an alternative procedure that involves imputing the missing data sequentially and computing appropriate importance sampling weights in many applications this new procedure works very well without the need for iterations sensitivity analysis influence analysis and updating with new data can be performed cheaply bayesian prediction and model selection can also be incorporated examples taken from a wide range of applications are used for illustration

markov chain monte carlo methods for computing bayes factors a comparative review the problem of calculating posterior probabilities for a collection of competing models and associated bayes factors continues to be a formidable challenge for applied bayesian statisticians current approaches that take advantage of modern markov chain monte carlo computing methods include those that attempt to sample over some form of the joint space created by the model indicators and the parameters for each model others that sample over the model space alone and still others that attempt to estimate the marginal likelihood of each model directly because the collection of these is equivalent to the collection of model probabilities themselves we review several methods and compare them in the context of three examples a simple regression example a more challenging hierarchical longitudinal model and a binary data latent variable model we find that the joint model parameter space search methods perform adequately but can be difficult to program and tune whereas the marginal likelihood methods o

bayesian analysis of binary and polychotomous response data abstract a vast literature in statistics biometrics and econometrics is concerned with the analysis of binary and polychotomous response data the classical approach fits a categorical response regression model using maximum likelihood and inferences about the model are based on the associated asymptotic theory the accuracy of classical confidence statements is questionable for small sample sizes in this article exact bayesian methods for modeling categorical response data are developed using the idea of data augmentation the general approach can be summarized as follows the probit regression model for binary outcomes is seen to have an underlying normal regression structure on latent continuous data values of the latent data can be simulated from suitable truncated normal distributions if the latent data are known then the posterior distribution of the parameters can be computed using standard results for normal linear models draws from this posterior are used to sample new latent data and t

a bayesian goodness of fit test and semiparametric generalization of logistic regression with measurement data summary logistic regression is a popular tool for risk analysis in medical and population health science with continuous response data it is common to create a dichotomous outcome for logistic regression analysis by specifying a threshold for positivity fitting a linear regression to the nondichotomized response variable assuming a logistic sampling model for the data has been empirically shown to yield more efficient estimates of odds ratios than ordinary logistic regression of the dichotomized endpoint we illustrate that risk inference is not robust to departures from the parametric logistic distribution moreover the model assumption of proportional odds is generally not satisfied when the condition of a logistic distribution for the data is violated leading to biased inference from a parametric logistic analysis we develop novel bayesian semiparametric methodology for testing goodness of fit of parametric logistic regression with continuous measurement data the testing procedures hold for any cutoff threshold and our approach simultaneously provides the ability to perform semiparametric risk estimation bayes factors are calculated using the savage dickey ratio for testing the null hypothesis of logistic regression versus a semiparametric generalization we propose a fully bayesian and a computationally efficient empirical bayesian approach to testing and we present methods for semiparametric estimation of risks relative risks and odds ratios when parametric logistic regression fails theoretical results establish the consistency of the empirical bayes test results from simulated data show that the proposed approach provides accurate inference irrespective of whether parametric assumptions hold or not evaluation of risk factors for obesity shows that different inferences are derived from an analysis of a real data set when deviations from a logistic distribution are permissible in a flexible semiparametric framework

bayesian inference for semiparametric binary regression abstract we propose a regression model for binary response data that places no structural restrictions on the link function except monotonicity and known location and scale predictors enter linearly we demonstrate bayesian inference calculations in this model by modifying the dirichlet process we obtain a natural prior measure over this semiparametric model and we use polya sequence theory to formulate this measure in terms of a finite number of unobserved variables we design a markov chain monte carlo algorithm for posterior simulation and apply the methodology to data on radiotherapy treatments for cancer

estimating a semiparametric asymmetric stochastic volatility model with a dirichlet process mixture in this paper we extend the parametric asymmetric stochastic volatility model asv where returns are correlated with volatility by flexibly modeling the bivariate distribution of the return and volatility innovations nonparametrically its novelty is in modeling the joint conditional return volatility distribution with an infinite mixture of bivariate normal distributions with mean zero vectors but having unknown mixture weights and covariance matrices this semiparametric asv model nests stochastic volatility models whose innovations are distributed as either normal or student t distributions plus the response in volatility to unexpected return shocks is more general than the fixed asymmetric response with the asv model the unknown mixture parameters are modeled with a dirichlet process prior this prior ensures a parsimonious finite posterior mixture that best represents the distribution of the innovations and a straightforward sampler of the conditional posteriors we develop a bayesian markov chain monte carlo sampler to fully characterize the parametric and distributional uncertainty nested model comparisons and out of sample predictions with the cumulative marginal likelihoods and one day ahead predictive log bayes factors between the semiparametric and parametric versions of the asv model shows the semiparametric model projecting more accurate empirical market returns a major reason is how volatility responds to an unexpected market movement when the market is tranquil expected volatility reacts to a negative positive price shock by rising initially declining but then rising when the positive shock is large however when the market is volatile the degree of asymmetry and the size of the response in expected volatility is muted in other words when times are good no news is good news but when times are bad neither good nor bad news matters with regards to volatility

independent and identically distributed monte carlo algorithms for semiparametric linear mixed models hybrid versions of independent and identically distributed weighted chinese restaurant wcr algorithms are developed for inference in semiparametric linear mixed models under minimal assumptions for the random effects distributions the wcr method of working with the posterior partition structure leads to rao blackwell estimates for higher order moments of random effects such as skewness and kurtosis and can be used to estimate densities for random effects a key feature of our approach is the manner in which we incorporate external estimates into our algorithms the use of such information leads to simplified computational procedures reduces the amount of user input required for specifying models and results in numerical stability and accuracy the resulting procedures are automated and can be readily used in standard statistical software our methods are tested by simulation and illustrated by application to a longitudinal study involving chronic renal disease

marginal likelihood from the gibbs output abstract in the context of bayes estimation via gibbs sampling with or without data augmentation a simple approach is developed for computing the marginal density of the sample data marginal likelihood given parameter draws from the posterior distribution consequently bayes factors for model comparisons can be routinely computed as a by product of the simulation hitherto this calculation has proved extremely challenging our approach exploits the fact that the marginal density can be expressed as the prior times the likelihood function over the posterior density this simple identity holds for any parameter value an estimate of the posterior density is shown to be available if all complete conditional densities used in the gibbs sampler have closed form expressions to improve accuracy the posterior density is estimated at a high density point and the numerical standard error of resulting estimate is derived the ideas are applied to probit regression and finite mixture models

bayesian nonparametric spatial modeling with dirichlet process mixing customary modeling for continuous point referenced data assumes a gaussian process that is often taken to be stationary when such models are fitted within a bayesian framework the unknown parameters of the process are assumed to be random so a random gaussian process results here we propose a novel spatial dirichlet process mixture model to produce a random spatial process that is neither gaussian nor stationary we first develop a spatial dirichlet process model for spatial data and discuss its properties because of familiar limitations associated with direct use of dirichlet process models we introduce mixing by convolving this process with a pure error process we then examine properties of models created through such dirichlet process mixing in the bayesian framework we implement posterior inference using gibbs sampling spatial prediction raises interesting questions but these can be handled finally we illustrate the approach using simulated data as well as a dataset involving precipitati

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

generalized poststratification and importance sampling for subsampled markov chain monte carlo estimation benchmark estimation is motivated by the goal of producing an approximation to a posterior distribution that is better than the empirical distribution function this is accomplished by incorporating additional information into the construction of the approximation we focus here on generalized poststratification the most successful implementation of benchmark estimation in our experience we develop generalized poststratification for settings where the source of the simulation differs from the posterior that is to be approximated this allows us to use the techniques in settings where it is advantageous to draw from a distribution different than the posterior whether for exploration of the data and or model for algorithmic simplicity for improved convergence of the simulation or for improved estimation of selected features of the posterior we develop an asymptotic in simulation size theory for the estimators providing conditions under which central limit theorems hold the central limit theorems

comment on transport processes on fractal structures the scaling exponents for the depleted sierpinski gasket derived by havlin and ben avraham ibid vol 16 p l483 7 1983 are shown to differ sharply from those derived by the method of given and mandelbrot ibid vol 16 p l565 9 1983 the discrepancy is analysed and it is shown that the argument used by havlin and ben avraham is incorrect

bayesian nonparametric functional data analysis through density estimation in many modern experimental settings observations are obtained in the form of functions and interest focuses on inferences on a collection of such functions we propose a hierarchical model that allows us to simultaneously estimate multiple curves nonparametrically by using dependent dirichlet process mixtures of gaussians to characterize the joint distribution of predictors and outcomes function estimates are then induced through the conditional distribution of the outcome given the predictors the resulting approach allows for flexible estimation and clustering while borrowing information across curves we also show that the function estimates we obtain are consistent on the space of integrable functions as an illustration we consider an application to the analysis of conductivity and temperature at depth data in the north atlantic

tracking eigenvalues to the frontier of moduli space i convergence and spectral accumulation abstract we consider the spectral behavior of the laplace beltrami operator associated to a class of singular perturbations of a riemannian metric on a complete manifold the class of perturbations generalizes the well known opening node perturbation of teichmuller theory in particular we recover results of l ji and m zworski j func anal 114 1993 412 420 and s wolpert invent math 108 1992 91 129 from our more general methods

nonparametric bayesian analysis for assessing homogeneity in k l contingency tables with fixed right margin totals abstract in this work i postulate a nonparametric bayesian model for data that can be accommodated in a contingency table with fixed right margin totals this data structure usually arises when comparing different groups regarding classification probabilities for a number of categories i assume that cell count vectors for each group are conditionally independent with multinomial distribution given vectors of classification probabilities in turn these vectors of probabilities are assumed to be a sample from a distribution f and the prior distribution of f is assumed to be a dirichlet process centered on a probability measure and with weight c i also assume a prior distribution for c as a way of obtaining a better control on the clustering structure induced by the dirichlet process i use this setting to assess homogeneity of classification probabilities and propose a bayes factor i derive exact expressions for the relevant quantities these can be directly computed when the number of rows k i

bayesian model selection in finite mixtures by marginal density decompositions we consider the problem of estimating the number of components d and the unknown mixing distribution in a finite mixture model in which d is bounded by some fixed finite number n our approach relies on the use of a prior over the space of mixing distributions with at most n components by decomposing the resulting marginal density under this prior we discover a weighted bayes factor method for consistently estimating d that can be implemented by an iid generalized weighted chinese restaurant gwcr monte carlo algorithm we also discuss a gibbs sampling method the blocked gibbs sampler for estimating d and also the mixing distribution we show that our resulting posterior is consistent and achieves the frequentist optimal op n 1 4 rate of estimation we compare the performance of the new gwcr model selection procedure with that of the akaike information criterion and the bayes information criterion implemented through an em algorithm applications of our methods to five real datasets and simulation

nonparametric hierarchical bayes via sequential imputations we consider the empirical bayes estimation of a distribution using binary data via the dirichlet process let d denote a dirichlet process with being a finite measure on 0 1 instead of having direct samples from an unknown random distribution f from d we assume that only indirect binomial data are observable this paper presents a new interpretation of lo s formula and thereby relates the predictive density of the observations based on a dirichlet process model to likelihoods of much simpler models as a consequence the log likelihood surface as well as the maximum likelihood estimate of c 0 1 is found when the shape of is assumed known together with a formula for the fisher information evaluated at the estimate the sequential imputation method of kong liu and wong is recommended for overcoming computational difficulties commonly encountered in this area the related approximation formulas are provided an analysis of the tack data of beckett and diaconis which motivated this study is supplemented to illustrate our methods