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

solving a singular system of two nonlinear odes abstract in this paper a new method is given in order to solve a singular system of two nonlinear ordinary differential equations having multiple solutions and we may give a special solution by imposing an additional condition representation of the exact solution is given in the form of series in the reproducing kernel space its approximate solution is obtained by truncating the series the error of the approximate solution is monotone decreasing in the sense of w 2 0 1 and the intrinsic merit of the method given in the paper lies in its speedy convergence some numerical examples are studied to demonstrate the accuracy of the present method

on bayesian consistency we consider a sequence of posterior distributions based on a data dependent prior which we shall refer to as a pseudoposterior distribution and establish simple conditions under which the sequence is hellinger consistent it is shown how investigations into these pseudo posteriors assist with the understanding of some true posterior distributions including polya trees the infinite dimensional exponential family and mixture models

multivariate adaptive regression splines a new method is presented for flexible regression modeling of high dimensional data the model takes the form of an expansion in product spline basis functions where the number of basis functions as well as the parameters associated with each one product degree and knot locations are automatically determined by the data this procedure is motivated by the recursive partitioning approach to regression and shares its attractive properties unlike recursive partitioning however this method produces continuous models with continuous derivatives it has more power and flexibility to model relationships that are nearly additive or involve interactions in at most a few variables in addition the model can be represented in a form that separately identifies the additive contributions and those associated with the different multivariable interactions

solving integro differential equations with cauchy kernel a simple method for solving the fredholm singular integro differential equations with cauchy kernel is proposed based on a new reproducing kernel space using a transformation and modifying the traditional reproducing kernel method the singular term is removed and the analytical representation of the exact solution is obtained in the form of series in the new reproducing kernel space the advantage of the approach lies in the fact that on the one hand by improving the definition of traditional inner product the representation of new reproducing kernel function becomes simple and requirement for image space of operator is weakened comparing with traditional reproducing kernel method on the other hand the approximate solution and its derivatives converge uniformly to the exact solution and its derivatives some examples are displayed to demonstrate the validity and applicability of the proposed method

approximate solution of a class of singular integral equations of second kind a simple method based on polynomial approximation of a function is employed to obtain approximate solution of a class of singular integral equations of the second kind for a hypersingular integral equation of the second kind this method avoids the complex function theoretic method and produces the known exact solution to prandtl s integral equation as a special case for a particular singular integro differential equation of the second kind this also produces an approximate solution which compares favourably with numerical results obtained by various galerkin methods the convergence of the method for both the equations is also established

composition algebras over a ring of fractions abstract using some theorems on composition algebras over rings of genus zero and elementary results from algebraic geometry all composition algebras over a ring of fractions related to the projective line over a field are enumerated and partly classified

composition algebras over rings of genus zero the theory of composition algebras over locally ringed spaces and some basic results from algebraic geometry are used to characterize composition algebras over open dense subschemes of curves of genus zero

grids and the arithmetics of jordan pairs abstract we prove that every 2 finitedimensional covering standard division grid of a jordan pair v over a henselian field canonically determines a norm on v this is used to classify maximal orders which contain a covering division grid of v we show that weakly separable orders always satisfy this condition and link their existence to ramification properties of v

spike and slab group lassos for grouped regression and sparse generalized additive models abstract we introduce the spike and slab group lasso ssgl for bayesian estimation and variable selection in linear regression with grouped variables we further extend the ssgl to sparse generali

cyclic compositions and trisotopies abstract cyclic compositions in the sense of springer t a springer oktaven jordan algebren und ausnahmegruppen universitat gottingen 1963 and knus merkurjev rost and tignol m a knus a merkurjev m rost j p tignol the book of involutions amer math soc colloq publ vol 44 amer math soc providence ri 1998 with a preface in french by j tits are investigated by means of cyclic trisotopies a concept originally due to albert a a albert a construction of exceptional jordan division algebras ann of math 2 67 1958 1 28 using the quadrupling of composition algebras we enumerate cyclic trisotopies and compositions in a rational manner i e without extending the base field we relate cyclic trisotopies explicitly to simple associative algebras of degree 3 with involution and to the tits process of cubic jordan algebras

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

derivations and automorphisms of jordan algebras in characteristic two a jordan algebra j over a field k of characteristic 2 becomes a 2 lie algebra l j with lie product x y x y and squaring x 2 x 2 we determine the precise ideal structure of l j in case j is simple finite dimensional and k is algebraically closed we also decide which of these algebras have smooth automorphism groups finally we study the derivation algebra of a reduced albert algebra j h3 o k and show that der j has a unique proper nonzero ideal vj isomorphic to l j k 1j with quotient der j v j independent of o on the group level this gives rise to a special isogeny between the automorphism group of j and that of the split albert algebra whose kernel is the infinitesimal group determined by vj

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

a new algorithm for a class of singular boundary value problems abstract in this paper we present a new algorithm to solve a class of singular boundary value problems in the reproducing kernel space w 2 3 0 1 the algorithm is efficiently applied to solving some model problems with the comparison between the numerical solutions and the exact solutions it is demonstrated by the numerical examples that this algorithm is of high precision

solving the generalized regularized long wave equation on the basis of a reproducing kernel space on the basis of a reproducing kernel space an iterative algorithm for solving the generalized regularized long wave equation is presented the analytical solution in the reproducing kernel space is shown in a series form and the approximate solution u n is constructed by truncating the series to n terms the convergence of u n to the analytical solution is also proved results obtained by the proposed method imply that it can be considered as a simple and accurate method for solving such evolution equations

new algorithm for a class of nonlinear integro differential equations in the reproducing kernel space this paper is concerned with the new algorithm for giving the approximate solution of a class of nonlinear integro differential equations in the reproducing kernel space two integro differential equations are solved using the algorithm it turns out that this algorithm is valid

isotopes of octonion algebras g2 torsors and triality octonion algebras over rings are in contrast to those over fields not determined by their norm forms octonion algebras whose norm is isometric to the norm q of a given algebra c are twisted forms of c by means of the aut c torsor o q o q aut c we show that over any commutative unital ring these twisted forms are precisely the isotopes c a b of c with multiplication given by x y xa by for unit norm octonions a b of c the link is provided by the triality phenomenon which we study from new and classical perspectives we then study these twisted forms using the interplay thus obtained between torsor geometry and isotope computations thus obtaining new results on octonion algebras over e g rings of laurent polynomials

solving singular two point boundary value problem in reproducing kernel space in this paper we present a new method for solving singular two point boundary value problem for certain ordinary differential equation having singular coefficients its exact solution is represented in the form of series in reproducing kernel space in the mean time the n term approximation u n x to the exact solution u x is obtained and is proved to converge to the exact solution some numerical examples are studied to demonstrate the accuracy of the present method results obtained by the method are compared with the exact solution of each example and are found to be in good agreement with each other

the exact solution for solving a class nonlinear operator equations in the reproducing kernel space in this paper the author will introduce the sobolev space w 1 2 w n firstly and then give the reproducing properties of the space w 1 2 w n as these results of those a class of the nonlinear operator equation n i 1 m i j 1 a i ju f is transformed to the n dimensional linear operator equation au f finally we also give the exact solution of this nonlinear operator equation

solving singular nonlinear second order periodic boundary value problems in the reproducing kernel space abstract in this paper we present a new method for solving a singular nonlinear second order periodic boundary value problem its analytical solution is represented in the form of series in the reproducing kernel space in the mean time the n term approximation u n x to the analytical solution u x is obtained and is proved to converge to the analytical solution some numerical examples are studied to demonstrate the accuracy of the present method results obtained by the method indicate the method is simple and effective

composition algebras over algebraic curves of genus zero we rephrase the classical theory of composition algebras over fields particularly the cayley dickson doubling process and zorn s vector matrices in the setting of locally ringed spaces fixing an arbitrary base field we use these constructions to classify composition algebras over complete smooth curves of genus zero applications are given to composition algebras over function fields of genus zero and polynomial rings

bayesian fractional posteriors we consider the fractional posterior distribution that is obtained by updating a prior distribution via bayes theorem with a fractional likelihood function a usual likelihood function raised to a fractional power first we analyze the contraction property of the fractional posterior in a general misspecified framework our contraction results only require a prior mass condition on certain kullback leibler kl neighborhood of the true parameter or the kl divergence minimizer in the misspecified case and obviate constructions of test functions and sieves commonly used in the literature for analyzing the contraction property of a regular posterior we show through a counterexample that some condition controlling the complexity of the parameter space is necessary for the regular posterior to contract rendering additional flexibility on the choice of the prior for the fractional posterior second we derive a novel bayesian oracle inequality based on a pac bayes inequality in misspecified models our derivation reveals several advantages of averaging based bayesian procedures over optimization based frequentist procedures as an application of the bayesian oracle inequality we derive a sharp oracle inequality in multivariate convex regression problems we also illustrate the theory in gaussian process regression and density estimation problems

how to solve the equation aubu cu f in this paper we first discuss the problem of how to solve a continuous linear operator equation in separable hilbert space we give the necessary and sufficient condition for the existence of solution and the method to decide the existence for solutions of the equation is also given if the solution exists we give the representation and approxima of the minimal normal solution of the equation and formula are obtained we conduct some research of how to solve the operator equation aubu cu f we give the factorization method and characteristic value method to solve such an equation

numerical solution of the fredholm singular integro differential equation with cauchy kernel by using taylor series expansion and galerkin method in this paper we present a taylor series expansion method for a class of fredholm singular integro differential equation with cauchy kernel this method uses the truncated taylor series polynomial of the unknown function and transforms the integro differential equation into an nth order linear ordinary differential equation with variable coefficients by galerkin method we use the orthogonal legendre polynomials as a basis for finding the approximate solution of nth order differential equation by the property of odd or even function we reduce the singularity of the integrals to the one point some numerical examples are also given to illustrate the efficiency and accuracy of the method

on the numerical treatment of the singular integral equation of the second kind here a numerical treatment for solving the integral equation of the second kind with cauchy kernel is presented the singular term has been removed and the solution in the legendre polynomial form has been used to obtain a system of linear algebraic equation this system is solved numerically

integro differential equation with cauchy kernel in this work we present a galerkin approach for solving the linear integro differential equation of the second kind with cauchy kernel we use the orthogonal basis functions legendre polynomials of the first and of the second kind

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

new algorithm for second order boundary value problems of integro differential equation this paper is concerned with a new algorithm for giving the analytical and approximate solutions of a class of boundary value problems in the reproducing kernel space the analytical solution u x and approximate solution u n x are represented in terms of series for any initial function u 1 x w 2 3 0 1 we prove u n x u x u n x u x u n x u x n two numerical examples are studied to demonstrate the accuracy of the present method results obtained by the method indicate the method is simple and effective

robust bayesian inference via coarsening the standard approach to bayesian inference is based on the assumption that the distribution of the data belongs to the chosen model class however even a small violation of this assumption can ha

adaptive bayesian multivariate density estimation with dirichlet mixtures summary we show that rate adaptive multivariate density estimation can be performed using bayesian methods based on dirichlet mixtures of normal kernels with a prior distribution on the kernel s covariance matrix parameter we derive sufficient conditions on the prior specification that guarantee convergence to a true density at a rate that is minimax optimal for the smoothness class to which the true density belongs no prior knowledge of smoothness is assumed the sufficient conditions are shown to hold for the dirichlet location mixture of normals prior with a gaussian base measure and an inverse wishart prior on the covariance matrix parameter locally h older

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

numerical solution of a singular integro differential equation this paper presents a method based on polynomial approximation using bernstein polynomial basis to obtain approximate numerical solution of a singular integro differential equation with cauchy kernel the numerical results obtained by the present method compares favorably with those obtained by various galerkin methods earlier in the literature also the convergence of the method is established rigorously for the studied integro differential equation

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

asymptotic solution of a class of strongly singular integral equations a class of strongly singular nonlinear integral equations is considered and sufficient conditions for the uniqueness of the solution and in the linear and homogeneous case the nonpositiveness of the associated eigenvalues are obtained when the singular integral is proportional to a small perturbation parameter a procedure for the construction of the uniformly valid leading term of the asymptotic solutions for both linear and nonlinear equations is presented the results are illustrated by means of an example that emerges in the analysis of microcrack growth in reinforced ceramics

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

generically algebraic jordan algebras over commutative rings abstract the theory of the generic minimum polynomial norm and trace is developed for quadratic jordan algebras which are finitely generated and projective modules over an arbitrary commutative base ring using scheme theoretic methods we recover with new proofs most of the classical theory over fields and also obtain a number of results which are new even in the classical setting

simplified reproducing kernel method and convergence order for linear volterra integral equations with variable coefficients abstract this paper proposes a simplified reproducing kernel method to solve the linear volterra integral equations with variable coefficients the main idea of the method is to establish a reproducing kernel direct space that can be used in volterra integral equations and in the first time this paper analyzes the convergence order and stability of the approximate solution then the uniform convergence of the numerical solution is proved and the time consuming schmidt orthogonalization process is avoided the proposed method is proved to be stable and is not less than the second order convergence the algorithm is proved to be feasible and stable through some numerical examples

albert algebras over curves of genus zero and one abstract albert algebras and other jordan algebras are constructed over curves of genus zero and one using a generalization of the tits process and the first tits construction due to achhammer

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

composition algebras over rings of fractions revisited abstract let k be a field of characteristic not two let f h x 0 x 1 k x 0 x 1 be an irreducible homogeneous polynomial and denote the ring of elements of degree zero in the homogeneous localization k x 0 x 1 f h by k x 0 x 1 f h for deg f h 3 it is proved that the composition algebras over k x 0 x 1 f h not containing zero divisors are defined over k and that there is at most one split composition algebra not defined over k for deg f h 4 all composition algebras over k x 0 x 1 f h are enumerated and partly classified

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

existence theorems of global solutions of initial value problems for nonlinear integrodifferential equations of mixed type in banach spaces and applications abstract this paper investigates the existence of global solutions for initial value problems of the nonlinear integrodifferential equations in banach spaces we obtain the existence of global solutions by schauder s fixed point theorem our results are the generalizations and improvements of recent results as applications of our results the existence of global solutions of two classes of third order mixed boundary value problem is obtained and an example of infinite system for nonlinear integrodifferential equations is worked out to illustrate our main results