minimum proper loss estimators for parametric models in this paper we propose a methodology for systematically deriving estimators minimizing proper loss functions defined on parametric statistical models by restricting ourselves to losses taking a functional form which allows for straightforward proofs of key properties proving propriety is tantamount to deriving a pairwise divergence quantity between probability measures admitting a natural interpretation of minimum proper loss estimators as divergence minimizers we show that proper losses of varying complexity can be readily constructed given propriety of losses taking a rudimentary form and that for many important models verifying desired properties in the rudimentary case is immediate as a special case we derive computationally tractable estimators requiring only the first and second moments verify strict propriety properties and empirically confirm their utility though parameter estimation tasks using both controlled simulations and real world meteorological network data sets comparisons against numerous standard estimators show that the proposed estimators are at least competitive with and often markedly superior to all standard references uniformly across tasks data sets and model classes suggesting a strong alternative to standard benchmarks in a wide variety of estimation problems

statistical models for autocorrelated count data a generalized linear mixed model is an increasingly popular choice for the modelling of correlated non normal responses in a regression setting a number of methods are currently available for fitting a generalized linear mixed model including monte carlo markov chain maximum likelihood algorithms approximate maximum likelihood pql iterative bias correction and others of interest in this paper is to compare the parameter estimation of the various methods in the modelling of a count data set the incidence of polio in the usa over the period 1970 1983 using a longlinear generalized linear mixed model with an autoregressive correlation structure despite the fact that all of these methods are considered valid modelling techniques we find that parameter estimates and standard errors differ substantially between analyses particularly in the estimation of the parameters describing the random effects distribution a small simulation study is helpful in understanding some of these differences the methods lead to reasonably similar predictions for future observations with small differences observed in some monthly counts

the estimation of age period and cohort effects for vital rates in models for vital rates which include effects due to age period and cohort there is aliasing due to a linear dependence among these three factors this dependence arises both when age and period intervals are equal and when they are not one solution to the dependence is to set an arbitrary constraint on the parameters estimable functions of the parameters are invariant to the particular constraint applied for evenly spaced intervals deviations from linearity are estimable but only a linear function of the three slopes is estimable when age and period intervals have different widths further aliasing occurs it is assumed that the number of deaths in the numerator of the rate equation has a poisson distribution the calculations are illustrated with data on mortality from prostate cancer among nonwhites in the u s

rates of convergence for minimum contrast estimators we shall present here a general study of minimum contrast estimators in a nonparametric setting although our results are also valid in the classical parametric case for independent observations these estimators include many of the most popular estimators in various situations such as maximum likelihood estimators least squares and other estimators of the regression function estimators for mixture models or deconvolution the main theorem relates the rate of convergence of those estimators to the entropy structure of the space of parameters optimal rates depending on entropy conditions are already known at least for some of the models involved and they agree with what we get for minimum contrast estimators as long as the entropy counts are not too large but under some circumstances large entropies or changes in the entropy structure due to local perturbations the resulting the rates are only suboptimal counterexamples are constructed which show that the phenomenon is real for non parametric maximum likelihood or regression this proves that under purely metric assumptions our theorem is optimal and that minimum contrast estimators happen to be suboptimal

finding inverse systems from coordinates abstract let i be a homogeneous ideal in r k x 0 x n such that r i is an artinian gorenstein ring a famous theorem of macaulay says that in this instance i is the ideal of polynomial differential operators with constant coefficients that cancel the same homogeneous polynomial f a major question related to this result is to be able to describe f in terms of the ideal i in this note we give a partial answer to this question by analyzing the case when i is the artinian reduction of the ideal of a reduced arithmetically gorenstein zero dimensional scheme p n we obtain f from the coordinates of the points of

a sequential particle filter method for static models summary particle filter methods are complex inference procedures which combine importance sampling and monte carlo schemes in order to explore consistently a sequence of multiple distributions of interest we show that such methods can also offer an efficient estimation

asymptotically optimal prediction for time varying data generating processes we develop a methodology referred to as kinetic prediction for predicting time series undergoing unknown changes in their data generating distributions based on kolmogorov tikhomirov s varepsilon entropy we propose a concept called varepsilon predictability that quantifies the size of a model class which can be parametric or nonparametric and the maximal number of abrupt structural changes that guarantee the achievability of asymptotically optimal prediction moreover for parametric distribution families we extend the aforementioned kinetic prediction with discretized function spaces to its counterpart with continuous function spaces and propose a sequential monte carlo based implementation we also extend our methodology for predicting smoothly varying data generating distributions under reasonable assumptions we prove that the average predictive performance converges almost surely to the oracle bound which corresponds to the case that the data generating distributions are known in advance the results also shed some light on the so called prediction inference dilemma various examples and numerical results are provided to demonstrate the wide applicability of our methodology

consistency of pseudolikelihood estimation of fully visible boltzmann machines a boltzmann machine is a classic model of neural computation and a number of methods have been proposed for its estimation most methods are plagued by either very slow convergence or asymptotic bias in the resulting estimates here we consider estimation in the basic case of fully visible boltzmann machines we show that the old principle of pseudolikelihood estimation provides an estimator that is computationally very simple yet statistically consistent

bayesian measures of model complexity and fit summary we consider the problem of comparing complex hierarchical models in which the number of parameters is not clearly defined using an information theoretic argument we derive a measure pd for the effective number of parameters in a model as the difference between the posterior mean of the deviance and the deviance at the posterior means of the parameters of interest in general pd approximately corresponds to the trace of the product of fisher s information and the posterior covariance which in normal models is the trace of the hat matrix projecting observations onto fitted values its properties in exponential families are explored the posterior mean deviance is suggested as a bayesian measure of fit or adequacy and the contributions of individual observations to the fit and complexity can give rise to a diagnostic plot of deviance residuals against leverages adding pd to the posterior mean deviance gives a deviance information criterion for comparing models which is related to other information criteria and has an approximate decision theoretic justification the procedure is illustrated in some examples and comparisons are drawn with alternative bayesian and classical proposals throughout it is emphasized that the quantities required are trivial to compute in a markov chain monte carlo analysis

consistency of objective bayes factors as the model dimension grows in the class of normal regression models with a finite number of regressors and for a wide class of prior distributions a bayesian model selection procedure based on the bayes factor is consistent casella and moreno j amer statist assoc 104 2009 1261 1271 however in models where the number of parameters increases as the sample size increases properties of the bayes factor are not totally understood here we study consistency of the bayes factors for nested normal linear models when the number of regressors increases with the sample size we pay attention to two successful tools for model selection schwarz ann statist 6 1978 461 464 approximation to the bayes factor and the bayes factor for intrinsic priors berger and pericchi j amer statist assoc 91 1996 109 122 moreno bertolino and racugno j amer statist assoc 93 1998 1451 1460 we find that the the schwarz approximation and the bayes factor for intrinsic priors are consistent when the rate of growth of the dimension of the bigger model is o n b for b 1 when b 1 the schwarz approximation is always inconsistent under the alternative while the bayes factor for intrinsic priors is consistent except for a small set of alternative models which is characterized

estimation of non normalized statistical models by score matching one often wants to estimate statistical models where the probability density function is known only up to a multiplicative normalization constant typically one then has to resort to markov chain monte carlo methods or approximations of the normalization constant here we propose that such models can be estimated by minimizing the expected squared distance between the gradient of the log density given by the model and the gradient of the log density of the observed data while the estimation of the gradient of log density function is in principle a very difficult non parametric problem we prove a surprising result that gives a simple formula for this objective function the density function of the observed data does not appear in this formula which simplifies to a sample average of a sum of some derivatives of the log density given by the model the validity of the method is demonstrated on multivariate gaussian and independent component analysis models and by estimating an overcomplete filter set for natural image data

pattern avoidance and rational smoothness of schubert varieties abstract letwbe an element of the weyl groupsn and letxwbe the schubert variety associated towin the flag manifoldsln c b lakshmibai and sandhya showed thatxwis smooth if and only ifwavoids the patterns 4231 and 3412 using two tests for rational smoothness from carrell and peterson we show that rational smoothness ofxwis characterized by pattern avoidance for typesbandcas well a key step in the proof of this result is a sequence of rules for factoring the poincare polynomials for the cohomology ring ofxw generalizing the work of gasharov patterns can also be used to characterize actual smoothness of schubert varieties

rationally smooth schubert varieties and inversion hyperplane arrangements abstract we show that an element w of a finite weyl group w is rationally smooth if and only if the hyperplane arrangement i associated to the inversion set of w is inductively free and the product d 1 1 d l 1 of the coexponents d 1 d l is equal to the size of the bruhat interval e w where e is the identity in w as part of the proof we describe exactly when a rationally smooth element in a finite weyl group has a chain billey postnikov decomposition for finite coxeter groups we show that chain billey postnikov decompositions are connected with certain modular coatoms of i

a score regression approach to assess calibration of continuous probabilistic predictions calibration the statistical consistency of forecast distributions and the observations is a central requirement for probabilistic predictions calibration of continuous forecasts is typically assessed using the probability integral transform histogram in this article we propose significance tests based on scoring rules to assess calibration of continuous predictive distributions for an ideal normal forecast we derive the first two moments of two commonly used scoring rules the logarithmic and the continuous ranked probability score this naturally leads to the construction of two unconditional tests for normal predictions more generally we propose a novel score regression approach where the individual scores are regressed on suitable functions of the predictive variance this conditional approach is applicable even for certain nonnormal predictions based on the dawid sebastiani score two case studies illustrate that the score regression approach has typically more power in detecting miscalibrated forecasts than the other approaches considered including a recently proposed technique based on conditional exceedance probability curves

predictive model assessment for count data we discuss tools for the evaluation of probabilistic forecasts and the critique of statistical models for count data our proposals include a nonrandomized version of the probability integral transform marginal calibration diagrams and proper scoring rules such as the predictive deviance in case studies we critique count regression models for patent data and assess the predictive performance of bayesian age period cohort models for larynx cancer counts in germany the toolbox applies in bayesian or classical and parametric or nonparametric settings and to any type of ordered discrete outcomes

the intrinsic bayes factor for model selection and prediction abstract in the bayesian approach to model selection or hypothesis testing with models or hypotheses of differing dimensions it is typically not possible to utilize standard noninformative or default prior distributions this has led bayesians to use conventional proper prior distributions or crude approximations to bayes factors in this article we introduce a new criterion called the intrinsic bayes factor which is fully automatic in the sense of requiring only standard noninformative priors for its computation and yet seems to correspond to very reasonable actual bayes factors the criterion can be used for nested or nonnested models and for multiple model comparison and prediction from another perspective the development suggests a general definition of a reference prior for model comparison

auxiliary mixture sampling for parameter driven models of time series of counts with applications to state space modelling we consider parameter driven models of time series of counts where the observations are assumed to arise from a poisson distribution with a mean changing over time according to a latent process estimation of these models is carried out within a bayesian framework using data augmentation and markov chain monte carlo methods we suggest a new auxiliary mixture sampler which possesses a gibbsian transition kernel where we draw from full conditional distributions belonging to standard distribution families only emphasis lies on application to state space modelling of time series of counts but we show that auxiliary mixture sampling may be applied to a wider range of parameter driven models including random effects models and panel data models based on the poisson distribution

smc 2 an efficient algorithm for sequential analysis of state space models we consider the generic problem of performing sequential bayesian inference in a state space model with observation process y state process x and fixed parameter theta an idealized approach would be to apply the iterated batch importance sampling ibis algorithm of chopin 2002 this is a sequential monte carlo algorithm in the theta dimension that samples values of theta reweights iteratively these values using the likelihood increments p y t y 1 t 1 theta and rejuvenates the theta particles through a resampling step and a mcmc update step in state space models these likelihood increments are intractable in most cases but they may be unbiasedly estimated by a particle filter in the x dimension for any fixed theta this motivates the smc 2 algorithm proposed in this article a sequential monte carlo algorithm defined in the theta dimension which propagates and resamples many particle filters in the x dimension the filters in the x dimension are an example of the random weight particle filter as in fearnhead et al 2010 on the other hand the particle markov chain monte carlo pmcmc framework developed in andrieu et al 2010 allows us to design appropriate mcmc rejuvenation steps thus the theta particles target the correct posterior distribution at each iteration t despite the intractability of the likelihood increments we explore the applicability of our algorithm in both sequential and non sequential applications and consider various degrees of freedom as for example increasing dynamically the number of x particles we contrast our approach to various competing methods both conceptually and empirically through a detailed simulation study included here and in a supplement and based on particularly challenging examples

on the optimal approximation rates for criss cross finite element spaces abstract let denote the triangulation of the plane obtained by multi integer translates of the four lines x 0 y 0 x y and x y by l k h we mean the space of all piecewise polynomials of degree k with respect to the scaled triangulation h having continuous partial derivatives of order on r 2 we show that the approximation properties of l k h are completely governed by those of the space spanned by the translates of all so called box splines contained in l k h combining this fact with fourier analysis techniques allows us to determine the optimal controlled approximation rates for the above subspace of box splines where is the largest degree of smoothness for which these spaces are dense as h tends to zero furthermore we study the question of local linear dependence of the translates of the box splines for the above criss cross triangulations

particle markov chain monte carlo methods summary markov chain monte carlo and sequential monte carlo methods have emerged as the two main tools to sample from high dimensional probability distributions although asymptotic convergence of markov chain monte carlo algorithms is ensured under weak assumptions the performance of these algorithms is unreliable when the proposal distributions that are used to explore the space are poorly chosen and or if highly correlated variables are updated independently we show here how it is possible to build efficient high dimensional proposal distributions by using sequential monte carlo methods this allows us not only to improve over standard markov chain monte carlo schemes but also to make bayesian inference feasible for a large class of statistical models where this was not previously so we demonstrate these algorithms on a non linear state space model and a levy driven stochastic volatility model

on the optimality of conditional expectation as a bregman predictor we consider the problem of predicting a random variable x from observations denoted by a random variable z it is well known that the conditional expectation e x z is the optimal l sup 2 predictor also known as the least mean square error predictor of x among all borel measurable functions of z in this orrespondence we provide necessary and sufficient conditions for the general loss functions under which the conditional expectation is the unique optimal predictor we show that e x z is the optimal predictor for all bregman loss functions blfs of which the l sup 2 loss function is a special case moreover under mild conditions we show that the blfs are exhaustive i e if for every random variable x the infimum of e f x y over all constants y is attained by the expectation e x then f is a blf

bayesian model comparison with the hyvarinen score computation and consistency the bayes factor is a widely used criterion in model comparison and its logarithm is a difference of out of sample predictive scores under the logarithmic scoring rule however when some of the ca

bandwidth choice for density derivatives an adaptation of least squares cross validation is proposed for bandwidth choice in the kernel estimation of the derivatives of a probability density the practicality of the method is demonstrated by an example and a simulation study theoretical justification is provided by an asymptotic optimality result

information geometry of boltzmann machines a boltzmann machine is a network of stochastic neurons the set of all the boltzmann machines with a fixed topology forms a geometric manifold of high dimension where modifiable synaptic weights of connections play the role of a coordinate system to specify networks a learning trajectory for example is a curve in this manifold it is important to study the geometry of the neural manifold rather than the behavior of a single network in order to know the capabilities and limitations of neural networks of a fixed topology using the new theory of information geometry a natural invariant riemannian metric and a dual pair of affine connections on the boltzmann neural network manifold are established the meaning of geometrical structures is elucidated from the stochastic and the statistical point of view this leads to a natural modification of the boltzmann machine learning rule

econometric models for count data with an application to the patents r d relationship this paper focuses on developing and adapting statistical models of counts nonnegative integers in the context of panel data and using them to analyze the relationship between patents and r d expenditures since a variety of other economic data come in the form of repeated counts of some individual actions or events the methodology should have wide applications the statistical models we develop are applications and generalizations of the poisson distribution two important issues are i given the panel nature of our data how can we allow for separate persistent individual fixed or random effects ii how does one introduce the equivalent of disturbances in the equation into the analysis of poisson and other discrete probability functions the first problem is solved by conditioning on the total sum of outcomes over the observed years while the second problem is solved by introducing an additional source of randomness allowing the poisson parameter to be itself randomly distributed and compounding the two distributions lastly we develop a test statistic for the presence of serial correlation when fixed effects estimators are used in nonlinear conditional models

assigning a value to a power likelihood in a general bayesian model summarybayesian robustness under model misspecification is a current area of active research among recent ideas is that of raising the likelihood function to a power in this paper we discuss the choice of appropriate power and provide examples

multivariate spline spaces in principle all of the properties and applications of multivariate spline functions can be derived from the basic theories in chapter 1 the expressions of the multivariate spline functions in 2 of chapter 1 are inconvenient to be applied directly because the expressions also rely on the solutions of the global conformality conditions so as to determine all the smooth cofactors

bregman divergence bounds and universality properties of the logarithmic loss a loss function measures the discrepancy between the true values and their estimated fits for a given instance of data in classification problems a loss function is said to be proper if a minimizer of the expected loss is the true underlying probability we show that for binary classification the divergence associated with smooth proper and convex loss functions is upper bounded by the kullback leibler kl divergence to within a normalization constant this implies that by minimizing the logarithmic loss associated with the kl divergence we minimize an upper bound to any choice of loss from this set as such the logarithmic loss is universal in the sense of providing performance guarantees with respect to a broad class of accuracy measures importantly this notion of universality is not problem specific enabling its use in diverse applications including predictive modeling data clustering and sample complexity analysis generalizations to arbitary finite alphabets are also developed the derived inequalities extend several well known f divergence results

vector bundles on complex projective spaces and systems of partial differential equations i on etablit et on etudie une relation entre l espace des solutions d un systeme d equations aux derivees partielles a coefficient constant et la cohomologie d un fibre vectoriel associe faisceau reflexif sur un espace projectif complexe

a bayesian generalized age period cohort power model for cancer projections age period cohort apc models are the state of art in cancer projections assessing past and recent trends and extrapolating mortality or incidence data into the future nordpred is a well established software assuming a poisson distribution for the counts and a log link or power link function with fixed power however its predictive performance is poor for sparse data bayesian models with log link function have been applied but they can lead to extreme estimates in this paper we address criticisms of the aforementioned models by providing bayesian formulations based on a power link and develop a generalized apc power link model which assumes a random rather than fixed power parameter in addition a power model with a fixed power parameter of five was formulated in the bayesian framework the predictive performance of the new models was evaluated on swiss lung cancer mortality data using model based estimates of observed periods results indicated that the generalized apc power link model provides best estimates for male and female lung cancer mortality the gender specific models were further applied to project lung cancer mortality in switzerland during the periods 2009 2013 and 2014 2018 copyright 2014 john wiley sons ltd

robust estimation under heavy contamination using unnormalized models contamination caused by outliers is inevitable in data analysis and robust statistical methods are often needed in this paper we develop a new approach for robust data analysis on the basis of scoring rules a scoring rule is a discrepancy measure to assess the quality of probabilistic forecasts we propose a simple method of estimating not only parameters in the statistical model but also the contamination ratio i e the ratio of outliers the outliers are detected based on the estimated contamination ratio for this purpose we use scoring rules with extended statistical models called unnormalized models regression problems are also considered we study complex heterogeneous contamination wherein the contamination ratio in a response variable may depend on covariate variables and propose a simple method to estimate a robust regression function and expected contamination ratio simulation studies demonstrate the effectiveness of our method

sequential monte carlo samplers summary we propose a methodology to sample sequentially from a sequence of probability distributions that are defined on a common space each distribution being known up to a normalizing constant these probability distributions are approximated by a cloud of weighted random samples which are propagated over time by using sequential monte carlo methods this methodology allows us to derive simple algorithms to make parallel markov chain monte carlo algorithms interact to perform global optimization and sequential bayesian estimation and to compute ratios of normalizing constants we illustrate these algorithms for various integration tasks arising in the context of bayesian inference

robust and efficient estimation by minimising a density power divergence a minimum divergence estimation method is developed for robust parameter estimation the proposed approach uses new density based divergences which unlike existing methods of this type such as minimum hellinger distance estimation avoid the use of nonparametric density estimation and associated complications such as bandwidth selection the proposed class of density power divergences is indexed by a single parameter which controls the trade off between robustness and efficiency the methodology affords a robust extension of maximum likelihood estimation for which 0 choices of near zero afford considerable robustness while retaining efficiency close to that of maximum likelihood

generic singularities of certain schubert varieties let g be a connected semisimple algebraic group b a borel subgroup t a maximal torus in b with weyl group w and q a subgroup containing b for w in w let x wq denote the schubert variety overline bwq q for y in w such that x yq subseteq x wq one knows that byq q admits a t stable transversal in x wq which we denote by cal n yq wq we prove that under certain hypotheses cal n yq wq is isomorphic to the orbit closure of a highest weight vector in a certain weyl module we also obtain a generalisation of this result under slightly weaker hypotheses further we prove that our hypotheses are satisfied when q is a maximal parabolic subgroup corresponding to a minuscule or cominuscule fundamental weight and x yq is an irreducible component of the boundary of x wq that is the complement of the open orbit of the stabiliser in g of x wq as a consequence we describe the singularity of x wq along byq q and obtain that the boundary of x wq equals its singular locus

coxeter and crystallographic arrangements are inductively free abstract using the classification of finite weyl groupoids we prove that crystallographic arrangements a large subclass of the class of simplicial arrangements which was recently defined are hereditarily inductively free in particular all crystallographic reflection arrangements are hereditarily inductively free among them the arrangement of type e 8 with little extra work we prove that also all coxeter arrangements are inductively free

a novel parameter estimation method for boltzmann machines we propose a novel estimator for a specific class of probabilistic models on discrete spaces such as the boltzmann machine the proposed estimator is derived from minimization of a convex risk function and can be constructed without calculating the normalization constant whose computational cost is exponential order we investigate statistical properties of the proposed estimator such as consistency and asymptotic normality in the framework of the estimating function small experiments show that the proposed estimator can attain comparable performance to the maximum likelihood expectation at a much lower computational cost and is applicable to high dimensional data

game theory maximum entropy minimum discrepancy and robust bayesian decision theory we describe and develop a close relationship between two problems that have customarily been regarded as distinct that of maximizing entropy and that of minimizing worst case expected loss using a formulation grounded in the equilibrium theory of zero sum games between decision maker and nature these two problems are shown to be dual to each other the solution to each providing that to the other although topsoe described this connection for the shannon entropy over 20 years ago it does not appear to be widely known even in that important special case we here generalize this theory to apply to arbitrary decision problems and loss functions we indicate how an appropriate generalized definition of entropy can be associated with such a problem and we show that subject to certain regularity conditions the above mentioned duality continues to apply in this extended context this simultaneously provides a possible rationale for maximizing entropy and a tool for finding robust bayes acts we also describe the essential identity between the problem of maximizing entropy and that of minimizing a related discrepancy or divergence between distributions this leads to an extension to arbitrary discrepancies of a well known minimax theorem for the case of kullback leibler divergence the redundancy capacity theorem of information theory for the important case of families of distributions having certain mean values specified we develop simple sufficient conditions and methods for identifying the desired solutions we use this theory to introduce a new concept of generalized exponential family linked to the specific decision problem under consideration and we demonstrate that this shares many of the properties of standard exponential families finally we show that the existence of an equilibrium in our game can be rephrased in terms of a pythagorean property of the related divergence thus generalizing previously announced results for kullback leibler and bregman divergences

model choice a minimum posterior predictive loss approach summary model choice is a fundamental and much discussed activity in the analysis of datasets nonnested hierarchical models introducing random effects may not be handled by classical methods bayesian approaches using predictive distributions can be used though the formal solution which includes bayes factors as a special case can be criticised we propose a predictive criterion where the goal is good prediction of a replicate of the observed data but tempered by fidelity to the observed values we obtain this criterion by minimising posterior loss for a given model and then for models under consideration selecting the one which minimises this criterion for a broad range of losses the criterion emerges as a form partitioned into a goodness of fit term and a penalty term we illustrate its performance with an application to a large dataset involving residential property transactions

fractional bayes factors for model comparison bayesian comparison of models is achieved simply by calculation of posterior probabilities of the models themselves however there are difficulties with this approach when prior information about the parameters of the various models is weak partial bayes factors offer a resoiution of the problem by setting aside part of the data as a training sampie the training sampie is used to obtain an initiai informative posterior distribution of the parameters in each model model comparison is then based on a bayes factor calculated from the remaining data properties of partial bayes factors are discussed particularly in the context of weak prior information and they are found to have advantages over other proposed methods of model comparison a new variant of the partial bayes factor the fractional bayes factor is advocated on grounds of consistency simplicity robustness and coherence

linear estimating equations for exponential families with application to gaussian linear concentration models abstract in many families of distributions maximum likelihood estimation is intractable because the normalization constant for the density which enters into the likelihood function is not easily available the score matching estimator 35 provides an alternative where this normalization constant is not required for an exponential family e g a gaussian linear concentration model the corresponding estimating equations become linear 2 36 and the score matching estimator is shown to be consistent and asymptotically normally distributed as the number of observations increase to infinity although not necessarily efficient for linear concentration models that are also linear in the covariance 37 we show that the score matching estimator is identical to the maximum likelihood estimator hence in such cases it is also efficient gaussian graphical models and graphical models with symmetries 32 form particularly interesting subclasses of linear concentration models and we investigate the potential use of the score matching estimator for this case

arrangements of ideal type abstract in 2006 sommers and tymoczko defined so called arrangements of ideal type a i stemming from ideals i in the set of positive roots of a reduced root system they showed in a case by case argument that a i is free if the root system is of classical type or g 2 and conjectured that this is also the case for all types this was established only very recently in a uniform manner by abe barakat cuntz hoge and terao the set of non zero exponents of the free arrangement a i is given by the dual of the height partition of the roots in the complement of i in the set of positive roots generalizing the shapiro steinberg kostant theorem which asserts that the dual of the height partition of the set of positive roots gives the exponents of the associated weyl group our first aim in this paper is to investigate a stronger freeness property of the a i we show that all a i are inductively free with the possible exception of some cases in type e 8 in the same paper from 2006 sommers and tymoczko define a poincare polynomial i t associated with each ideal i which generalizes the poincare polynomial w t for the underlying weyl group w solomon showed that w t satisfies a product decomposition depending on the exponents of w for any coxeter group w sommers and tymoczko showed in a case by case analysis in types a n b n and c n and some small rank exceptional types that a similar factorization property holds for the poincare polynomials i t generalizing the formula of solomon for w t they conjectured that their multiplicative formula for i t holds in all types in our second aim to investigate this conjecture further the same inductive tools we develop to obtain inductive freeness of the a i are also employed here we also show that this conjecture holds inductively in almost all instances with only a small number of possible exceptions

training products of experts by minimizing contrastive divergence it is possible to combine multiple latent variable models of the same data by multiplying their probability distributions together and then renormalizing this way of combining individual expert models makes it hard to generate samples from the combined model but easy to infer the values of the latent variables of each expert because the combination rule ensures that the latent variables of different experts are conditionally independent when given the data a product of experts poe is therefore an interesting candidate for a perceptual system in which rapid inference is vital and generation is unnecessary training a poe by maximizing the likelihood of the data is difficult because it is hard even to approximate the derivatives of the renormalization term in the combination rule fortunately a poe can be trained using a different objective function called contrastive divergence whose derivatives with regard to the parameters can be approximated accurately and efficiently examples are presented of contrastive divergence learning using several types of expert on several types of data

blind separation of mixture of independent sources through a quasi maximum likelihood approach we propose two methods for separating mixture of independent sources without any precise knowledge of their probability distribution they are obtained by considering a maximum likelihood ml solution corresponding to some given distributions of the sources and relaxing this assumption afterward the first method is specially adapted to temporally independent non gaussian sources and is based on the use of nonlinear separating functions the second method is specially adapted to correlated sources with distinct spectra and is based on the use of linear separating filters a theoretical analysis of the performance of the methods has been made a simple procedure for optimally choosing the separating functions is proposed further in the second method a simple implementation based on the simultaneous diagonalization of two symmetric matrices is provided finally some numerical and simulation results are given illustrating the performance of the method and the good agreement between the experiments and the theory

information geometry of u boost and bregman divergence we aim at an extension of adaboost to u boost in the paradigm to build a stronger classification machine from a set of weak learning machines a geometric understanding of the bregman divergence defined by a generic convex function u leads to the u boost method in the framework of information geometry extended to the space of the finite measures over a label set we propose two versions of u boost learning algorithms by taking account of whether the domain is restricted to the space of probability functions in the sequential step we observe that the two adjacent and the initial classifiers are associated with a right triangle in the scale via the bregman divergence called the pythagorean relation this leads to a mild convergence property of the u boost algorithm as seen in the expectation maximization algorithm statistical discussions for consistency and robustness elucidate the properties of the u boost methods based on a stochastic assumption for training data

bayesian prediction of spatial count data using generalized linear mixed models summary spatial weed count data are modeled and predicted using a generalized linear mixed model combined with a bayesian approach and markov chain monte carlo informative priors for a data set with sparse sampling are elicited using a previously collected data set with extensive sampling furthermore we demonstrate that so called langevin hastings updates are useful for efficient simulation of the posterior distributions and we discuss computational issues concerning prediction

bounds on the deficit in the logarithmic sobolev inequality abstract the deficit in the logarithmic sobolev inequality for the gaussian measure is considered and estimated by means of transport and information theoretic distances

approximate bayesian inference for latent gaussian models by using integrated nested laplace approximations summary structured additive regression models are perhaps the most commonly used class of models in statistical applications it includes among others generalized linear models generalized additive models smoothing spline models state space models semiparametric regression spatial and spatiotemporal models log gaussian cox processes and geostatistical and geoadditive models we consider approximate bayesian inference in a popular subset of structured additive regression models latent gaussian models where the latent field is gaussian controlled by a few hyperparameters and with non gaussian response variables the posterior marginals are not available in closed form owing to the non gaussian response variables for such models markov chain monte carlo methods can be implemented but they are not without problems in terms of both convergence and computational time in some practical applications the extent of these problems is such that markov chain monte carlo sampling is simply not an appropriate tool for routine analysis we show that by using an integrated nested laplace approximation and its simplified version we can directly compute very accurate approximations to the posterior marginals the main benefit of these approximations is computational where markov chain monte carlo algorithms need hours or days to run our approximations provide more precise estimates in seconds or minutes another advantage with our approach is its generality which makes it possible to perform bayesian analysis in an automatic streamlined way and to compute model comparison criteria and various predictive measures so that models can be compared and the model under study can be challenged

efficient probabilistic forecasts for counts summary efficient probabilistic forecasts of integer valued random variables are derived the optimality is achieved by estimating the forecast distribution non parametrically over a given broad model class and proving asymptotic non parametric efficiency in that setting the method is developed within the context of the integer auto regressive class of models which is a suitable class for any count data that can be interpreted as a queue stock birth and death process or branching process the theoretical proofs of asymptotic efficiency are supplemented by simulation results that demonstrate the overall superiority of the non parametric estimator relative to a misspecified parametric alternative in large but finite samples the method is applied to counts of stock market iceberg orders a subsampling method is used to assess sampling variation in the full estimated forecast distribution and a proof of its validity is given

bayesianly justifiable and relevant frequency calculations for the applied statistician a common reaction among applied statisticians is that the bayesian statistician s energies in an applied problem must be directed at the a priori elicitation of one model specification from which an optimal design and all inferences follow automatically by applying bayes s theorem to calculate conditional distributions of unknowns given knowns i feel however that the applied bayesian statistician s tool kit should be more extensive and include tools that may be usefully labeled frequency calculations three types of bayesianly justifiable and relevant frequency calculations are presented using examples to convey their use for the applied statistician

existence and uniqueness of proper scoring rules to discuss the existence and uniqueness of proper scoring rules one needs to extend the associated entropy functions as sublinear functions to the conic hull of the prediction set in some natural function spaces such as the lebesgue lp spaces over rd the positive cones have empty interior entropy functions defined on such cones have directional derivatives only which typically exist on large subspaces and behave similarly to gradients certain entropies may be further extended continuously to open cones in normed spaces containing signed densities the extended densities are g teaux differentiable except on a negligible set and have everywhere continuous subgradients due to the supporting hyperplane theorem we introduce the necessary framework from analysis and algebra that allows us to give an affirmative answer to the titular question of the paper as a result of this we give a formal sense in which entropy functions have uniquely associated proper scoring rules we illustrate our framework by studying the derivatives and subgradients of the following three prototypical entropies shannon entropy hyvarinen entropy and quadratic entropy

generalized exponents of a free arrangement of hyperplanes and shepherd todd brieskorn formula we define an n arrangement as a finite family of hyperplanes through the origin in c 1 in 11 and 12 we studied the free arrangement and defined its structure sequence their definitions will be given again in sect 2 in this article we say the generalized exponents instead of the structure sequence let d o d 1 d be the generalized exponents of a free n arrangement x let xj u h our main result is hex

non gaussian ornstein uhlenbeck based models and some of their uses in financial economics non gaussian processes of ornstein uhlenbeck ou type offer the possibility of capturing important distributional deviations from gaussianity and for flexible modelling of dependence structures this paper develops this potential drawing on and extending powerful results from probability theory for applications in statistical analysis their power is illustrated by a sustained application of ou processes within the context of finance and econometrics we construct continuous time stochastic volatility models for financial assets where the volatility processes are superpositions of positive ou processes and we study these models in relation to financial data and theory

econometric analysis of realized volatility and its use in estimating stochastic volatility models summary the availability of intraday data on the prices of speculative assets means that we can use quadratic variation like measures of activity in financial markets called realized volatility to study the stochastic properties of returns here under the assumption of a rather general stochastic volatility model we derive the moments and the asymptotic distribution of the realized volatility error the difference between realized volatility and the discretized integrated volatility which we call actual volatility these properties can be used to allow us to estimate the parameters of stochastic volatility models without recourse to the use of simulation intensive methods

strictly proper scoring rules prediction and estimation scoring rules assess the quality of probabilistic forecasts by assigning a numerical score based on the predictive distribution and on the event or value that materializes a scoring rule is proper if the forecaster maximizes the expected score for an observation drawn from the distributionf if he or she issues the probabilistic forecast f rather than g f it is strictly proper if the maximum is unique in prediction problems proper scoring rules encourage the forecaster to make careful assessments and to be honest in estimation problems strictly proper scoring rules provide attractive loss and utility functions that can be tailored to the problem at hand this article reviews and develops the theory of proper scoring rules on general probability spaces and proposes and discusses examples thereof proper scoring rules derive from convex functions and relate to information measures entropy functions and bregman divergences in the case of categorical variables we prove a rigorous version of the

polynomial solutions to constant coefficient differential equations let d 1 d r c x 1 x n be constant coefficient differnetial operators with zero constant term let s f c x 1 x n d j f 0 for all 1 j r be the space of polynomial solutions to the system of simultaneous differential equations d j f 0 it is proved that s is a module over d v the ring of differential operators on the affine scheme v with coordinate ring c x 1 x n if v is smooth and irreducible then s is a simple d v module s 1 d v and the generators for d v yield an algorithm for obtaining a basis for s

coherent dispersion criteria for optimal experimental design we characterize those coherent design criteria which depend only on the dispersion matrix assumed proper and nonsingular of the state of nature which may be a parameter vector or a set of future observables and describe the associated decision problems connections are established with the classical approach to optimal design theory for the normal linear model based on concave functions of the information matrix implications of the theory for more general models are also considered

second order efficiency of minimum contrast estimators in a curved exponential family this paper presents a sufficient condition for second order efficiency of an estimator the condition is easily checked in the case of minimum contrast estimators the a minimum contrast estimator is defined and proved to be second order efficient for every a 0 a 1 the fisher scoring method is also considered in the light of second order efficiency it is shown that a contrast function is associated with the second order tensor and the affine connection this fact leads us to prove the above assertions in the differential geometric framework due to amari

new goodness of fit diagnostics for conditional discrete response models abstract this paper proposes new specification tests for conditional models with discrete responses which are key to apply efficient maximum likelihood methods to obtain consistent estimates of partial effects and to get appropriate predictions of the probability of future events in particular we test the static and dynamic ordered choice model specifications and can cover infinite support distributions for e g count data the traditional approach for specification testing of discrete response models is based on probability integral transforms of a jittered discrete data which leads to continuous uniform i i d series under the true conditional distribution then standard specification testing techniques for continuous variables could be applied to the transformed series but the extra randomness from jitters affects the power properties of these methods we investigate in this paper an alternative transformation based only on original discrete data that avoids any randomization we analyze the asymptotic properties of goodness of fit tests based on this new transformation and explore the properties in finite samples of a bootstrap algorithm to approximate the critical values of test statistics which are model and parameter dependent we show analytically and in simulations that our approach dominates the methods based on randomization in terms of power we apply the new tests to models of the monetary policy conducted by the federal reserve

tests for detecting overdispersion in poisson regression models abstract poisson regression models are widely used in analyzing count data this article develops tests for detecting extra poisson variation in such situations the tests can be obtained as score tests against arbitrary mixed poisson alternatives and are generalizations of tests of fisher 1950 and collings and margolin 1985 accurate approximations for computing significance levels are given and the power of the tests against negative binomial alternatives is compared with those of the pearson and deviance statistics one way to test for extra poisson variation is to fit models that parametrically incorporate and then test for the absence of such variation within the models for example negative binomial models can be used in this way cameron and trivedi 1986 lawless 1987a the tests in this article require only the poisson model to be fitted two test statistics are developed that are motivated partly by a desire to have good distributional approximations for computing significance levels simu

a basis for polynomial solutions to systems of linear constant coefficient pde s abstract letkrepresent either the real or the complex numbers letpk k 1 2 rbe constant coefficient with coefficients fromk polynomials innvariables and letn m u x k x m pk x1 xn u x 0 k 1 2 rbe the set of all polynomial solutions of degree m to this system of partial differential equations we solve the problem of finding an easily computed basis for the vector spacen m to do this we use a certain associative and commutative algebra defined overk namelyk k 1 2 n where pk 0 k 1 r and 1m1 2m2 nmn 0 m1 mn m 1 let the vector spacek mequal the span overkof 1m1 2m2 nmn m1 mn m we show how the expression mj 0 x1 1 xn n j j can be used to find an easily computed basis forn m

probabilistic forecasts calibration and sharpness summary probabilistic forecasts of continuous variables take the form of predictive densities or predictive cumulative distribution functions we propose a diagnostic approach to the evaluation of predictive performance that is based on the paradigm of maximizing the sharpness of the predictive distributions subject to calibration calibration refers to the statistical consistency between the distributional forecasts and the observations and is a joint property of the predictions and the events that materialize sharpness refers to the concentration of the predictive distributions and is a property of the forecasts only a simple theoretical framework allows us to distinguish between probabilistic calibration exceedance calibration and marginal calibration we propose and study tools for checking calibration and sharpness among them the probability integral transform histogram marginal calibration plots the sharpness diagram and proper scoring rules the diagnostic approach is illustrated by an assessment and ranking of probabilistic forecasts of wind speed at the stateline wind energy centre in the us pacific northwest in combination with cross validation or in the time series context our proposal provides very general nonparametric alternatives to the use of information criteria for model diagnostics and model selection

smooth points of t stable varieties in g b and the peterson map let g be a connected semi simple algebraic group defined over an algebraically closed field k and let t b p be respectively a maximal torus a borel subgroup and a parabolic subgroup of g inspired by a beautiful result of dale peterson describing the singular locus of a schubert variety in g b we characterize the t fixed points in the singular locus of an arbitrary irreducible t stable subvariety of g p a t variety for short peterson s result cf the deformation theorem 1 says that if k then a schubert variety x g b is smooth at a t fixed point x if and only if it is smooth at every t fixed point y x in the bruhat chevalley order on the fixed point set x t and all the limits c x x lim z x t z x z c c t of the zariski tangent spaces t z x of x coincide as c varies over the set of all t stable curves in x with c t x y where y x using this peterson showed that if g is simply laced and defined over then every rationally smooth point of a schubert variety in g b is smooth more generally the deformation c x x is defined for any k variety x with a t action provided c is what we call good i e c is a curve of the form c overline tz where z is a smooth point of x x t and x c t our first main result theorem 1 4 says that if x x is an attractive fixed point then x is smooth at x if and only if there exist at least two good c containing x such that c x x te x x where te x x denotes the span of the tangent lines of the t stable curves in x containing x in addition if x is cohen macaulay at x and c x x te x x for even one good c then x is smooth at x our second main result theorem 1 6 says that if x is a t variety in g p where g is simply laced then c x x te x x for each good c this is not true for general g but when g has no g2 factors then c x x is always contained in the linear span of the reduced tangent cone to x at x these results lead to several descriptions of the smooth fixed points of a t variety in g p and in particular they give simple proofs of peterson s results valid for any algebraically closed field we also show cf example 7 1 that there can exist t stable subvarieties in g b where g is simply laced which have rationally smooth t fixed points in their singular loci

jensen divergence based on fisher s information the measure of jensen fisher divergence between probability distributions is introduced and its theoretical grounds set up this quantity in contrast to the remaining jensen divergences grasps the fluctuations of the probability distributions because it is controlled by the local fisher information which is a gradient functional of the distribution so it is appropriate and informative when studying the similarity of distributions mainly for those having oscillatory character the new jensen fisher divergence shares with the jensen shannon divergence the following properties non negativity additivity when applied to an arbitrary number of probability densities symmetry under exchange of these densities vanishing under certain conditions and definiteness even when these densities present non common zeros moreover the jensen fisher divergence is shown to be expressed in terms of the relative fisher information as the jensen shannon divergence does in terms of the kullback leibler or relative shannon entropy finally the jensen shannon and jensen fisher divergences are compared for the following three large non trivial and qualitatively different families of probability distributions the sinusoidal generalized gamma like and rakhmanov hermite distributions which are closely related to the quantum mechanical probability densities of numerous physical systems

certain reflexive sheaves on ⁿ_ and a problem in approximation theory this paper establishes a link between certain local problems in the theory of splines and properties of vector bundles and reflexive sheaves on complex projective spaces