information theoretic determination of minimax rates of convergence we present some general results determining minimax bounds on statistical risk for density estimation based on certain information theoretic considerations these bounds depend only on metric entropy conditions and are used to identify the minimax rates of convergence

on the uniform consistency of bayes estimates for multinomial probabilities a k sided die is thrown n times to estimate the probabilities 1 k of landing on the various sides the mle of is the vector of empirical proportions p p 1 p k consider a set of bayesians that put uniformly positive prior mass on all reasonable subsets of the parameter space their posterior distributions will be uniformly concentrated near p sharp bounds are given using entropy these bounds apply to all sample sequences there are no exceptional null sets

adaptive nonparametric bayesian inference using location scale mixture priors we study location scale mixture priors for nonparametric statistical problems including multivariate regression density estimation and classification we show that a rate adaptive procedure can be obtained if the prior is properly constructed in particular we show that adaptation is achieved if a kernel mixture prior on a regression function is constructed using a gaussian kernel an inverse gamma bandwidth and gaussian mixing weights

spineurs operateurs de dirac et variations de metriques in this article a geometric process to compare spinors for different metrics is constructed it makes possible the extension to spinor fields of a variant of the lie derivative called the metric lie derivative giving a geometric approach to a construction originally due to yvette kosmann the comparison of spinor fields for two different riemannian metrics makes the study of the variation of dirac operators feasible for this it is crucial to take into account the fact that the bundle in which the sections acted upon by the dirac operators take their values is changing we also give the formulas for the change in the eigenvalues of the dirac operator we conclude by giving a few cases in which an eigenvalue is stationary

on a spin conformal invariant on manifolds with boundary let m be an n dimensional connected compact manifold with non empty boundary equipped with a riemannian metric g a spin structure and a chirality operator we define and study some properties of a spin conformal invariant given by lambda rm min m partial m underset overline g in g rm inf lambda 1 pm overline g rm vol m overline g frac 1 n where lambda 1 pm overline g is the smallest eigenvalue of the dirac operator under the chiral bag boundary condition mathbb b pm overline g more precisely we show that if n 2 then lambda rm min m partial m leq lambda rm min mathbb s n partial mathbb s n

smooth yamabe invariant and surgery we prove a surgery formula for the smooth yamabe invariant sigma m of a compact manifold m assume that n is obtained from m by surgery of codimension at least 3 we prove the existence of a positive number lambda n depending only on the dimension n of m such that sigma n geq min sigma m lambda n

on the consistency of bayes estimates on etudie les proprietes de frequence des regles de bayes avec une attention particuliere a la consistence

a constructive representation of univariate skewed distributions we introduce a general perspective on the introduction of skewness into symmetric distributions through inverse probability integral transformations we provide a constructive representation of skewed distributions where the skewing mechanism and the original symmetric distributions are specified separately we study the effects of the skewing mechanism on e g modality tail behavior and the amount of skewness generated the representation is used to introduce novel classes of skewed distributions where we induce certain prespecified characteristics through particular choices of the skewing mechanism finally we use a bayesian linear regression framework to compare the new classes with some existing distributions in the context of two empirical examples

exact mean integrated squared error an exact and easily computable expression for the mean integrated squared error mise for the kernel estimator of a general normal mixture density is given for gaussian kernels of arbitrary order this provides a powerful new way of understanding density estimation which complements the usual tools of simulation and asymptotic analysis the family of normal mixture densities is very flexible and the formulae derived allow simple exact analysis for a wide variety of density shapes a number of applications of this method giving important new insights into kernel density estimation are presented among these is the discovery that the usual asymptotic approximations to the mise can be quite inaccurate especially when the underlying density contains substantial fine structure and also strong evidence that the practical importance of higher order kernels is surprisingly small for moderate sample sizes 1 introduction substantial research has been devoted to kernel density estimation this is because it provides a simple yet appealing context in which to study problems and issues that arise in all types of nonparametric curve estimation this includes regression spectral density and hazard estimation and also a variety of other estimators including histograms splines and orthogonal series three important and useful tools for understanding the behavior of nonparametric curve estimators are asymptotic analysis simulation and numerical calculation of error criteria each of these methods provides many useful insights into the complicated structure present in the study of curve estimation however each has its limitations as well rhe strength of asymptotic analysis is that it frequently allows simultaneous study of many different specific examples through general results applying to entire classes of settings the weakness of asymptotics is that they only describe behavior in the limit this is still very useful in many situations because the asymptotics describe the actual situation quite well however it is less useful when the asymptotics have not yet kicked in that is in studying situations where the asymptotically dominant effect has not taken over yet perhaps the biggest drawback to asymptotics is that it is very difficult to determine in a given situation which of these possibilities is occurring

nonlinear dirac equations with critical nonlinearities on compact spin manifolds we study some basic analytical problems for nonlinear dirac equations involving critical sobolev exponents on compact spin manifolds their solutions are obtained as critical points of certain strongly indefinite functionals defined on h1 2 spinors with critical growth we prove the existence of a non trivial solution for the brezis nirenberg type problem when the dimension m of the manifold is larger than 3 we also prove a global compactness result for the associated palais smale sequences and the regularity of l2mm 1 weak solutions

rationalizability and correlated equilibria we discuss the unity between the two standard approaches to noncooperative solution concepts for games the decision theoretic approach starts from the assumption that the rationality of the players is common knowledge this leads to the notion of correlated rationalizability it is shown that correlated rationalizability is equivalent to a posteriori equilibrium a refinement of subjective correlated equilibrium hence a decision theoretic justification for the equilibrium approach to game theory is provided an analogous equivalence result is proved between independent rationalizability which is the appropriate concept if each player believes that the others act independently and conditionally independent a posteriori equilibrium a characterization of nash equilibrium is also provided

the markov moment problem and de finetti s theorem part ii the markov moment problem is to characterize sequences admitting the representation s n 0 1 x nf x dx where f x is a probability density on 0 1 and 0 f x c for almost all x there are well known characterizations through complex systems of non linear inequalities on s n necessary and sufficient linear conditions are the following s 0 1 and for all and here is the forward difference operator this result is due to hausdorff we give a new proof with some ancillary results for example characterizing monotone densities then we make the connection to de finetti s theorem with characterizations of the mixing measure

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

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

a spinorial analogue of aubin s inequality let m g be a compact riemannian spin manifold of dimension 2 for any metric tilde g conformal to g we denote by tilde lambda the first positive eigenvalue of the dirac operator on m tilde g sigma we show that rm inf tilde g in g tilde lambda rm vol m tilde g 1 n leq n 2 rm vol s n 1 n this inequality is a spinorial analogue of aubin s inequality an important inequality in the solution of the yamabe problem the inequality is already known in the case n 3 and in the case n 2 ker d 0 our proof also works in the remaining case n 2 ker d 0 with the same method we also prove that any conformal class on a riemann surface contains a metric with 2 tilde lambda 2 leq tilde mu where tilde mu denotes the first positive eigenvalue of the laplace operator

a conformal lower bound for the smallest eigenvalue of the dirac operator and killing spinors on a riemannian spin manifold we give a lower bound for the square of the eigenvalues of the dirac operator by the smallest eigenvalue of the conformal laplacian the yamabe operator we prove in the limiting case that the eigenspinor field is a killing spinor i e parallel with respect to a natural connection in particular if the scalar curvature is positive the eigenspinor field is annihilated by harmonic forms and the metric is einstein

existence uniqueness and a constructive solution algorithm for a class of finite markov moment problems we consider a class of finite markov moment problems with an arbitrary number of positive and negative branches we show criteria for the existence and uniqueness of solutions and we characterize in detail the nonunique solution families moreover we present a constructive algorithm to solve the moment problems numerically and prove that the algorithm computes the right solution

conditioning likelihood and coherence a review of some foundational concepts 2000 conditioning likelihood and coherence a review of some foundational concepts journal of the american statistical association vol 95 no 452 pp 1340 1346

dirac harmonic maps we introduce a functional that couples the nonlinear sigma model with a spinor field open image in new window in two dimensions it is conformally invariant the critical points of this functional are called dirac harmonic maps we study some geometric and analytic aspects of such maps in particular a removable singularity theorem

stationary solutions for the 2d critical dirac equation with kerr nonlinearity abstract in this paper we prove the existence of an exponentially localized stationary solution for a two dimensional cubic dirac equation it appears as an effective equation in the description of nonlinear waves for some condensed matter bose einstein condensates and nonlinear optics optical fibers systems the nonlinearity is of kerr type that is of the form 2 and thus not lorenz invariant we solve compactness issues related to the critical sobolev embedding h 1 2 r 2 c 2 l 4 r 2 c 4 thanks to a particular radial ansatz our proof is then based on elementary dynamical systems arguments

misspecification in infinite dimensional bayesian statistics we consider the asymptotic behavior of posterior distributions if the model is misspecified given a prior distribution and a random sample from a distribution p 0 which may not be in the support of the prior we show that the posterior concentrates its mass near the points in the support of the prior that minimize the kullback leibler divergence with respect to p 0 an entropy condition and a prior mass condition determine the rate of convergence the method is applied to several examples with special interest for infinite dimensional models these include gaussian mixtures nonparametric regression and parametric models

on generating functions of hausdorff moment sequences the class of generating functions for completely monotone se quences moments of finite positive measures on 0 1 has an elegant char acterization as the class of pick functions analytic and positive on 1 we establish this and another such characterization and develop a variety of consequences in particular we characterize generating functions for moments of convex and concave probability distribution functions on 0 1 also we provide a simple analytic proof that for any real p and r with p 0 the fuss catalan or raney numbers r pn r pn r

on rates of convergence for posterior distributions in infinite dimensional models this paper introduces a new approach to the study of rates of convergence for posterior distributions it is a natural extension of a recent approach to the study of bayesian consistency in particular we improve on current rates of convergence for models including the mixture of dirichlet process model and the random bernstein polynomial model

local sensitivity of posterior expectations we investigate the degree to which posterior expectations are sensitive to prior distributions using a local method based on functional differentiation invariance considerations suggest a family of norms which can be used to measure perturbations to the prior the sensitivity measure is seen to depend heavily on the choice of norm asymptotic results suggest which norm will yield the most useful results in practice we find that to maintain asymptotically sensible behaviour it is necessary to reduce the richness of the class of prior perturbations as the dimension of the parameter space increases jeffreys prior is characterized as the prior to which inference is least sensitive

a reference bayesian test for nested hypotheses and its relationship to the schwarz criterion abstract to compute a bayes factor for testing h 0 0 in the presence of a nuisance parameter priors under the null and alternative hypotheses must be chosen as in bayesian estimation an important problem has been to define automatic or reference methods for determining priors based only on the structure of the model in this article we apply the heuristic device of taking the amount of information in the prior on equal to the amount of information in a single observation then after transforming to be null orthogonal to we take the marginal priors on to be equal under the null and alternative hypotheses doing so and taking the prior on to be normal we find that the log of the bayes factor may be approximated by the schwarz criterion with an error of order o p n rather than the usual error of order o p 1 this result suggests the schwarz criterion should provide sensible approximate solutions to bayesian testing problems at least when the hypotheses are nested when

filtered variate prior distributions for histogram smoothing abstract we develop prior distributions for histogram inference favoring smooth population frequencies that is probability vectors with small differences for neighboring categories we give a theory of prior random probability vectors representable as a linear transform or filter of a standard random probability vector or equivalently a random weighted average of nonrandom smooth probability vectors promising methods of prior assessment are given based on elicitation of a list of typically smooth probability vectors the empirical moments of which can then be matched by the mean vector and variance matrix of a constructed continuous type filtered variate prior distribution

risk bounds for model selection via penalization performance bounds for criteria for model selection are devel oped using recent theory for sieves the model selection criteria are based on an empirical loss or contrast function with an added penalty term moti vated by empirical process theory and roughly proportional to the number of parameters needed to describe the model divided by the number of ob servations most of our examples involve density or regression estimation settings and we focus on the problem of estimating the unknown density or regression function we show that the quadratic risk of the minimum penal ized empirical contrast estimatoris bounded by an index of the accuracy of the sieve this accuracy index quanties the trade off among the candidate models between the approximation error and parameter dimension relative to sample size if we choose a list of models which exhibit good approximation prop erties with respect to different classes of smoothness the estimator can be simultaneously minimax rate optimal in each of those classes this is what is usually called adaptation the type of classes of smoothness in which one gets adaptation depends heavily on the list of models if too many models are involved in order to get accurate approximation of many wide classes of functions simultaneously it may happen that the estimator is only approx

lower eigenvalue estimates for dirac operators we derive upper eigenvalue estimates for generalized dirac operators on closed riemannian manifolds in the case of the classical dirac operator the estimates on the first eigenvalues are sharp for spheres of constant curvature

probability inequalities for likelihood ratios and convergence rates of sieve mles let y 1 y n be independent identically distributed with density p 0 and let f be a space of densities we show that the supremum of the likelihood ratios i 1 n p y i p 0 y i where the supremum is over p f with p l 2 p 0 1 2 2 e is exponentially small with probability exponentially close to 1 the exponent is proportional to ne 2 the only condition required for this to hold is that e exceeds a value determined by the bracketing hellinger entropy of f a similar inequality also holds if we replace f by f n and p 0 by q n where q n is an approximation to p 0 in a suitable sense these results are applied to establish rates of convergence of sieve mles furthermore weak conditions are given under which the optimal rate e n defined by h e n f ne n 2 where h f is the hellinger entropy of f is nearly achievable by sieve estimators

the spinorial tau invariant and 0 dimensional surgery let m be a compact manifold with a metric g and with a fixed spin structure chi let lambda 1 g be the first non negative eigenvalue of the dirac operator on m g chi we set tau m chi sup inf lambda 1 g where the infimum runs over all metrics g of volume 1 in a conformal class g 0 on m and where the supremum runs over all conformal classes g 0 on m let m chi be obtained from m chi by 0 dimensional surgery we prove that tau m chi geq tau m chi as a corollary we can calculate tau m chi for any riemann surface m

bayesian estimation of the spectral density of a time series this article describes a bayesian approach to estimating the spectral density of a stationary time series a nonparametric prior on the spectral density is described through bernstein polynomials because the actual likelihood is very complicated a pseudoposterior distribution is obtained by updating the prior using the whittle likelihood a markov chain monte carlo algorithm for sampling from this posterior distribution is described that is used for computing the posterior mean variance and other statistics a consistency result is established for this pseudoposterior distribution that holds for a short memory gaussian time series and under some conditions on the prior to prove this asymptotic result a general consistency theorem of schwartz is extended for a triangular array of independent nonidentically distributed observations this extension is also of independent interest a simulation study is conducted to compare the proposed method with some existing methods the method is illustrated with

frequentist accuracy of bayesian estimates in the absence of relevant prior experience popular bayesian estimation techniques usually begin with some form of uninformative prior distribution intended to have minimal inferential influence bayes rule will still produce nice looking estimates and credible intervals but these lack the logical force attached to experience based priors and require further justification this paper concerns the frequentist assessment of bayes estimates a simple formula is shown to give the frequentist standard deviation of a bayesian point estimate the same simulations required for the point estimate also produce the standard deviation exponential family models make the calculations particularly simple and bring in a connection to the parametric bootstrap

distribution functions of means of a dirichlet process let be a random probability measure chosen by a dirichlet process on r b with parameter and such that x dx turns out to be a finite random variable the main concern of this paper is the statement of a suitable expression for the distribution function of that random variable such an expression is deduced through an extension of a procedure based on the use of generalized stieltjes transforms

bayesian posteriors for arbitrarily rare events we study how much data a bayesian observer needs to correctly infer the relative likelihoods of two events when both events are arbitrarily rare each period either a blue die or a red die is tossed the two dice land on side 1 with unknown probabilities p 1 and q 1 which can be arbitrarily low given a data generating process where p 1 c q 1 we are interested in how much data are required to guarantee that with high probability the observer s bayesian posterior mean for p 1 exceeds 1 c times that for q 1 if the prior densities for the two dice are positive on the interior of the parameter space and behave like power functions at the boundary then for every 0 there exists a finite n so that the observer obtains such an inference after n periods with probability at least 1 whenever n p 1 n the condition on n and p 1 is the best possible the result can fail if one of the prior densities converges to zero exponentially fast at the boundary

fully nonparametric regression for bounded data using dependent bernstein polynomials abstractwe propose a novel class of probability models for sets of predictor dependent probability distributions with bounded domain the proposal extends the dirichlet bernstein prior for single density estimation by using dependent stick breaking processes a general model class and two simplified versions are discussed in detail appealing theoretical properties such as continuity association structure marginal distribution large support and consistency of the posterior distribution are established for all models the behavior of the models is illustrated using simulated and real life data the simulated data are also used to compare the proposed methodology to existing methods supplementary materials for this article are available online

accelerated hazards model based on parametric families generalized with bernstein polynomials a transformed bernstein polynomial that is centered at standard parametric families such as weibull or log logistic is proposed for use in the accelerated hazards model this class provides a convenient way towards creating a bayesian nonparametric prior for smooth densities blending the merits of parametric and nonparametric methods that is amenable to standard estimation approaches for example optimization methods in sas or r can yield the posterior mode and asymptotic covariance matrix this novel nonparametric prior is employed in the accelerated hazards model which is further generalized to time dependent covariates the proposed approach fares considerably better than previous approaches in simulations data on the effectiveness of biodegradable carmustine polymers on recurrent brain malignant gliomas is investigated

the consistency of posterior distributions in nonparametric problems we give conditions that guarantee that the posterior probability of every hellinger neighborhood of the true distribution tends to 1 almost surely the conditions are 1 a requirement that the prior not put high mass near distributions with very rough densities and 2 a requirement that the prior put positive mass in kullback leibler neighborhoods of the true distribution the results are based on the idea of approximating the set of distributions with a finite dimensional set of distributions with sufficiently small hellinger bracketing metric entropy we apply the results to some examples

new approaches to bayesian consistency we use martingales to study bayesian consistency we derive sufficient conditions for both hellinger and kullback leibler consistency which do not rely on the use of a sieve alternative sufficient conditions for hellinger consistency are also found and demonstrated on examples

bayesian regression with nonparametric heteroskedasticity abstract this paper studies large sample properties of a semiparametric bayesian approach to inference in a linear regression model the approach is to model the distribution of the regression error term by a normal distribution with the variance that is a flexible function of covariates the main result of the paper is a semiparametric bernstein von mises theorem under misspecification even when the distribution of the regression error term is not normal the posterior distribution of the properly recentered and rescaled regression coefficients converges to a normal distribution with the zero mean and the variance equal to the semiparametric efficiency bound

consistency of bayesian procedures for variable selection it has long been known that for the comparison of pairwise nested models a decision based on the bayes factor produces a consistent model selector in the frequentist sense here we go beyond the usual consistency for nested pairwise models and show that for a wide class of prior distributions including intrinsic priors the corresponding bayesian procedure for variable selection in normal regression is consistent in the entire class of normal linear models we find that the asymptotics of the bayes factors for intrinsic priors are equivalent to those of the schwarz bic criterion also recall that the jeffreys lindley paradox refers to the well known fact that a point null hypothesis on the normal mean parameter is always accepted when the variance of the conjugate prior goes to infinity this implies that some limiting forms of proper prior distributions are not necessarily suitable for testing problems intrinsic priors are limits of proper prior distributions and for finite sample sizes they have been proved to behave extremely well for variable selection in regression a consequence of our results is that for intrinsic priors lindley s paradox does not arise

an asymptotic property of model selection criteria probability models are estimated by use of penalized log likelihood criteria related to akaike 1973 information criterion aic and minimum description length mdl the accuracies of the density estimators are shown to be related to the tradeoff between three terms the accuracy of approximation the model dimension and the descriptive complexity of the model classes the asymptotic risk is determined under conditions on the penalty term and is shown to be minimax optimal for some cases as an application we show that the optimal rate of convergence is simultaneously achieved for log densities in sobolev spaces w sub 2 sup s u without knowing the smoothness parameter s and norm parameter u in advance applications to neural network models and sparse density function estimation are also provided

asymptotic behavior of bayes estimates under possibly incorrect models we prove that the posterior distribution in a possibly incorrect parametric model a s concentrates in a strong sense on the set of pseudotrue parameters determined by the true distribution as a consequence we obtain in the case of a unique pseudotrue parameter the strong consistency of pseudo bayes estimators w r t general loss functions further we present a simple example based on normal distributions and having two different pseudotrue parameters where pseudo bayes estimators have an essentially different asymptotic behavior than the pseudomaximum likelihood estimator while the mle is strongly consistent the sequence of posterior means is strongly inconsistent and a s almost all its accumulation points are not pseudotrue finally we give conditions under which a pseudo bayes estimator for a unique pseudotrue parameter has an asymptotic normal distribution

convergence rates of posterior distributions we consider the asymptotic behavior of posterior distributions and bayes estimators for infinite dimensional statistical models we give general results on the rate of convergence of the posterior measure these are applied to several examples including priors on finite sieves log spline models dirichlet processes and interval censoring

steady state learning and nash equilibrium we study the steady states of a system in which players learn about the strategies their opponents are playing by updating their bayesian priors in light of their observations players are matched at random to play a fixed extensive form game and each player observes the realized actions in his own matches but not the intended off path play of his opponents or the realized actions in other matches because players are assumed to live finite lives there are steady states in which learning continually takes place if lifetimes are long and players are very patient the steady state distribution of actions approximates that of a nash equilibrium

rates of convergence for the gaussian mixture sieve gaussian mixtures provide a convenient method of densityestimation that lies somewhere between parametric models and kernel densityestimators when the number of components of the mixture is allowed to increase as sample size increases the model is called a mixture sieve we establish a bound on the rate of convergence in hellinger distance for densityestimation using the gaussian mixture sieve assuming that the true density is itself a mixture of gaussians the underlying mixing measure of the true densityis not necessarilyassumed to have finite support computing the rate involves some delicate calculations since the size of the sieve as measured bybracketing entropy and the saturation rate cannot be found using standard methods when the mixing measure has compact support using kn n 2 3 log n 1 3 components in the mixture yields a rate of order log n 1 6 n1 6 for every 0 the rates depend heavilyon the tail behavior of the true density the sensitivity to the tail behavior is diminished byusing a robust sieve which includes a long tailed component in the

harmonic maps to buildings and singular perturbation theory the notion of a uni versal building associated with a point in the hitchin base is introduced the universal building is a building equipped with a harmonic map from the universal cover of the given riemann surface that is initial among harmonic maps which induce the given cameral cover of the riemann surface the notion of a versal building is obtained by relaxing the uniqueness condition in the definition of an initial object in the rank one case the universal building is the leaf space of the quadratic differential defining the point in the hitchin base the main conjectures of this paper are 1 a versal building always exists 2 the asymptotics of the riemann hilbert correspondence and the non abelian hodge correspondence are controlled by the harmonic map associated to any versal building 3 spectral networks arise as inverse images of singularities of the versal building and 4 versal buildings encode the data of a 3d calabi yau category whose space of stability conditions has a connected component that contains the hitchin base the main theorem establishes the existence of the universal building conjecture 3 as well as the riemann hilbert part of conjecture 2 in the case of the rank two example introduced in the seminal work of berk nevins roberts on higher order stokes phenomena it is also shown that the asymptotics of the riemann hilbert correspondence are always controlled by a harmonic map to a certain building which is constructed as the asymptotic cone of a symmetric space

convergence rates for density estimation with bernstein polynomials mixture models for density estimation provide a very useful set up for the bayesian or the maximum likelihood approach for a density on the unit interval mixtures of beta densities form a flexible model the class of bernstein densities is a much smaller subclass of the beta mixtures defined by bernstein polynomials which can approximate any continuous density a bernstein polynomial prior is obtained by putting a prior distribution on the class of bernstein densities the posterior distribution of a bernstein polynomial prior is consistent under very general conditions in this article we present some results on the rate of convergence of the posterior distribution if the underlying distribution generating the data is itself a bernstein density then we show that the posterior distribution converges at nearly parametric rate log n sqrt n for the hellinger distance if the true density is not of the bernstein type we show that the posterior converges at a rate n 1 3 log n 5 6 provided that the true density is twice differentiable and bounded away from 0 similar rates are also obtained for sieve maximum likelihood estimates these rates are inferior to the pointwise convergence rate of a kernel type estimator we show that the bayesian bootstrap method gives a proxy for the posterior distribution and has a convergence rate at par with that of the kernel estimator

rates of convergence of posterior distributions we compute the rate at which the posterior distribution concentrates around the true parameter value the spaces we work in are quite general and include infinite dimensional cases the rates are driven by two quantities the size of the space as measured by bracketing entropy and the degree to which the prior concentrates in a small ball around the true parameter we consider two examples

consistency of bernstein polynomial posteriors a bernstein prior is a probability measure on the space of all the distribution functions on 0 1 under very general assumptions it selects absolutely continuous distribution functions whose densities are mixtures of known beta densities the bemstein prior is of interest in bayesian nonparametric inference with continuous data we study the consistency of the posterior from a bernstein prior we first show that under mild assumptions the posterior is weakly consistent for any distribution function p o on 0 1 with continuous and bounded lebesgue density with slightly stronger assumptions on the prior the posterior is also hellinger consistent this implies that the predictive density from a bernstein prior which is a bayesian density estimate converges in the hellinger sense to the true density assuming that it is continuous and bounded we also study a sieve maximum likelihood version of the density estimator and show that it is also hellinger consistent under weak assumptions when the order of the bernstein polynomial i e the number of components in the beta distribution mixture is truncated we show that under mild restrictions the posterior concentrates on the set of pseudotrue densities finally we study the behaviour of the predictive density numerically and we also study a hybrid bayes maximum likelihood density estimator