sharp constants for moser trudinger inequalities on spheres in complex space ℂn the main results of this paper concern sharp constants for the moser trudinger inequalities on spheres in complex space c n we derive moser trudinger inequalities for smooth functions and holomorphic functions with different sharp constants see theorem 1 1 the sharp moser trudinger inequalities under consideration involve the complex tangential gradients for the functions and thus we have shown here such inequalities in the cr setting though there is a close connection in spirit between inequalities proven here on complex spheres and those on the heisenberg group for functions with compact support in any finite domain proven earlier by the same authors 17 derivation of the sharp constants for moser trudinger inequalities on complex spheres are more complicated and difficult to obtain than on the heisenberg group variants of moser onofri type inequalities are also given on complex spheres as applications of our sharp inequalities see theorems 1 2 and 1 3 one of the key ingredients in deriving the main theorems is a sharp representation formula for functions on the complex spheres in terms of complex tangential gradients see theorem 1 4 c 2004 wiley periodicals inc

normalized random measures driven by increasing additive processes this paper introduces and studies a new class of nonparametric prior distributions random probability distribution functions are constructed via normalization of random measures driven by increasing additive processes in particular we present results for the distribution of means under both prior and posterior conditions and via the use of strategic latent variables undertake a full bayesian analysis our class of priors includes the well known and widely used mixture of a dirichlet process

a sharpened moser pohozaev trudinger inequality with mean value zero in r2 abstract let r 2 be a smooth bounded domain and q x be a polynomial with q 0 0 then under some hypothesis on q x there holds sup u 2 d x 1 u d x 0 e 2 u 2 q 0 q u 2 d x d x a sufficient condition will be given to assure that the above inequality does not hold furthermore the existence of the extremal functions will be derived

do we need hundreds of classifiers to solve real world classification problems we evaluate 179 classifiers arising from 17 families discriminant analysis bayesian neural networks support vector machines decision trees rule based classifiers boosting bagging stacking random forests and other ensembles generalized linear models nearest neighbors partial least squares and principal component regression logistic and multinomial regression multiple adaptive regression splines and other methods implemented in weka r with and without the caret package c and matlab including all the relevant classifiers available today we use 121 data sets which represent the whole uci data base excluding the large scale problems and other own real problems in order to achieve significant conclusions about the classifier behavior not dependent on the data set collection the classifiers most likely to be the bests are the random forest rf versions the best of which implemented in r and accessed via caret achieves 94 1 of the maximum accuracy overcoming 90 in the 84 3 of the data sets however the difference is not statistically significant with the second best the svm with gaussian kernel implemented in c using libsvm which achieves 92 3 of the maximum accuracy a few models are clearly better than the remaining ones random forest svm with gaussian and polynomial kernels extreme learning machine with gaussian kernel c5 0 and avnnet a committee of multi layer perceptrons implemented in r with the caret package the random forest is clearly the best family of classifiers 3 out of 5 bests classifiers are rf followed by svm 4 classifiers in the top 10 neural networks and boosting ensembles 5 and 3 members in the top 20 respectively

sharp sobolev inequalities on the sphere and the moser trudinger inequality where de denotes normalized surface measure v is the conformal gradient and q 2n n 2 a modern folklore theorem is that by taking the infinitedimensional limit of this inequality one obtains the gross logarithmic sobolev inequality for gaussian measure which determines nelson s hypercontractive estimates for the hermite semigroup see 8 one observes using conformal invariance that the above inequality is equivalent to the sharp sobolev inequality on rn for which boundedness and extremal functions can be easily calculated using dilation invariance and geometric symmetrization the roots here go back to hardy and littlewood the advantage of casting the problem on the sphere is that the role of the constants is evident and one is led immediately to the conjecture that this inequality should hold whenever possible for example 2 q 0o if n 2 this is in fact true and will be demonstrated in section 2 a clear question at this point is what is the situation in dimension 2 two important arguments 25 26 27 dealt with this issue both motivated by geometric variational problems because q goes to infinity for dimension 2 the appropriate function space is the exponential class responding in part

updating schemes correlation structure blocking and parameterization for the gibbs sampler in this paper many convergence issues concerning the implementation of the gibbs sampler are investigated exact computable rates of convergence for gaussian target distributions are obtained different random and non random updating strategies and blocking combinations are compared using the rates the effect of dimensionality and correlation structure on the convergence rates are studied some examples are considered to demonstrate the results for a gaussian image analysis problem several updating strategies are described and compared for problems in bayesian linear models several possible parameterizations are analysed in terms of their convergence rates characterizing the optimal choice

valid post selection inference it is common practice in statistical data analysis to perform data driven variable selection and derive statistical inference from the resulting model such inference enjoys none of the guarantees that classical statistical theory provides for tests and confidence intervals when the model has been chosen a priori we propose to produce valid post selection inference by reducing the problem to one of simultaneous inference and hence suitably widening conventional confidence and retention intervals simultaneity is required for all linear functions that arise as coefficient estimates in all submodels by purchasing simultaneity insurance for all possible submodels the resulting post selection inference is rendered universally valid under all possible model selection procedures this inference is therefore generally conservative for particular selection procedures but it is always less conservative than full scheffe protection importantly it does not depend on the truth of the selected submodel and hence it produces valid inference even in wrong models we describe the structure of the simultaneous inference problem and give some asymptotic results

extremal functions for moser s inequality let q be a bounded smooth domain in ri and u x a c1 function with compact support in q moser s inequality states that there is a constant co depending only on the dimension n such that 1 wn1 n j ee n u dx co where iqi is the lebesgue measure of q and wn 1 the surface area of the unit ball in rn we prove in this paper that there are extremal functions for this inequality in other words we show that the sup j nw u dx u e wj 11vuh1n 1 is attained earlier results include carleson chang 1986 q is a ball in any dimension and flucher 1992 q is any domain in 2 dimensions

bayesian regression trees for high dimensional prediction and variable selection abstractdecision tree ensembles are an extremely popular tool for obtaining high quality predictions in nonparametric regression problems unmodified however many commonly used decision tree ensemble methods do not adapt to sparsity in the regime in which the number of predictors is larger than the number of observations a recent stream of research concerns the construction of decision tree ensembles that are motivated by a generative probabilistic model the most influential method being the bayesian additive regression trees bart framework in this article we take a bayesian point of view on this problem and show how to construct priors on decision tree ensembles that are capable of adapting to sparsity in the predictors by placing a sparsity inducing dirichlet hyperprior on the splitting proportions of the regression tree prior we characterize the asymptotic distribution of the number of predictors included in the model and show how this prior can be easily incorporated into existing markov chai

asymptotic equivalence of bayes cross validation and widely applicable information criterion in singular learning theory in regular statistical models the leave one out cross validation is asymptotically equivalent to the akaike information criterion however since many learning machines are singular statistical models the asymptotic behavior of the cross validation remains unknown in previous studies we established the singular learning theory and proposed a widely applicable information criterion the expectation value of which is asymptotically equal to the average bayes generalization loss in the present paper we theoretically compare the bayes cross validation loss and the widely applicable information criterion and prove two theorems first the bayes cross validation loss is asymptotically equivalent to the widely applicable information criterion as a random variable therefore model selection and hyperparameter optimization using these two values are asymptotically equivalent second the sum of the bayes generalization error and the bayes cross validation error is asymptotically equal to 2 n where is the real log canonical threshold and n is the number of training samples therefore the relation between the cross validation error and the generalization error is determined by the algebraic geometrical structure of a learning machine we also clarify that the deviance information criteria are different from the bayes cross validation and the widely applicable information criterion

bayesian inference for logistic models using polya gamma latent variables we propose a new data augmentation strategy for fully bayesian inference in models with binomial likelihoods the approach appeals to a new class of polya gamma distributions which are constructed in detail a variety of examples are presented to show the versatility of the method including logistic regression negative binomial regression nonlinear mixed effect models and spatial models for count data in each case our data augmentation strategy leads to simple effective methods for posterior inference that 1 circumvent the need for analytic approximations numerical integration or metropolis hastings and 2 outperform other known data augmentation strategies both in ease of use and in computational efficiency all methods including an efficient sampler for the polya gamma distribution are implemented in the r package bayeslogit supplementary materials for this article are available online

bayesian regression tree ensembles that adapt to smoothness and sparsity ensembles of decision trees are a useful tool for obtaining flexible estimates of regression functions examples of these methods include gradient boosted decision trees random forests and bayesian classification and regression trees two potential shortcomings of tree ensembles are their lack of smoothness and their vulnerability to the curse of dimensionality we show that these issues can be overcome by instead considering sparsity inducing soft decision trees in which the decisions are treated as probabilistic we implement this in the context of the bayesian additive regression trees framework and illustrate its promising performance through testing on benchmark data sets we provide strong theoretical support for our methodology by showing that the posterior distribution concentrates at the minimax rate up to a logarithmic factor for sparse functions and functions with additive structures in the high dimensional regime where the dimensionality of the covariate space is allowed to grow nearly exponentially in the sample size our method also adapts to the unknown smoothness and sparsity levels and can be implemented by making minimal modifications to existing bayesian additive regression tree algorithms

conformal deformation of metrics on s 2 soit d un domaine lisse de s 2 et k une fonction lisse definie sur d alors pour que le probleme aux limites de neumann u k exp 2u 1 sur d u n 0 sur d ait une solution u on a a quand aire d max x d k x 0

moser trudinger and beckner onofri s inequalities on the cr sphere we derive sharp moser trudinger inequalities on the cr sphere the rst type is in the adams form for powers of the sublaplacian and for general spectrally dened operators on the space of cr pluriharmonic functions we will then obtain the sharp beckner onofri inequality for crpluriharmonic functions on the sphere and as a consequence a sharp logarithmic hardy littlewood sobolev inequality in the form given by carlen and loss

comparison theorems for double splittings of k monotone matrices abstract in this paper using the knowledge of matrices which leave a cone invariant in r n comparison theorems for two double splittings of a k monotone matrix are obtained these results generalize the corresponding ones that are introduced in shen and huang 2006 1 for double splittings of monotone matrices

estimates for green function and poisson kernels of higher order dirichlet boundary value problems pointwise estimates are derived for the kernels associated to the polyharmonic dirichlet problem on bounded smooth domains as a consequence one obtains optimal weighted l p l q regularity estimates for weights involving the distance function

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

a widely applicable bayesian information criterion a statistical model or a learning machine is called regular if the map taking a parameter to a probability distribution is one to one and if its fisher information matrix is always positive definite if otherwise it is called singular in regular statistical models the bayes free energy which is defined by the minus logarithm of bayes marginal likelihood can be asymptotically approximated by the schwarz bayes information criterion bic whereas in singular models such approximation does not hold recently it was proved that the bayes free energy of a singular model is asymptotically given by a generalized formula using a birational invariant the real log canonical threshold rlct instead of half the number of parameters in bic theoretical values of rlcts in several statistical models are now being discovered based on algebraic geometrical methodology however it has been difficult to estimate the bayes free energy using only training samples because an rlct depends on an unknown true distribution in the present paper we define a widely applicable bayesian information criterion wbic by the average log likelihood function over the posterior distribution with the inverse temperature 1 log n where n is the number of training samples we mathematically prove that wbic has the same asymptotic expansion as the bayes free energy even if a statistical model is singular for or unrealizable by a statistical model since wbic can be numerically calculated without any information about a true distribution it is a generalized version of bic onto singular statistical models

bayesian cart model search abstract in this article we put forward a bayesian approach for finding classification and regression tree cart models the two basic components of this approach consist of prior specification and stochastic search the basic idea is to have the prior induce a posterior distribution that will guide the stochastic search toward more promising cart models as the search proceeds such models can then be selected with a variety of criteria such as posterior probability marginal likelihood residual sum of squares or misclassification rates examples are used to illustrate the potential superiority of this approach over alternative methods

a bayesian cart algorithm a stochastic search form of classification and regression tree cart analysis breiman et al 1984 is proposed motivated by a bayesian model an approximation to a probability distribution over the space of possible trees is explored using reversible jump markov chain monte carlo methods green 1995

log linear bayesian additive regression trees for multinomial logistic and count regression models we introduce bayesian additive regression trees bart for log linear models including multinomial logistic regression and count regression with zero inflation and overdispersion bart has been app

greedy function approximation a gradient boosting machine function estimation approximation is viewed from the perspective of numerical optimization in function space rather than parameter space a connection is made between stagewise additive expansions and steepest descent minimization a general gradient descent boosting paradigm is developed for additive expansions based on any fitting criterion specific algorithms are presented for least squares least absolute deviation and huber m loss functions for regression and multiclass logistic likelihood for classification special enhancements are derived for the particular case where the individual additive components are regression trees and tools for interpreting such treeboost models are presented gradient boosting of regression trees produces competitive highly robust interpretable procedures for both regression and classification especially appropriate for mining less than clean data connections between this approach and the boosting methods of freund and shapire and friedman hastie and tibshirani are discussed

a sharp form of the moser trudinger inequality on a compact riemannian surface in this paper a sharp form of the moser trudinger inequality is established on a compact riemannian surface via the method of blow up analysis and the existence of an extremal function for such an inequality is proved

reversible jump markov chain monte carlo computation and bayesian model determination markov chain monte carlo methods for bayesian computation have until recently been restricted to problems where the joint distribution of all variables has a density with respect to some fixed standard underlying measure they have therefore not been available for application to bayesian model determination where the dimensionality of the parameter vector is typically not fixed this paper proposes a new framework for the construction of reversible markov chain samplers that jump between parameter subspaces of differing dimensionality which is flexible and entirely constructive it should therefore have wide applicability in model determination problems the methodology is illustrated with applications to multiple change point analysis in one and two dimensions and to a bayesian comparison of binomial experiments

comparison theorems for weak splittings of bounded operators comparison theorems for weak splittings of bounded operators are presented these theorems extend the classical comparison theorem for regular splittings of matrices by varga the less known result by wo nicki and the recent results for regular and weak regular splittings of matrices by neumann and plemmons elsner and lanzkron rose and szyld the hypotheses of the theorems presented here are weaker and the theorems hold for general banach spaces and rather general cones hypotheses are given which provide strict inequalities for the comparisons it is also shown that the comparison theorem by alefeld and volkmann applies exclusively to monotone sequences of iterates and is not equivalent to the comparison of the spectral radius of the iteration operators

adams inequalities for bi laplacian and extremal functions in dimension four let be a smooth oriented bounded domain be the sobolev space and be the first eigenvalue of the bi laplacian operator 2 then for any a 0a o we have and the above supremum is infinity when a o this strengthens adams inequality in dimension 4 d adams a sharp inequality of j moser for high order derivatives ann of math 128 1988 365 398 where he proved the above inequality holds for a 0 moreover we prove that for sufficiently small a an extremal function for the above inequality exists as a special case of our results we thus show that there exists with such that this establishes the existence of an extremal function of the original adams inequality in dimension 4

kahler einstein metrics with positive scalar curvature in this paper we prove that the existence of kahler einstein metrics implies the stability of the underlying kahler manifold in a suitable sense in particular this disproves a long standing conjecture that a compact kahler manifold admits kahler einstein metrics if it has positive first chern class and no nontrivial holomorphic vector fields we will also establish an analytic criterion for the existence of kahler einstein metrics our arguments also yield that the analytic criterion is satisfied on stable kahler manifolds provided that the partial c 0 estimate posed in t6 is true

covariance structure of the gibbs sampler with applications to the comparisons of estimators and augmentation schemes summary we study the covariance structure of a markov chain generated by the gibbs sampler with emphasis on data augmentation when applied to a bayesian missing data problem the gibbs sampler produces two natural approximations for the posterior distribution of the parameter vector the empirical distribution based on the sampled values of the parameter vector and a mixture of complete data posteriors we prove that rao blackwellization causes a one lag delay for the autocovariances among dependent samples obtained from data augmentation and consequently the mixture approximation produces estimates with smaller variances than the empirical approximation the covariance structure results are used to compare different augmentation schemes it is shown that collapsing and grouping random components in a gibbs sampler with two or three components usually result in more efficient sampling schemes

bayesian generalized additive models for location scale and shape for zero inflated and overdispersed count data frequent problems in applied research preventing the application of the classical poisson log linear model for analyzing count data include overdispersion an excess of zeros compared to the poisson distribution correlated responses as well as complex predictor structures comprising nonlinear effects of continuous covariates interactions or spatial effects we propose a general class of bayesian generalized additive models for zero inflated and overdispersed count data within the framework of generalized additive models for location scale and shape where semiparametric predictors can be specified for several parameters of a count data distribution as standard options for applied work we consider the zero inflated poisson the negative binomial and the zero inflated negative binomial distribution the additive predictor specifications rely on basis function approximations for the different types of effects in combination with gaussian smoothness priors we develop bayesian inference based on markov c

convergence and comparison theorems for double splittings of matrices in this paper some convergence theorems for the double splitting of a monotone matrix or a hermitian positive definite matrix are presented two comparison theorems for two double splittings of a monotone matrix are obtained meanwhile we establish a new sufficient condition for convergence of the gauss seidel double sor method for an h matrix