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

rates of convergence for bayes and maximum likelihood estimation for mixture of normal densities we study the rates of convergence of the maximum likelihood estimator mle and posterior distribution in density estimation problems where the densities are location or location scale mixtures of normal distributions with the scale parameter lying between two positive numbers the true density is also assumed to lie in this class with the true mixing distribution either compactly supported or having sub gaussian tails we obtain bounds for hellinger bracketing entropies for this class and from these bounds we deduce the convergence rates of sieve mles in hellinger distance the rate turns out to be log n n where k 1 is a constant that depends on the type of mixtures and the choice of the sieve next we consider a dirichlet mixture of normals as a prior on the unknown density we estimate the prior probability of a certain kullback leibler type neighborhood and then invoke a general theorem that computes the posterior convergence rate in terms the growth rate of the hellinger entropy and the concentration rate of the prior the posterior distribution is also seen to converge at the rate log n n in where k now depends on the tail behavior of the base measure of the dirichlet process

bayesian aspects of some nonparametric problems we study the bayesian approach to nonparametric function estimation problems such as nonparametric regression and signal estimation we consider the asymptotic properties of bayes procedures for conjugate gaussian priors we show that so long as the prior puts nonzero measure on the very large parameter set of interest then the bayes estimators are not satisfactory more specifically we show that these estimators do not achieve the correct minimax rate over norm bounded sets in the parameter space thus all bayes estimators for proper gaussian priors have zero asymptotic efficiency in this minimax sense we then present a class of priors whose bayes procedures attain the optimal minimax rate of convergence these priors may be viewed as compound or hierarchical mixtures of suitable gaussian distributions

on totally real cubic fields the authors have constructed a table of the 26440 nonconjugate totally real cubic number fields of discriminant d 500000 thereby extending the existing table of fields with d 100000 by i 0 angell 1 serious defects in angell s table are pointed out for each field running number discriminant coefficients of a generating polynomial integral basis class number and a fundamental pair of units are listed the article contains statistics about the following subjects distribution of class numbers fields in which every norm positive unit is totally positive nonconjugate fields with the same discriminant fields with noncyclic class group the fields are tabulated by means of a method due to davenport and heilbronn 7 8 which leads to a unique normalized generating polynomial the given units are chosen so that the fundamental parallelogram of the unit lattice determined by the corresponding vectors in the logarithmic space is reduced

relevant statistics for bayesian model choice the choice of the summary statistics that are used in bayesian inference and in particular in approximate bayesian computation algorithms has bearings on the validation of the resulting inference those statistics are nonetheless customarily used in approximate bayesian computation algorithms without consistency checks we derive necessary and sufficient conditions on summary statistics for the corresponding bayes factor to be convergent namely to select the true model asymptotically those conditions which amount to the expectations of the summary statistics differing asymptotically under the two models are quite natural and can be exploited in approximate bayesian computation settings to infer whether or not a choice of summary statistics is appropriate via a monte carlo validation

on bayesian supremum norm contraction rates building on ideas from castillo and nickl 6 a method is pro vided to study nonparametric bayesian posterior convergence rates when strong measures of distances such as the sup norm are considered in particular we show that likelihood methods can achieve optimal minimax sup norm rates in density estimation on the unit interval the introduced methodology is used to prove that commonly used families of prior distri butions on densities namely log density priors and dyadic random density histograms can indeed achieve optimal sup norm rates of convergence new results are also derived in the gaussian white noise model as a further il lustration of the presented techniques ams 2000 subject classications primary 62g20 secondary 62g05 62g07

functions realized by consistent sequential machines a sequential machine is consistent iff when both the input and output strings of symbols are regarded as representing numerical quantities according to conventional interpretations quantitatively more significant symbols occurring earlier than less then the machine may be regarded as calculating an ordinary numerical function for all possible input strings consistent sequential machines have not been of significant interest to automata theorists but they may have applications in signal processing the relationships holding among i input output alphabets and ii structures of consistent sequential machines and iii properties of numerical functions thus realized are explored in connection with ii the finite state and permutation free properties are of particular interest in connection with iii the properties of monotonicity invertibility piecewise linearity and differentiability are of particular interest synthesis methods are not significantly treated

rates of contraction of posterior distributions based on gaussian process priors we derive rates of contraction of posterior distributions on nonparametric or semiparametric models based on gaussian processes the rate of contraction is shown to depend on the position of the true parameter relative to the reproducing kernel hilbert space of the gaussian process and the small ball probabilities of the gaussian process we determine these quantities for a range of examples of gaussian priors and in several statistical settings for instance we consider the rate of contraction of the posterior distribution based on sampling from a smooth density model when the prior models the log density as a fractionally integrated brownian motion we also consider regression with gaussian errors and smooth classification under a logistic or probit link function combined with various priors

approximation dans les espaces metriques et theorie de l estimation we investigate the relations between the speed of estimation and the metric structure of the parameter space especially in the case when its metric dimension is infinite given some distance d on generally hellinger distance in the case of n i i d variables we consider the minimax risk for n observations r n q mathop inf sup limits t n theta in theta text ie theta d q theta t n t n being any estimate of we shall look for functions r such that for positive constants c 1 q and c 2 q c 1 r q n r n q c2 r q n r n is the speed of estimation and we shall show under fairly general conditions including i i d variables and regular cases of markov chains and stationnary gaussian processes that r n is determined up to multiplicative constants by the metric structure of we shall also give a construction for some sort of universal estimates the risk of which is bounded by c 2 r q n in all cases where the preceding theory applies

asymptotic normality of semiparametric and nonparametric posterior distributions in this article it is shown that the marginal semiparametric and nonparametric posterior distributions for a parameter of interest behave like an ordinary parametric posterior distribution this in practice provides support of the utility of marginal semiparametric and nonparametric posterior distributions in particular the marginal semiparametric and nonparametric posterior distributions are asymptotically normal and centered at the corresponding maximum likelihood estimates mles or posterior means with covariance matrix the inverse of the fisher information additionally the semiparametric and nonparametric mles for the parameter of interest and the marginal posterior means are asymptotically normal and centered at the true parameter with the same covariance matrix the results are a semiparametric version and a nonparametric version of the parametric bayesian central limit theorem that establish a connection between the semiparametric and the nonparametric bayesian inference and their frequentis

a semiparametric bernstein von mises theorem for gaussian process priors this paper is a contribution to the bayesian theory of semiparametric estimation we are interested in the so called bernstein von mises theorem in a semiparametric framework where the unknown quantity is f with the parameter of interest and f an infinite dimensional nuisance parameter two theorems are established one in the case with no loss of information and one in the information loss case with gaussian process priors the general theory is applied to three specific models the estimation of the center of symmetry of a symmetric function in gaussian white noise a time discrete functional data analysis model and cox s proportional hazards model in all cases the range of application of the theorems is investigated by using a family of gaussian priors parametrized by a continuous parameter

recovering gradients from sparsely observed functional data summary the recovery of gradients of sparsely observed functional data is a challenging ill posed inverse problem given observations of smooth curves e g growth curves at isolated time points the aim is to provide estimates of the underlying gradients or growth velocities to address this problem we develop a bayesian inversion approach that models the gradient in the gaps between the observation times by a tied down brownian motion conditionally on its values at the observation times the posterior mean and covariance kernel of the growth velocities are then found to have explicit and computationally tractable representations in terms of quadratic splines the hyperparameters in the prior are specified via nonparametric empirical bayes with the prior precision matrix at the observation times estimated by constrained l1 minimization the infinitessimal variance of the brownian motion prior is selected by cross validation the approach is illustrated using both simulated and real data examples

bernstein von mises theorems for gaussian regression with increasing number of regressors this paper brings a contribution to the bayesian theory of nonparametric and semiparametric estimation we are interested in the asymptotic normality of the posterior distribution in gaussian linear regression models when the number of regressors increases with the sample size two kinds of bernstein von mises theorems are obtained in this framework nonparametric theorems for the parameter itself and semiparametric theorems for functionals of the parameter we apply them to the gaussian sequence model and to the regression of functions in sobolev and c classes in which we get the minimax convergence rates adaptivity is reached for the bayesian estimators of functionals in our applications

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

on convergence of posterior distributions a general asymptotic theory of estimation was developed by ibragimov and has minskii under certain conditions on the normalized likelihood ratios in an earlier work the present authors studied the limiting behaviour of the posterior distributions under the general setup of ibragimov and has minskii in particular they obtained a necessary condition for the convergence of a suitably centered and normalized posterior to a constant limit in terms of the limiting likelihood ratio process in this paper it is shown that this condition is also sufficient to imply the posterior convergence some related results are also presented

a dirichlet process gaussian state machine model for change detection in transient processes abstractthe ability to detect incipient and critical changes in real world process esessential for system integrity assurance is currently impeded by the mismatch between the key assumption of stationarity underlying most change detection methods and the nonlinear and nonstationary transient dynamics of most real world processes the current approaches are slow or outright unable to detect qualitative changes in the behaviors that lead to anomalies we present a dirichlet process gaussian state machine dpgsm model to represent dynamic intermittency which is one of the most ubiquitous real world transient behaviors the dpgsm model treats a signal as a random walk among a dirichlet process mixture of gaussian clusters hypothesis tests and a numerical scheme based on this nonparametric representation were developed to detect subtle changes in the transient intermittent dynamics experimental investigations suggest that the dpgsm approach can consistently detect incipient critical changes in intermi

on adaptive posterior concentration rates tion of holder balls and that moreover achieve our lower bound we analyse the consequences in terms of asymptotic behaviour of poste rior credible balls as well as frequentist minimax adaptive estimation our results are appended with an upper bound for the contraction rate under an arbitrary loss in a generic regular experiment the up per bound is attained for certain sieve priors and enables to extend our results to density estimation

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

contiguity of the whittle measure for a gaussian time series for a stationary time series whittle constructed a likelihood for the spectral density based on the approximate independence of the discrete fourier transforms of the data at certain frequencies whittle s likelihood has been widely used in the literature for con structing estimators in this paper we show that for a gaussian time series the whittle measure is mutually contiguous with the actual distribution of the data as a consequence most asymptotic properties of estimators and test statistics derived under the whittle measure can be carried over to the actual distribution

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

posterior convergence rates of dirichlet mixtures at smooth densities we study the rates of convergence of the posterior distribution for bayesian density estimation with dirichlet mixtures of normal distributions as the prior the true density is assumed to be twice continuously differentiable the bandwidth is given a sequence of priors which is obtained by scaling a single prior by an appropriate order in order to handle this problem we derive a new general rate theorem by considering a countable covering of the parameter space whose prior probabilities satisfy a summability condition together with certain individual bounds on the hellinger metric entropy we apply this new general theorem on posterior convergence rates by computing bounds for hellinger bracketing entropy numbers for the involved class of densities the error in the approximation of a smooth density by normal mixtures and the concentration rate of the prior the best obtainable rate of convergence of the posterior turns out to be equivalent to the well known frequentist rate for integrated mean squared error n 2 5 up to a logarithmic factor

large sample inference for log spline models let f be a continuous and positive unknown density on a known compact interval mathscr y let f denote the distribution function of f and let q f 1 denote its quantile function a finite parameter exponential family model based on b splines is constructed maximum likelihood estimation of the parameters of the model based on a random sample of size n from f yields estimates hat f f and hat q of f f and q respectively under mild conditions if the number of parameters tends to infinity in a suitable manner as n rightarrow infty these estimates achieve the optimal rate of convergence the asymptotic behavior of the corresponding confidence bounds is also investigated in particular it is shown that the standard errors of hat f and hat q are asymptotically equal to those of the usual empirical distribution function and empirical quantile function

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

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

convergence rates of posterior distributions for non i i d observations we consider the asymptotic behavior of posterior distributions and bayes estimators based on observations which are required to be neither independent nor identically distributed we give general results on the rate of convergence of the posterior measure relative to distances derived from a testing criterion we then specialize our results to independent nonidentically distributed observations markov processes stationary gaussian time series and the white noise model we apply our general results to several examples of infinite dimensional statistical models including nonparametric regression with normal errors binary regression poisson regression an interval censoring model whittle estimation of the spectral density of a time series and a nonlinear autoregressive model

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