some aspects of polya tree distributions for statistical modelling the definition and elementary properties of polya tree distributions are reviewed two theorems are presented showing that polya trees can be constructed to concentrate arbitrarily closely about any desired pdf and that polya tree priors can put positive mass in every relative entropy neighborhood of every positive density with finite entropy thereby satisfying a consistency condition such theorems are false for dirichlet processes models are constructed combining partially specified polya trees with other information such as monotonicity or unimodality it is shown how to compute bounds on posterior expectations over the class of all priors with the given specifications a numerical example is given a theorem of diaconis and freedman about dirichlet processes is generalized to polya trees allowing polya trees to be the models for errors in regression problems finally empirical bayes models using dirichlet processes are generalized to polya trees an example from berry and christensen is reanalyzed with a polya tree model

spatially dependent polya tree modeling for survival data with the proliferation of spatially oriented time to event data spatial modeling in the survival context has received increased recent attention a traditional way to capture a spatial pattern is to introduce frailty terms in the linear predictor of a semiparametric model such as proportional hazards or accelerated failure time we propose a new methodology to capture the spatial pattern by assuming a prior based on a mixture of spatially dependent polya trees for the baseline survival in the proportional hazards model thanks to modern markov chain monte carlo mcmc methods this approach remains computationally feasible in a fully hierarchical bayesian framework we compare the spatially dependent mixture of polya trees mpt approach to the traditional spatial frailty approach and illustrate the usefulness of this method with an analysis of iowan breast cancer survival data from the surveillance epidemiology and end results seer program of the national cancer institute our method provides better goodness of fit over the traditional alternatives as measured by log pseudo marginal likelihood lpml the deviance information criterion dic and full sample score fss statistics

approximation of a laminated microstructure for a rotationally invariant double well energy density we give error estimates for the approximation of a laminated microstructure which minimizes the energy int omega phi nabla v x dx for a rotationally invariant double well energy density phi a we present error estimates for the convergence of the deformation in l 2 the convergence of directional derivatives of the deformation in the twin planes the weak convergence of the deformation gradient the convergence of the microstructure or young measure of the deformation gradients and the convergence of nonlinear integrals of the deformation gradient

a bayesian semiparametric survival model with longitudinal markers we consider inference for data from a clinical trial of treatments for metastatic prostate cancer patients joined the trial with diverse prior treatment histories the resulting heterogeneous patient population gives rise to challenging statistical inference problems when trying to predict time to progression on different treatment arms inference is further complicated by the need to include a longitudinal marker as a covariate to address these challenges we develop a semiparametric model for joint inference of longitudinal data and an event time the proposed approach includes the possibility of cure for some patients the event time distribution is based on a nonparametric polya tree prior for the longitudinal data we assume a mixed effects model incorporating a regression on covariates in a nonparametric event time model in general and for a polya tree model in particular is a challenging problem we exploit the fact that the covariate itself is a random variable we achieve an implementation of the desired regression by factoring the joint model for the event time and the longitudinal outcome into a marginal model for the event time and a regression of the longitudinal outcomes on the event time i e we implicitly model the desired regression by modeling the reverse conditional distribution

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

bayesian semiparametric median regression modeling median regression models become an attractive alternative to mean regression models when employing flexible families of distributions for the errors classical approaches are typically algorithmic with desirable properties emerging asymptotically however nonparametric error models may be most attractive in the case of smaller sample sizes where parametric specifications are difficult to justify hence a bayesian approach enabling exact inference given the observed data may be appealing in this context there is little bayesian work we develop two fully bayesian modeling approaches employing mixture models for the errors in a median regression model the associated families of error distributions allow for increased variability skewness and flexible tail behavior the first family is semiparametric with extra variability captured nonparametrically through mixing and skewness handled parametrically the second family a fully nonparametric one includes all unimodal densities on the real line with

numerical approximation of the solution of variational problem with a double well potential variational problems with a double well potential are not lower semicontinuous and can fail to attain a minimum value rather the gradients of minimizing sequences do not converge pointwise and can have oscillations however the gradients do converge in the weak topology i e their local spatial averages converge such functionals arise in the description of equilibria of crystals or other ordered materials stable configurations for solid crystals which have symmetry related martensitic energy wells have a fine scale microstructure which can be related to the oscillations that energy minimizing sequences for the bulk energy exhibit an analysis is given of approximation methods for variational problems with a double well potential to give a rigorous justification for the use of such numerical methods to model the behavior of this class of solid crystals

estimates for numerical approximations of rank one convex envelopes summary we present a convergence analysis of an algorithm for the numerical computation of the rank one convex envelope of a function f mathrm m m times n rightarrow mathbb r a rate of convergence for the scheme is established and numerical experiments are presented to illustrate the analytical results and applications of the algorithm

the relaxation of non quasiconvex variational integrals we show that the steepest descent algorithm in connection with wiggly energies yields minimizing sequences that converge to a global minimum of the associated non quasiconvex variational integrals we introduce a multi level infinite dimensional variant of the steepest descent algorithm designed to compute complex microstructures by forming non smooth minimizers from the smooth initial guesses we apply this multi level method to the minimization of the variational problems associated with martensitic branching

markov chains for exploring posterior distributions several markov chain methods are available for sampling from a posterior distribution two important examples are the gibbs sampler and the metropolis algorithm in addition several strategies are available for constructing hybrid algorithms this paper outlines some of the basic methods and strategies and discusses some related theoretical and practical issues on the theoretical side results from the theory of general state space markov chains can be used to obtain convergence rates laws of large numbers and central limit theorems for estimates obtained from markov chain methods these theoretical results can be used to guide the construction of more efficient algorithms for the practical use of markov chain methods standard simulation methodology provides several variance reduction techniques and also give guidance on the choice of sample size and allocation

semiparametric regression modeling with mixtures of berkson and classical error with application to fallout from the nevada test site nous construisons des methodes bayesiennes pour une modelisation semi parametrique d une fonction de regression monotone quand les predicteurs sont mesures avec une erreur classique une erreur de berkson ou les deux de telles methodes demandent une distribution pour le predicteur non observe variable latente distribution que nous modelisons aussi de facon semi parametrique de telles combinaisons de methodes semi parametriques pour la reponse a des doses aussi bien que pour la distribution de la variable latente n ont pas ete etudiees dans la litterature sur les erreurs de mesure quelle que soit la forme de l erreur de mesure de plus nos methodes proposent une nouvelle approche des problemes ou l erreur de mesure combine une composante de berkson et une composante classique ces methodes sont generales mais nous les developpons autour d une application particuliere qui est l etude des maladies de la thyroide en relation avec les radiations venant du site de test du nevada nous utilisons ces donnees pour illustrer nos methodes lesquelles suggerent une estimation ponctuelle moyenne a posteriori du risque relatif a des doses pres de deux fois plus elevee que celle obtenue par les analyses precedentes mais suggere aussi une bien plus grande incertitude sur le risque relatif

w1 p quasiconvexity and variational problems for multiple integrals abstract variational problems for the multiple integral i u g u x dx where r m and u r n are studied a new condition on g called w1 p quasiconvexity is introduced which generalizes in a natural way the quasiconvexity condition of c b morrey it being shown in particular to be necessary for sequential weak lower semicontinuity of i in w 1 p r n and for the existence of minimizers for certain related integrals counterexamples are given concerning the weak continuity properties of jacobians in w 1 p r n p n m an existence theorem for nonlinear elastostatics is proved under optimal growth hypotheses

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

on the numerical analysis of non convex variational problems we discuss a numerical method for finding young measure valued minimizers of non convex variational problems to have any hope of a convergence theorem one must work in a setting where the minimizer is unique and minimizing sequences converge strongly this paper has two main goals i we specify a method for producing strongly convergent minimizing sequences despite the failure of strict convexity and ii we show how uniqueness of the young measure can be parlayed into a numerical convergence theorem the treatment of ii is done in the setting of two model problems one involving scalar valued functions and a multiwell energy the other from micromagnetics

numerical analysis of oscillations in multiple well problems variational problems which fail to be convex occur often in the study of ordered materials such as crystals in these problems the energy density for the material has multiple potential wells in this paper we study multiple well problems by first determining the analytic properties of energy minimizing sequences and then by estimating the continuous problem by an approximation using piecewise linear finite elements we show that even when there is no minimizer of the energy the approximations still take on a predictable structure

numerical analysis of oscillations in a nonconvex problem related to image selective smoothing we study some numerical properties of a nonconvex variational problem which arises as the continuous limit of a discrete optimization method designed for the smoothing of images with preservation of discontinuities the functional that has to be minimized fails to attain a minimum value instead minimizing sequences develop gradient oscillations which allow them to reduce the value of the functional the oscillations of the gradient exhibit analogies with microstructures in ordered materials the pattern of the oscillations is analysed numerically by using discrete parametrized measures

numerical approximations in variational problems with potential wells in this paper some numerical aspects of variational problems which fail to be convex are studied it is well known that for such a problem in general the infimum of the energy the functional that has to be minimized fails to be attained instead minimizing sequences develop oscillations which allow them to decrease the energy it is shown that there exists a minimizes for an approximation of the problem and the oscillations in the minimizing sequence are analyzed it is also shown that these minimizing sequences choose their gradients in the vicinity of the wells with a probability which tends to be constant an estimate of the approximate deformation as it approximates a measure and some numerical results are also given

the multivariate beta process and an extension of the polya tree model we introduce a novel stochastic process that we term the multivariate beta process the process is defined for modelling dependent random probabilities and has beta marginal distributions we use this process to define a probability model for a family of unknown distributions indexed by covariates the marginal model for each distribution is a polya tree prior an important feature of the proposed prior is the easy centring of the nonparametric model around any parametric regression model we use the model to implement nonparametric inference for survival distributions the nonparametric model that we introduce can be adopted to extend the support of prior distributions for parametric regression models

modelling accelerated failure time with a dirichlet process the relationship between survival times and covariates is modelled via the accelerated failure time model with a dirichlet process prior assumed for the underlying baseline survival fucntion emphasis is placed on estimating the survival distribution for future individuals with given covariates a method of estimating regression coefficients which uses the marginal distribution for observed censored data is given

on the symmetric square unstable local transfer we prove the fundamental lemma for spherical functions with respect to the natural induction lifting frompgl 2 topgl 3 which appears as the unstable counterpart of the stable symmetric square lifting fromsl 2 topgl 3 see iv for an introduction to this project and vi for the final results thus spherical functions onpgl 2 andpgl 3 which correspond to each other by satisfying an elementary representation theoretic relation are shown to have matching orbital integrals the proof of this local statement is based on an application of the global trace formula

bayesian semiparametric proportional odds models methodology for implementing the proportional odds regression model for survival data assuming a mixture of finite polya trees mpt prior on baseline survival is presented extensions to frailties and generalized odds rates are discussed although all manner of censoring and truncation can be accommodated we discuss model implementation regression diagnostics and model comparison for right censored data an advantage of the mpt model is the relative ease with which predictive densities survival and hazard curves are generated much discussion is devoted to practical implementation of the proposed models and a novel mcmc algorithm based on an approximating parametric normal model is developed a modest simulation study comparing the small sample behavior of the mpt model to a rank based estimator and a real data example is presented

on the symmetric square twisted trace formula abstract we compute a trace formula for a test function on pgl 3 a twisted by the outer automorphism g j t g 1 j j 0 1 1 1 0 the resulting formula is then compared with trace formulae for h h 0 0 sl 2 and h 1 pgl 2 and matching functions f 0 and f 1 thereof we obtain a trace formula identity which plays a key role in our study of the symmetric square lifting from h a to g a our formulae are remarkably simple due to the introduction of a new concept of a regular function this eliminates the singular and weighted integrals in the trace formulae

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

a bayesian semiparametric accelerated failure time model a bayesian semiparametric approach is described for an accelerated failure time model the error distribution is assigned a polya tree prior and the regression parameters a noninformative hierarchical prior two cases are considered the first assumes error terms are exchangeable the second assumes that error terms are partially exchangeable a markov chain monte carlo algorithm is described to obtain a predictive distribution for a future observation given both uncensored and censored data

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

on the symmetric square orbital integrals j tg 1j for g in g the elements fi 8 of g f are called stably a conjugate if there is h in g f resp g with h a h1 conjugacy and stable conjugacy is defined analogously for h f and iii f on omitting a in sect 1 we show that the map n described in 1 5 below is a well defined bijection called the norm map from the set of stable a conjugacy classes in g f to the set of stable conjugacy classes in it f we also define a surjection n1 from the subset of this set described in 1 2 3 4 to the set of stable conjugacy classes in hi f in the case 4 n1 actually relates conjugacy classes let be an element of g f the set of eigenvalues of a is of the form 2 1 2 1 1 if 6a 1 put n 5 1 and ni 1 2 if 6a 6 is a non trivial unipotent let n6 and ni be the non trivial unipotent classes 3 if the eigenvalues 2 1 2 1 of 6a 6 are distinct let n6 be the class in h f determined by the eigenvalues 2 2 1 and ni6 the class in hi f with eigenvalues 4 1

combining information from several experiments with nonparametric priors summary this paper considers combining information from several experiments when the experiments can be summarised via a parameter value the structure of this set of parameters in terms of independence exchangeability partial exchangeability etc is assumed to be unknown and a finite number of possible structures are entertained each with an associated prior weight representing the degree of belief in that structure crucial is the criterion by which these structures are selected the final inference for the parameter values is taken to be the average with respect to the posterior weights of the values obtained from each structure this is performed within a bayesian nonparametric framework where the form of the prior distribution for the parameters is unrestricted therefore we do not assume that the distributions associated with a partial structure are from the same family different types of experiment suggest different types of distributions of parameters associated with each type of experiment

on the symmetric square applications of a trace formula in this paper we prove the existence of the symmetric square lifting of admissible and of automorphic representations from the group sl 2 to the group pgl 3 complete local results are obtained relating the character of an sl 2 packet with the twisted character of self contragredient pgl 3 modules our global results relate packets of cuspidal representations of sl 2 with a square integrable component and self contragredient automorphic pgl 3 modules with a component coming from a square integrable one the sharp results which concern sl 2 rather than gl 2 are afforded by the usage of the trace formula the surjectivity and injectivity of the correspondence implies that any self contragredient automorphic pgl 3 module as above is a lift and that the space of cuspidal sl 2 modules with a square integrable component admits multiplicity one theorem and rigidity strong multiplicity one theorem for packets and not for individual representations

semiparametric bayesian analysis of structural equation models with fixed covariates latent variables play the most important role in structural equation modeling in almost all existing structural equation models sems it is assumed that the distribution of the latent variables is normal as this assumption is likely to be violated in many biomedical researches a semiparametric bayesian approach for relaxing it is developed in this paper in the context of sems with covariates we provide a general bayesian framework in which a semiparametric hierarchical modeling with an approximate truncation dirichlet process prior distribution is specified for the latent variables the stick breaking prior and the blocked gibbs sampler are used for efficient simulation in the posterior analysis the developed methodology is applied to a study of kidney disease in diabetes patients a simulation study is conducted to reveal the empirical performance of the proposed approach supplementary electronic material for this paper is available in wiley interscience at http www mrw interscience wiley com suppmat 1097 0258 suppmat

analysis of the finite element approximation of microstructure in micromagnetics the solution to nonconvex variational problems is often characterized by microstructure the variational problem for the magnetization field in micromagnetics has a nonconvex constraint and energy minimizing sequences of magnetization fields can have oscillations whose scale converges to zero but whose amplitude remains finite it is shown that the finite element approximation of the magnetization field for this variational problem does not converge pointwise as the mesh is refined but that nonlinear functions of the approximate magnetization fields converge weakly or equivalently all local spatial averages of nonlinear functions of the approximate magnetization field converge as the mesh is refined which implies the convergence as the mesh is refined of the probability distribution of the approximate magnetization fields in local spatial domains a norm is given to measure the convergence of this microstructure and we prove a rate of convergence in this norm

semiparametric bayesian survival analysis using models with log linear median summary we present a novel semiparametric survival model with a log linear median regression function as a useful alternative to existing semiparametric models our large model class has many important practical advantages including interpretation of the regression parameters via the median and the ability to address heteroscedasticity we demonstrate that our modeling technique facilitates the ease of prior elicitation and computation for both parametric and semiparametric bayesian analysis of survival data we illustrate the advantages of our modeling as well as model diagnostics via a reanalysis of a small cell lung cancer study results of our simulation study provide further support for our model in practice

on the symmetric square definitions and lemmas we define the symmetric square lifting for admissible and automorphic representations from the group h h 0 sl 2 to the group g pgl 3 and derive its basic properties this lifting is defined by means of shintani character relations the definition is suggested by the computation of orbital integrals stable and unstable in our on the symmetric square orbital integrals math ann 279 1987 173 193 it is compatible with dual group homomorphisms 0 h g and 1 h 1 g where h 1 pgl 2 the lifting is proven for induced trivial and special representations and both spherical functions and orthogonality relations of characters are studied

on the convergence of finite element approximations of a relaxed variational problem the accuracy of finite element approximations to a convex but not strictly convex variational problem is considered convergence is proved for a finite element approximation of a particular vector field related to the solution in a special one dimensional case 0 h convergence is shown for a piecewise linear approximation of the derivative h denotes the size of each element domain numerical results are also presented for this one dimensional case

assessment and propagation of model uncertainty in most examples of inference and prediction the expression of uncertainty about unknown quantities y on the basis of known quantities x is based on a model m that formalizes assumptions about how x and y are related m will typically have two parts structural assumptions s such as the form of the link function and the choice of error distribution in a generalized linear model and parameters whose meaning is specific to a given choice of s it is common in statistical theory and practice to acknowledge parametric uncertainty about given a particular assumed structure s it is less common to acknowledge structural uncertainty about s itself a widely used approach involves enlisting the aid of x to specify a plausible single best choice s for s and then proceeding as if s were known to be correct in general this approach fails to assess and propagate structural uncertainty fully and may lead to miscalibrated uncertainty assessments about y given x when miscalibration occurs it will often result in understatement of inferential or predictive uncertainty about y leading to inaccurate scientific summaries and overconfident decisions that do not incorporate sufficient hedging against uncertainty in this paper i discuss a bayesian approach to solving this problem that has long been available in principle but is only now becoming routinely feasible by virtue of recent computational advances and examine its implementation in examples that involve forecasting the price of oil and estimating the chance of catastrophic failure of the us space shuttle

bayesian nonparametric meta analysis using polya tree mixture models summary a common goal in meta analysis is estimation of a single effect measure using data from several studies that are each designed to address the same scientific inquiry because studies are typically conducted in geographically disperse locations recent developments in the statistical analysis of meta analytic data involve the use of random effects models that account for study to study variability attributable to differences in environments demographics genetics and other sources that lead to heterogeneity in populations stemming from asymptotic theory study specific summary statistics are modeled according to normal distributions with means representing latent true effect measures a parametric approach subsequently models these latent measures using a normal distribution which is strictly a convenient modeling assumption absent of theoretical justification to eliminate the influence of overly restrictive parametric models on inferences we consider a broader class of random effects distributions we develop a novel hierarchical bayesian nonparametric polya tree mixture ptm model we present methodology for testing the ptm versus a normal random effects model these methods provide researchers a straightforward approach for conducting a sensitivity analysis of the normality assumption for random effects an application involving meta analysis of epidemiologic studies designed to characterize the association between alcohol consumption and breast cancer is presented which together with results from simulated data highlight the performance of ptms in the presence of nonnormality of effect measures in the source population

regression analysis using dependent polya trees many commonly used models for linear regression analysis force overly simplistic shape and scale constraints on the residual structure of data we propose a semiparametric bayesian model for regression analysis that produces data driven inference by using a new type of dependent polya tree prior to model arbitrary residual distributions that are allowed to evolve across increasing levels of an ordinal covariate e g time in repeated measurement studies by modeling residual distributions at consecutive covariate levels or time points using separate but dependent polya tree priors distributional information is pooled while allowing for broad pliability to accommodate many types of changing residual distributions we can use the proposed dependent residual structure in a wide range of regression settings including fixed effects and mixed effects linear and nonlinear models for cross sectional prospective and repeated measurement data a simulation study illustrates the flexibility of our novel semiparametric regression model to accurately capture evolving residual distributions in an application to immune development data on immunoglobulin g antibodies in children our new model outperforms several contemporary semiparametric regression models based on a predictive model selection criterion copyright 2013 john wiley sons ltd

survival analysis with median regression models abstract the median is a simple and meaningful measure for the center of a long tailed survival distribution to examine the covariate effects on survival a natural alternative to the usual mean regression model is to regress the median of the failure time variable or a transformation thereof on the covariates in this article we propose semiparametric procedures to make inferences for such median regression models with possibly censored observations our proposals can be implemented efficiently using a simulated annealing algorithm numerical studies are conducted to show the advantage of the new procedures over some recently developed methods for the accelerated failure time model a special type of mean regression models in the survival analysis the proposals discussed in the article are illustrated with a lung cancer data set

mixtures of polya trees for flexible spatial frailty survival modelling mixtures of polya trees offer a very flexible nonparametric approach for modelling time to event data many such settings also feature spatial association that requires further sophistication either at the point level or at the lattice level in this paper we combine these two aspects within three competing survival models obtaining a data analytic approach that remains computationally feasible in a fully hierarchical bayesian framework using markov chain monte carlo methods we illustrate our proposed methods with an analysis of spatially oriented breast cancer survival data from the surveillance epidemiology and end results program of the national cancer institute our results indicate appreciable advantages for our approach over competing methods that impose unrealistic parametric assumptions ignore spatial association or both

modeling regression error with a mixture of polya trees we model the error distribution in the standard linear model as a mixture of absolutely continuous polya trees constrained to have median 0 by considering a mixture we smooth out the partitioning effects of a simple polya tree and the predictive error density has a derivative everywhere except 0 the error distribution is centered around a standard parametric family of distributions and thus may be viewed as a generalization of standard models in which important data driven features such as skewness and multimodality are allowed by marginalizing the polya tree exact inference is possible up to markov chain monte carlo error

numerical analysis of oscillations in nonconvex problems we study numerically the pattern of the minimizing sequences of nonconvex problems which do not admit a minimizer