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

posterior consistency of gaussian process prior for nonparametric binary regression consider binary observations whose response probability is an unknown smooth function of a set of covariates suppose that a prior on the response probability function is induced by a gaussian process mapped to the unit interval through a link function in this paper we study consistency of the resulting posterior distribution if the covariance kernel has derivatives up to a desired order and the bandwidth parameter of the kernel is allowed to take arbitrarily small values we show that the posterior distribution is consistent in the l 1 distance as an auxiliary result to our proofs we show that under certain conditions a gaussian process assigns positive probabilities to the uniform neighborhoods of a continuous function this result may be of independent interest in the literature for small ball probabilities of gaussian processes

variational bayes solution of linear neural networks and its generalization performance it is well known that in unidentifiable models the bayes estimation provides much better generalization performance than the maximum likelihood ml estimation however its accurate approximation by markov chain monte carlo methods requires huge computational costs as an alternative a tractable approximation method called the variational bayes vb approach has recently been proposed and has been attracting attention its advantage over the expectation maximization em algorithm often used for realizing the ml estimation has been experimentally shown in many applications nevertheless it has not yet been theoretically shown in this letter through analysis of the simplest unidentifiable models we theoretically show some properties of the vb approach we first prove that in three layer linear neural networks the vb approach is asymptotically equivalent to a positive part james stein type shrinkage estimation then we theoretically clarify its free energy generalization error and training error comparing them with those of the ml estimation and the bayes estimation we discuss the advantage of the vb approach we also show that unlike in the bayes estimation the free energy and the generalization error are less simply related with each other and that in typical cases the vb free energy well approximates the bayes one while the vb generalization error significantly differs from the bayes one

testing single index restrictions with a focus on average derivatives abstract this paper considers a situation where the violation of a single index restriction is a concern only to the extent that it causes bias to the estimates of the average derivatives we propose a method to construct tests that concentrate their asymptotic powers upon only such interesting alternatives the test has a 1 2 limiting distribution under the null hypothesis and even accommodates the case where the parameter estimates have a convergence rate slower than n as in the case of maximum score estimation the testing procedure can be easily modified when the main interest lies in average increment effects of binary covariates multivariate average derivatives or linear restrictions other than those of average derivatives results from monte carlo experiments show that the asymptotic theory is a reasonable approximation of the finite sample distributions and an application of our methods to female labor market participation illustrates the importance of this non omnibus approach

moduli spaces and formal operads let mg l be the moduli space of stable algebraic curves of genus g with l marked points with the operations that relate the different moduli spaces identifying marked points the family mg l g l is a modular operad of projective smooth deligne mumford stacksm in this paper we prove that the modular operad of singular chains s m q is formal so it is weakly equivalent to the modular operad of its homology h m q as a consequence the up to homotopy algebras of these two operads are the same to obtain this result we prove a formality theorem for operads analogous to the deligne griffiths morgan sullivan formality theorem the existence of minimal models of modular operads and a characterization of formality for operads which shows that formality is independent of the ground field

testing the order of a model using locally conic parametrization population mixtures and stationary arma processes in this paper we address the problem of testing hypotheses using the likelihood ratio test statistic in nonidentifiable models with application to model selection in situations where the parametrization for the larger model leads to nonidentifiability in the smaller model we give two major applications the case where the number of populations has to be tested in a mixture and the case of stationary arma p q processes where the order p q has to be tested we give the asymptotic distribution for the likelihood ratio test statistic when testing the order of the model in the case of order selection for armas the asymptotic distribution is invariant with respect to the parameters generating the process a locally conic parametrization is a key tool in deriving the limiting distributions it allows one to discover the deep similarity between the two problems

the lambrechts stanley model of configuration spaces we prove the validity over mathbb r of a commutative differential graded algebra model of configuration spaces for simply connected closed smooth manifolds answering a conjecture of lambrechts stanley we get as a result that the real homotopy type of such configuration spaces only depends on the real homotopy type of the manifold we moreover prove if the dimension of the manifold is at least 4 that our model is compatible with the action of the fulton macpherson operad weakly equivalent to the little disks operad when the manifold is framed we use this more precise result to get a complex computing factorization homology of framed manifolds our proofs use the same ideas as kontsevich s proof of the formality of the little disks operads

remarks on chern simons invariants the perturbative chern simons theory is studied in a finite dimensional version or assuming that the propagator satisfies certain properties as is the case e g with the propagator defined by axelrod and singer it turns out that the effective bv action is a function on cohomology with shifted degrees that solves the quantum master equation and is defined modulo certain canonical transformations that can be characterized completely out of it one obtains invariants

renyi divergence and kullback leibler divergence renyi divergence is related to renyi entropy much like kullback leibler divergence is related to shannon s entropy and comes up in many settings it was introduced by renyi as a measure of information that satisfies almost the same axioms as kullback leibler divergence and depends on a parameter that is called its order in particular the renyi divergence of order 1 equals the kullback leibler divergence we review and extend the most important properties of renyi divergence and kullback leibler divergence including convexity continuity limits of sigma algebras and the relation of the special order 0 to the gaussian dichotomy and contiguity we also show how to generalize the pythagorean inequality to orders different from 1 and we extend the known equivalence between channel capacity and minimax redundancy to continuous channel inputs for all orders and present several other minimax results

shape from silhouette across time part i theory and algorithms shape from silhouette sfs is a shape reconstruction method which constructs a 3d shape estimate of an object using silhouette images of the object the output of a sfs algorithm is known as the visual hull vh traditionally sfs is either performed on static objects or separately at each time instant in the case of videos of moving objects in this paper we develop a theory of performing sfs across time estimating the shape of a dynamic object with unknown motion by combining all of the silhouette images of the object over time we first introduce a one dimensional element called a bounding edge to represent the visual hull we then show that aligning two visual hulls using just their silhouettes is in general ambiguous and derive the geometric constraints in terms of bounding edges that govern the alignment to break the alignment ambiguity we combine stereo information with silhouette information and derive a temporal sfs algorithm which consists of two steps 1 estimate the motion of the objects over time visual hull alignment and 2 combine the silhouette information using the estimated motion visual hull refinement the algorithm is first developed for rigid objects and then extended to articulated objects in the part ii of this paper we apply our temporal sfs algorithm to two human related applications 1 the acquisition of detailed human kinematic models and 2 marker less motion tracking

introducing a new problem shape from silhouette when the relative positions of the viewpoints is unknown 3d shapes can be reconstructed from 2d silhouettes by back projecting them from the corresponding viewpoints and intersecting the resulting solid cones however in many practical cases as observing an aircraft or an asteroid the positions of the viewpoints with respect to the object are not known in these cases the relative position of the solid cones is not known and the intersection cannot be performed the purpose of this paper is introducing and stating in a theoretical framework the problem of understanding 3d shapes from silhouettes when the relative positions of the viewpoints are unknown the results presented provide a first insight into the problem in particular the case of orthographic viewing directions parallel to the same plane is thoroughly discussed and sets of inequalities are presented which allow determining objects compatible with the silhouettes

reconstruction of 3 d geometry using 2 d profiles and a geometric prior model a method has been developed to reconstruct three dimensional 3 d surfaces from two dimensional 2 d projection data it is used to produce individualized boundary element models consisting of thorax and lung surfaces for electro and magnetocardiographic inverse problems two orthogonal projections are utilized a geometrical prior model built using segmented magnetic resonance images is deformed according to profiles segmented from projection images in the authors method virtual x ray images of the prior model are first constructed by simulating real x ray imaging the 2 d profiles of the model are segmented from the projections and elastically matched with the profiles segmented from patient data the displacement vectors produced by the elastic 2 d matching are back projected onto the 3 d surface of the prior model finally the model is deformed using the back projected vectors two different deformation methods are proposed the accuracy of the method is validated by a simulation the average reconstruction error of a thorax and lungs was 1 22 voxels corresponding to about 5 mm

what s next an interactive next best view approach shape from silhouettes algorithms use either a surface or a volumetric approach a problem with the volumetric approach is that in general we know neither the accuracy of the reconstruction nor where to locate new viewpoints for improving the accuracy in this paper we present a general approach to interactive object specific volumetric algorithms based on a necessary condition for the best possible reconstruction to have been performed the outlined approach can be applied to any class of objects as an example of this approach an interactive algorithm has been implemented for convex polyhedra

a test of non identifying restrictions and confidence regions for partially identified parameters abstract we propose an easily implementable test of the validity of a set of theoretical restrictions on the relationship between economic variables which do not necessarily identify the data generating process the restrictions can be derived from any model of interactions allowing censoring and multiple equilibria when the restrictions are parameterized the test can be inverted to yield confidence regions for partially identified parameters thereby complementing other proposals primarily chernozhukov et al chernozhukov v hong h tamer e 2007 estimation and confidence regions for parameter sets in econometric models econometrica 75 1243 1285

how far 3d shapes can be understood from 2d silhouettes each 2d silhouette of a 3d unknown object o constrains o inside the volume obtained by back projecting the silhouette from the corresponding viewpoint a set of silhouettes specifies a boundary volume r obtained by intersecting the volumes due to each silhouette r more or less closely approximates o depending on the viewpoints and the object itself this approach to the reconstruction of 3d objects is usually referred to as volume intersection this correspondence addresses the problem of inferring the shape of the unknown object o from the reconstructed object r for doing this the author divides the points of the surface of r into hard points which belong to the surface of any possible object originating r and soft points which may or may not belong to o the author considers two cases in the first case r is the closest approximation of o which can be obtained from its silhouettes i e its visual hull in the second case r is a generic reconstructed object in both cases the author supplies necessary and sufficient conditions for a point to be hard and gives rules for computing the hard surfaces

robust structure and motion from outlines of smooth curved surfaces this paper addresses the problem of estimating the motion of a camera as it observes the outline or apparent contour of a solid bounded by a smooth surface in successive image frames in this context the surface points that project onto the outline of an object depend on the viewpoint and the only true correspondences between two outlines of the same object are the projections of frontier points where the viewing rays intersect in the tangent plane of the surface in turn the epipolar geometry is easily estimated once these correspondences have been identified given the apparent contours detected in an image sequence a robust procedure based on ransac and a voting strategy is proposed to simultaneously estimate the camera configurations and a consistent set of frontier point projections by enforcing the redundancy of multiview epipolar geometry the proposed approach is in principle applicable to orthographic weak perspective and affine projection models experiments with nine real image sequences are presented for the orthographic projection case including a quantitative comparison with the ground truth data for the six data sets for which the latter information is available sample visual hulls have been computed from all image sequences for qualitative evaluation

singularities affect dynamics of learning in neuromanifolds the parameter spaces of hierarchical systems such as multilayer perceptrons include singularities due to the symmetry and degeneration of hidden units a parameter space forms a geometrical manifold called the neuromanifold in the case of neural networks such a model is identified with a statistical model and a riemannian metric is given by the fisher information matrix however the matrix degenerates at singularities such a singular structure is ubiquitous not only in multilayer perceptrons but also in the gaussian mixture probability densities arma time series model and many other cases the standard statistical paradigm of the cramer rao theorem does not hold and the singularity gives rise to strange behaviors in parameter estimation hypothesis testing bayesian inference model selection and in particular the dynamics of learning from examples prevailing theories so far have not paid much attention to the problem caused by singularity relying only on ordinary statistical theories developed for regular nonsingular models only recently have researchers remarked on the effects of singularity and theories are now being developed this article gives an overview of the phenomena caused by the singularities of statistical manifolds related to multilayer perceptrons and gaussian mixtures we demonstrate our recent results on these problems simple toy models are also used to show explicit solutions we explain that the maximum likelihood estimator is no longer subject to the gaussian distribution even asymptotically because the fisher information matrix degenerates that the model selection criteria such as aic bic and mdl fail to hold in these models that a smooth bayesian prior becomes singular in such models and that the trajectories of dynamics of learning are strongly affected by the singularity causing plateaus or slow manifolds in the parameter space the natural gradient method is shown to perform well because it takes the singular geometrical structure into account the generalization error and the training error are studied in some examples

model averaging in semiparametric estimation of treatment effects choosing the covariates and functional form of the propensity score is an important choice for estimating treatment effects this paper proposes a data driven way of averaging the estimators over candidate specifications to resolve the specification uncertainty in the propensity score weighting estimation of the att the proposed procedures minimize the estimated mse of the att estimator in a local asymptotic framework we formulate model averaging as a statistical decision problem in a limit experiment and derive an averaging scheme that is bayes optimal with respect to a given prior the averaging estimator outperforms selection estimators and the estimators in any of the candidate models in terms of bayes asymptotic mse our monte carlo studies illustrate the size of the mse gains we apply the averaging procedure to evaluate the effect of a labor market program

optimal aggregation of classifiers in statistical learning classification can be considered as nonparametric estimation of sets where the risk is defined by means of a specific distance between sets associated with misclassification error it is shown that the rates of convergence of classifiers depend on two parameters the complexity of the class of candidate sets and the margin parameter the dependence is explicitly given indicating that optimal fast rates approaching o n 1 can be attained where n is the sample size and that the proposed classifiers have the property of robustness to the margin the main result of the paper concerns optimal aggregation of classifiers we suggest a classifier that automatically adapts both to the complexity and to the margin and attains the optimal fast rates up to a logarithmic factor

boundedly pivotal structural change tests in continuous updating gmm with strong weak identification and completely unidentified cases abstract this paper develops structural change tests in the continuous updating gmm framework that are robust to weak identification we propose likelihood ratio like anderson rubin 1949 estimation of the parameters of a single equation in a complete system of stochastic equations annals of mathematical statistics 20 46 63 and kleinbergen 2005 testing parameters in gmm without assuming that they are identified manuscript brown university types of tests since the limits of the test statistics are not nuisance parameter free bounds for the limit of the test statistics are derived the bounds are nuisance parameter free and robust to identification problems simulations show that the anderson rubin 1949 and kleinbergen 2005 testing parameters in gmm without assuming that they are identified manuscript brown university type of tests have very good small sample properties in the case of weak instruments the sup lm test of andrews 1993a tests for parameter instability and structural change with unknown change point econometrica 61 821 856 rejects the true null of parameter stability more than the nominal level and also has low power in the weak instrument setup

finite sample properties of linear model identification in this paper we consider the finite sample properties of prediction error methods using a quadratic criterion function for system identification the problem we pose is how many data points are required to guarantee with high probability that the expected value of the quadratic identification criterion is close to its empirical mean value the sample sizes are obtained using risk minimization theory which provides uniform probabilistic bounds on the difference between the expected value of the squared prediction error and its empirical mean evaluated on a finite number of data points the bounds are very general no assumption is made about the true system belonging to the model class and the noise sequence is not assumed to be uniformly bounded further analysis shows that in order to maintain a given bound on the deviation the number of data points needed grows no faster than quadratically with the number of parameters for fir and arx models

two stage plans for estimating the inverse of a monotone function this study investigates two stage plans based on nonparametric procedures for estimating an inverse regression function at a given point specifically isotonic regression is used at stage one to obtain an initial estimate followed by another round of isotonic regression in the vicinity of this estimate at stage two it is shown that such two stage plans accelerate the convergence rate of one stage procedures and are superior to existing two stage procedures that use local parametric approximations at stage two when the available budget is moderate and or the regression function is ill behaved both wald and likelihood ratio type confidence intervals for the threshold value of interest are investigated and the latter are recommended in applications due to their simplicity and robustness the developed plans are illustrated through a comprehensive simulation study and an application to car fuel efficiency data

a simple nonparametric approach to estimating the distribution of random coefficients in structural models abstract we explore least squares and likelihood nonparametric mixtures estimators of the joint distribution of random coefficients in structural models the estimators fix a grid of heterogeneous parameters and estimate only the weights on the grid points an approach that is computationally attractive compared to alternative nonparametric estimators we provide conditions under which the estimated distribution function converges to the true distribution in the weak topology on the space of distributions we verify most of the consistency conditions for three discrete choice models we also derive the convergence rates of the least squares nonparametric mixtures estimator under additional restrictions we perform a monte carlo study on a dynamic programming model

smoothed quantile regression for panel data abstract this paper studies fixed effects estimation of quantile regression models for panel data under an asymptotic framework where both the numbers of individuals and time periods grow at the same rate we show that the fixed effects estimator for the smoothed objective function has a limiting normal distribution with a bias in the mean and provide the analytic form of the asymptotic bias we propose a one step bias correction estimator based on the analytic bias formula obtained from the asymptotic analysis importantly our results cover the case that observations are dependent over time we illustrate the effects of the bias correction through simulations

on the bootstrap of the maximum score estimator this paper shows that the bootstrap does not consistently estimate the asymptotic distribution of the maximum score estimator the theory developed also applies to other estimators within a cube root convergence class for some single parameter estimators in this class the results suggest a simple method for inference based upon the bootstrap

the visual hull of smooth curved objects the visual hull is a geometric entity that relates the shape of an object to its silhouettes or shadows this paper develops the theory of the visual hull of generic smooth objects we show that the visual hull can be constructed using surfaces which partition the viewpoint space of the aspect graph of the object the surfaces are those generated by the visual events tangent crossing and triple point an analysis based on the shape of the object at the tangency points of these surfaces allows pruning away many surfaces and patches not relevant to the construction an algorithm for computing the visual hull is outlined

generalised relaxed radon transform gr2t for robust inference this paper introduces the generalised relaxed radon transform gr 2t as an extension to the generalised radon transform grt 1 this new modelling allows us to define a new framework for robust inference the resulting objective functions are probability density functions that can be chosen differentiable and that can be optimised using gradient methods one of this cost function is already widely used in the forms of the hough transform and generalised projection based m estimator and it is interpreted as a conditional density function on the latent variables of interest in addition the joint density function of the latent variables is also proposed as a cost function and it has the advantage of including a prior about the latent variable several applications including lines detection in images and volume reconstruction from silhouettes captured from multiple views are presented to underline the versatility of this framework

convex clustering via l1 fusion penalization summary we study the large sample behaviour of a convex clustering framework which minimizes the sample within cluster sum of squares under an l1 fusion constraint on the cluster centroids this recently proposed approach has been gaining in popularity however its asymptotic properties have remained mostly unknown our analysis is based on a novel representation of the sample clustering procedure as a sequence of cluster splits determined by a sequence of maximization problems we use this representation to provide a simple and intuitive formulation for the population clustering procedure we then demonstrate that the sample procedure consistently estimates its population analogue and we derive the corresponding rates of convergence the proof conducts a careful simultaneous analysis of a collection of m estimation problems whose cardinality grows together with the sample size on the basis of the new perspectives gained from the asymptotic investigation we propose a key post processing modification of the original clustering framework we show both theoretically and empirically that the resulting approach can be successfully used to estimate the number of clusters in the population using simulated data we compare the proposed method with existing number of clusters and modality assessment approaches and obtain encouraging results we also demonstrate the applicability of our clustering method to the detection of cellular subpopulations in a single cell virology study

likelihood ratio of unidentifiable models and multilayer neural networks this paper discusses the behavior of the maximum likelihood estimator mle in the case that the true parameter cannot be identified uniquely among many statistical models with unidentifiability neural network models are the main concern of this paper it is known in some models with unidentifiability that the asymptotics of the likelihood ratio of the mle has an unusually larger order using the framework of locally conic models put forth by dacunha castelle and gassiat as a generalization of hartigan s idea a useful sufficient condition of such larger orders is derived this result is applied to neural network models and a larger order is proved if the true function is given by a smaller model also under the condition that the model has at least two redundant hidden units a log n lower bound for the likelihood ratio is derived

dynamics of learning in multilayer perceptrons near singularities the dynamical behavior of learning is known to be very slow for the multilayer perceptron being often trapped in the plateau it has been recently understood that this is due to the singularity in the parameter space of perceptrons in which trajectories of learning are drawn the space is riemannian from the point of view of information geometry and contains singular regions where the riemannian metric or the fisher information matrix degenerates this paper analyzes the dynamics of learning in a neighborhood of the singular regions when the true teacher machine lies at the singularity we give explicit asymptotic analytical solutions trajectories both for the standard gradient sgd and natural gradient ngd methods it is clearly shown in the case of the sgd method that the plateau phenomenon appears in a neighborhood of the critical regions where the dynamical behavior is extremely slow the analysis of the ngd method is much more difficult because the inverse of the fisher information matrix diverges we conquer the difficulty by introducing the blow down technique used in algebraic geometry the ngd method works efficiently and the state converges directly to the true parameters very quickly while it staggers in the case of the sgd method the analytical results are compared with computer simulations showing good agreement the effects of singularities on learning are thus qualitatively clarified for both standard and ngd methods

a note on density model size testing let f sub k sub k spl ges 1 be a nested family of parametric classes of densities with finite vapnik chervonenkis dimension let f be a probability density belonging to f sub k sup where k sup is the unknown smallest integer such that f spl isin f sub k given a random sample x sub 1 x sub n drawn from f an integer k sub 0 spl ges 1 and a real number spl alpha spl isin 0 1 we introduce a new simple explicit spl alpha level consistent testing procedure of the null hypothesis h sub 0 k sup k sub 0 versus the alternative h sub 1 k sup spl ne k sub 0 our method is inspired by the combinatorial tools developed in devroye and lugosi and it includes a wide range of density models such as mixture models neural networks or exponential families

stochastic calculus for fractional brownian motion with hurst exponent h a rough path method by analytic extension the d dimensional fractional brownian motion fbm for short b t b 1 t b d t t r with hurst exponent 0 1 is a d dimensional centered self similar gaussian process with covariance e b i s b i t 1 2 i j lsl 2 ltl 2 t s 2 the long standing problem of defining a stochastic integration with respect to fbm and the related problem of solving stochastic differential equations driven by fbm has been addressed successfully by several different methods although in each case with a restriction on the range of either d or the case corresponds to the usual stochastic integration with respect to brownian motion while most computations become singular when gets under various threshhold values due to the growing irregularity of the trajectories as 0 we provide here a new method valid for any d and for 1 4 by constructing an approximation e t e 0 of fbm which allows to define iterated integrals and then applying the geometric rough path theory the approximation relies on the definition of an analytic process z on the cut plane z c r of which fbm appears to be a boundary value and allows to understand very precisely the well known see 5 but as yet a little mysterious divergence of levy s area for 1 4

an infinite dimensional geometric structure on the space of all the probability measures equivalent to a given one let m be the set of all probability densities equivalent to a given reference probability measure this set is thought of as the maximal regular i e with strictly positive densities dominated statistical model for each f m we define 1 a banach space l f with unit ball v f and 2 a mapping s f from a subset u f of m onto v f in such a way that the system s f u f f m is an affine atlas on m moreover each parametric exponential model dominated by is a finite dimensional affine submanifold and each parametric statistical model dominated by with a suitable regularity is a submanifold the global geometric framework given by the manifold structure adds some insight to the so called geometric theory of statistical models in particular the present paper gives some of the developments connected with the fisher information metrics rao and the hilbert bundle introduced by amari

m kontsevich s graph complex and the grothendieck teichmuller lie algebra we show that the zeroth cohomology of m kontsevich s graph complex is isomorphic to the grothendieck teichmuller lie algebra mathfrak grt 1 the map is explicitly described this result has applications to deformation quantization and duflo theory we also compute the homotopy derivations of the gerstenhaber operad they are parameterized by mathfrak grt 1 up to one class or two depending on the definitions more generally the homotopy derivations of the non unital e n operads may be expressed through the cohomology of a suitable graph complex our methods also give a second proof of a result of h furusho stating that the pentagon equation for mathfrak grt 1 elements implies the hexagon equation

on the rationality homotopy type of configuration spaces in this article we prove that the rational homotopy type of the configuration space of n distinct points on a smooth projective variety x over c is determined by the rational cohomology ring of x together with its canonical orientation class this strengthens a previous result of fulton and macpherson fm see previous issue in this volume a few words are due on the history of differential graded algebras by a dga we shall mean a graded commutative differential graded algebra fulton and macpherson fm gave a model f n for the space f x n by resolving the singularities of the inclusion f x n c xn and using a result of morgan m on the rational homotopy type of smooth quasiprojective varieties over c surprisingly the dga f n depended not only on h x and its orientation class but also on the chern classes of x fulton suggested that one should be able to find another explicit model of f x n independent of chern classes the dga used here e n was found independently by the author and by totaro t totaro s idea was to study the leray spectral sequence of the inclusion f x n c xn by a weight filtration argument he showed that the only nontrivial differential in this spectral sequence is d2m the e2 term with the differential d2m is isomorphic to e n thus by a result of deligne d2 h e n h f x n as rings finally the relations 1 3 of theorem 1 2 are much older than any of the results mentioned above they were discovered by f cohen c see also clma in the study of configuration spaces of ir what happens could be described by saying that h f x n for a smooth projective variety x is related to h f lr2m n in the simplest possible way

tailor made tests for goodness of fit to semiparametric hypotheses 1 introduction the practice of statistical testing plays several roles inempirical research these roles range from the careful assessment of theevidence against speci c scienti c hypotheses to the judgment of whetheran estimated model displays decent goodness of t to the empirical data the paradigmatic situation we consider is one where the investigator viewssome departures from the hypothesized model as being of primary impor tance with others of interest if su ciently gross but otherwise secondary for instance low frequency departures from a signal hypothesized to beconstant might be considered of interest even if of low amplitude whilehigh frequency departures are less so unless they are of high amplitude the optimal testing of a simple hypothesis against a simple alternativeis the cornerstone of modern statistical theory however there is no clearnotion of optimality for more complicated situations the h ajek le camasymptotic theory proved that there exist strong concepts of asymptotic ef ciency in parametric estimation these ideas have been extended to semi parametric models see 3 14 22 however there is no compelling sense ofan asymptotically optimum test in either the parametric or the semipara metric asymptotic theories save for some simple one parameter hypotheses

using a monotone single index model to stabilize the propensity score in missing data problems and causal inference the augmented inverse weighting method is one of the most popular methods for estimating the mean of the response in causal inference and missing data problems an important component of this method is the propensity score popular parametric models for the propensity score include the logistic probit and complementary log log models a common feature of these models is that the propensity score is a monotonic function of a linear combination of the explanatory variables to avoid the need to choose a model we model the propensity score via a semiparametric single index model in which the score is an unknown monotonic nondecreasing function of the given single index under this new model the augmented inverse weighting estimator aiwe of the mean of the response is asymptotically linear semiparametrically efficient and more robust than existing estimators moreover we have made a surprising observation the inverse probability weighting and aiwes based on a correctly specified parametric model may have worse performance than their counterparts based on a nonparametric model a heuristic explanation of this phenomenon is provided a real data example is used to illustrate the proposed methods

on the problem in model selection of neural network regression in overrealizable scenario in considering a statistical model selection of neural networks and radial basis functions under an overrealizable case the problem of unidentifiability emerges because the model selection criterion is an unbiased estimator of the generalization error based on the training error this article analyzes the expected training error and the expected generalization error of neural networks and radial basis functions in overrealizable cases and clarifies the difference from regular models for which identifiability holds as a special case of an overrealizable scenario we assumed a gaussian noise sequence as training data in the least squares estimation under this assumption we first formulated the problem in which the calculation of the expected errors of unidentifiable networks is reduced to the calculation of the expectation of the supremum of the 2 process under this formulation we gave an upper bound of the expected training error and a lower bound of the expected generalization error where the generalization is measured at a set of training inputs furthermore we gave stochastic bounds on the training error and the generalization error the obtained upper bound of the expected training error is smaller than in regular models and the lower bound of the expected generalization error is larger than in regular models the result tells us that the degree of overfitting in neural networks and radial basis functions is higher than in regular models correspondingly it also tells us that the generalization capability is worse than in the case of regular models the article may be enough to show a difference between neural networks and regular models in the context of the least squares estimation in a simple situation this is a first step in constructing a model selection criterion in an overrealizable case further important problems in this direction are also included in this article

parameter identifiability in statistical machine learning a review this review examines the relevance of parameter identifiability for statistical models used in machine learning in addition to defining main concepts we address several issues of identifiability closely related to machine learning showing the advantages and disadvantages of state of the art research and demonstrating recent progress first we review criteria for determining the parameter structure of models from the literature this has three related issues parameter identifiability parameter redundancy and reparameterization second we review the deep influence of identifiability on various aspects of machine learning from theoretical and application viewpoints in addition to illustrating the utility and influence of identifiability we emphasize the interplay among identifiability theory machine learning mathematical statistics information theory optimization theory information geometry riemann geometry symbolic computation bayesian inference algebraic geometry and others finally we present a new perspective together with the associated challenges

parametric rate inference for one sided differentiable parameters suppose one has a collection of parameters indexed by a possibly infinite dimensional set given data generated from some distribution the objective is to estimate the maximal parameter in this

on the geometry of feedforward neural network error surfaces many feedforward neural network architectures have the property that their overall input output function is unchanged by certain weight permutations and sign flips in this paper the geometric structure of these equioutput weight space transformations is explored for the case of multilayer perceptron networks with tanh activation functions similar results hold for many other types of neural networks it is shown that these transformations form an algebraic group isomorphic to a direct product of weyl groups results concerning the root spaces of the lie algebras associated with these weyl groups are then used to derive sets of simple equations for minimal sufficient search sets in weight space these sets which take the geometric forms of a wedge and a cone occupy only a minute fraction of the volume of weight space a separate analysis shows that large numbers of copies of a network performance function optimum weight vector are created by the action of the equioutput transformation group and that these copies all lie on the same sphere some implications of these results for learning are discussed

new dependence coefficients examples and applications to statistics to measure the dependence between a real valued random variable x and a algebra we consider four distances between the conditional distribution function of x given and the distribution function of x the coefficients obtained are weaker than the corresponding mixing coefficients and may be computed in many situations in particular we show that they are well adapted to functions of mixing sequences iterated random functions and dynamical systems starting from a new covariance inequality we study the mean integrated square error for estimating the unknown marginal density of a stationary sequence we obtain optimal rates for kernel estimators as well as projection estimators on a well localized basis under a minimal condition on the coefficients using recent results we show that our coefficients may be also used to obtain various exponential inequalities a concentration inequality for lipschitz functions and a berry esseen type inequality

estimation and confidence regions for parameter sets in econometric models this paper develops a framework for performing estimation and inference in econometric models with partial identification focusing particularly on models characterized by moment inequalities and equalities applications of this framework include the analysis of game theoretic models revealed preference restrictions regressions with missing and corrupted data auction models structural quantile regressions and asset pricing models specifically we provide estimators and confidence regions for the set of minimizers i of an econometric criterion function q in applications the criterion function embodies testable restrictions on economic models a parameter value that describes an economic model satisfies these restrictions if q attains its minimum at this value interest therefore focuses on the set of minimizers called the identified set we use the inversion of the sample analog q n of the population criterion q to construct estimators and confidence regions for the identified set and develop consistency rates of convergence and inference results for these estimators and regions to derive these results we develop methods for analyzing the asymptotic properties of sample criterion functions under set identification

a note on appointment scheduling with piecewise linear cost functions we consider the problem of determining the optimal schedules for a given sequence of jobs on a single processor the objective is to minimize the expected total cost incurred by job waiting and processor idling where the job processing times are random variables it is known in the prior literature that if the processing times are integers and the costs are linear functions satisfying a mild condition then the problem can be solved in a polynomial number of expected cost evaluations in this work we extend the result to piecewise linear cost functions which include many useful objective functions in practice our analysis explores the hidden dual network flow structure of the appointment scheduling problem and thus greatly simplifies that of prior work we also find the number of samples needed to compute a near optimal solution when only some independent samples of processing times are known

computational geometric methods in volumetric intersection for 3d reconstruction abstract generation and analysis of 3d models is an important task that enables automatic inspection placement and manipulation of physical objects by robotic systems this work presents a new approach to the generation of 3d models of arbitrarily shaped physical objects the models are generated from intensity images obtained from known view positions the objects are modelled by contours on successive parallel planes the model becomes more refined as more images are used the contour model supports the ability to generate the surface of an object by means of tessellation we have developed a model that supports efficient intersection not only to perform reconstruction but also to support inclusion testing and collision detection we have implemented this algorithm using images of real objects the algorithm s time complexity is linear with the number of views used and with the number of objects reconstructed the resolution of the algorithm in reconstructing the contours is limited only by machine precision constraints this is because the intersection points of the contours are determined analytically by solving sets of simultaneous equations which describe the line segments that compose the contours

efficient shrinkage in parametric models this paper introduces shrinkage for general parametric models we show how to shrink maximum likelihood estimators towards parameter subspaces de ned by general nonlinear restrictions we derive the asymptotic distribution and risk of the generalized shrinkage estimator using a local asymptotic framework we show that if the shrinkage dimension exceeds two the asymptotic risk of the shrinkage estimator is strictly less than that of the mle this reduction holds globally in the parameter space we show that the reduction in asymptotic risk is substantial even for moderately large values of the parameters the formula simplify in a very convenient way in the context of high dimensional models we derive a simple bound for the asymptotic risk we also provide a new large sample minimax e ciency bound we use the concept of local asymptotic minimax bounds a generalization of the conventional asymptotic minimax bounds the

the visual hull concept for silhouette based image understanding many algorithms for both identifying and reconstructing a 3 d object are based on the 2 d silhouettes of the object in general identifying a nonconvex object using a silhouette based approach implies neglecting some features of its surface as identification clues the same features cannot be reconstructed by volume intersection techniques using multiple silhouettes of the object this paper addresses the problem of finding which parts of a nonconvex object are relevant for silhouette based image understanding for this purpose the geometric concept of visual hull of a 3 d object is introduced this is the closest approximation of object s that can be obtained with the volume intersection approach it is the maximal object silhouette equivalent to s i e which can be substituted for s without affecting any silhouette only the parts of the surface of s that also lie on the surface of the visual hull can be reconstructed or identified using silhouette based algorithms the visual hull depends not only on the object but also on the region allowed to the viewpoint two main viewing regions result in the external and internal visual hull in the former case the viewing region is related to the convex hull of s in the latter it is bounded by s the internal visual hull also admits an interpretation not related to silhouettes algorithms for computing visual hulls are presented and their complexity analyzed in general the visual hull of a 3 d planar face object turns out to be bounded by planar and curved patches

likelihood ratio tests for monotone functions we study the problem of testing for equality at a fixed point in the setting of nonparametric estimation of a monotone function the likelihood ratio test for this hypothesis is derived in the particular case of interval censoring or current status data and its limiting distribution is obtained the limiting distribution is that of the integral of the difference of the squared slope processes corresponding to a canonical version of the problem involving brownian motion t 2 and greatest convex minorants thereof inversion of the family of tests yields pointwise confidence intervals for the unknown distribution function we also study the behavior of the statistic under local and fixed alternatives 1 introduction we shall consider likelihood ratio tests and the corresponding confidence intervals in a class of problems involving nonparametric estimation of a monotone function the problem in each case involves testing the null hypothesis h 0 that the monotone function has a particular value at afixed point t 0 in the domain of the function of course with each testing problem there is a related problem of finding confidence intervals here are some examples of the problems we have in mind

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

difficulty of singularity in population coding fisher information has been used to analyze the accuracy of neural population coding this works well when the fisher information does not degenerate but when two stimuli are presented to a population of neurons a singular structure emerges by their mutual interactions in this case the fisher information matrix degenerates and the regularity condition ensuring the cramer rao paradigm of statistics is violated an animal shows pathological behavior in such a situation we present a novel method of statistical analysis to understand information in population coding in which algebraic singularity plays a major role the method elucidates the nature of the pathological case by calculating the fisher information we then suggest that synchronous firing can resolve singularity and show a method of analyzing the binding problem in terms of the fisher information our method integrates a variety of disciplines in population coding such as nonregular statistics bayesian statistics singularity in algebraic geometry and synchronous firing under the theme of fisher information

dynamics of learning near singularities in layered networks we explicitly analyze the trajectories of learning near singularities in hierarchical networks such as multilayer perceptrons and radial basis function networks which include permutation symmetry of hidden nodes and show their general properties such symmetry induces singularities in their parameter space where the fisher information matrix degenerates and odd learning behaviors especially the existence of plateaus in gradient descent learning arise due to the geometric structure of singularity we plot dynamic vector fields to demonstrate the universal trajectories of learning near singularities the singularity induces two types of plateaus the on singularity plateau and the near singularity plateau depending on the stability of the singularity and the initial parameters of learning the results presented in this letter are universally applicable to a wide class of hierarchical models detailed stability analysis of the dynamics of learning in radial basis function networks and multilayer perceptrons will be presented in separate work