sensitivity to distance and baseline distributions in forecast evaluation scoring rules can provide incentives for truthful reporting of probabilities and evaluation measures for the probabilities after the events of interest are observed often the space of events is ordered and an evaluation relative to some baseline distribution is desired scoring rules typically studied in the literature and used in practice do not take account of any ordering of events and they evaluate probabilities relative to a default baseline distribution in this paper we construct rich families of scoring rules that are strictly proper thereby encouraging truthful reporting are sensitive to distance thereby taking into account ordering of events and incorporate a baseline distribution relative to which the value of a forecast is measured in particular we extend the power and pseudospherical families of scoring rules to allow for sensitivity to distance with or without a specified baseline distribution

synchronization in a semiclassical kuramoto model synchronization is a ubiquitous phenomenon occurring in social biological and technological systems when the internal rythms of their constituents are adapted to be in unison as a result of their coupling this natural tendency towards dynamical consensus has spurred a large body of theoretical and experimental research in recent decades the kuramoto model constitutes the most studied and paradigmatic framework in which to study synchronization in particular it shows how synchronization appears as a phase transition from a dynamically disordered state at some critical value for the coupling strength between the interacting units the critical properties of the synchronization transition of this model have been widely studied and many variants of its formulations have been considered to address different physical realizations however the kuramoto model has been studied only within the domain of classical dynamics thus neglecting its applications for the study of quantum synchronization phenomena based on a system bath approach and within the feynman path integral formalism we derive equations for the kuramoto model by taking into account the first quantum fluctuations we also analyze its critical properties the main result being the derivation of the value for the synchronization onset this critical coupling increases its value as quantumness increases as a consequence of the possibility of tunneling that quantum fluctuations provide

a criterion for f subnormality in this note a criterion for f subnormality is obtained inspired by a subnormality criterion proposed by h wielandt see theorem 3 in math z 138 1974 199 203

optimal grouping for group minimax hypothesis testing bayesian hypothesis testing and minimax hypothesis testing represent extreme instances of detection in which the prior probabilities of the hypotheses are either completely and precisely known or are completely unknown group minimax also known as gamma minimax is a robust intermediary between bayesian and minimax hypothesis testing that allows for coarse or partial advance knowledge of the hypothesis priors by using information on sets in which the prior lies existing work on group minimax however does not consider the question of how to define the sets or groups of priors it is assumed that the groups are given in this paper we propose a novel intermediate detection scheme formulated through the quantization of the space of prior probabilities that optimally determines groups and also representative priors within the groups we show that when viewed from a quantization perspective group minimax amounts to determining centroids with a minimax bayes risk error divergence distortion criterion the appropriate bregman divergence for this task in addition the optimal partitioning of the space of prior probabilities is a bregman voronoi diagram together the optimal grouping and representation points are an epsilon net with respect to bayes risk error divergence and permit a rate distortion type asymptotic analysis of detection performance with the number of groups examples of detecting signals corrupted by additive white gaussian noise and of distinguishing exponentially distributed signals are presented

the statistical strength of nonlocality proofs there exist numerous proofs of bell s theorem stating that quantum mechanics is incompatible with local realistic theories of nature here the strength of such nonlocality proofs is defined in terms of the amount of evidence against local realism provided by the corresponding experiments statistical considerations show that the amount of evidence should be measured by the kullback leibler kl or relative entropy divergence the statistical strength of the following proofs is determined bell s original proof and peres optimized variant of it and proofs by clauser horne shimony and holt chsh hardy mermin and greenberger horne and zeilinger ghz the ghz proof is at least four and a half times stronger than all other proofs while of the two party proofs the one of chsh is the strongest

geometry of deformed exponential families invariant dually flat and conformal geometries an information geometrical foundation is established for the deformed exponential families of probability distributions two different types of geometrical structures an invariant geometry and a flat geometry are given to a manifold of a deformed exponential family the two different geometries provide respective quantities such as deformed free energies entropies and divergences the class belonging to both the invariant and flat geometries at the same time consists of exponential and mixture families theq families are characterized from the viewpoint of the invariant and flat geometries the q exponential family is a unique class that has the invariant and flat geometries in the extended class of positive measures furthermore it is the only class of which the riemannian metric is conformally connected with the invariant fisher metric

generalized quantum anharmonic oscillator using an operator ordering approach we construct a generalized expression for the normal ordering of a a m for integral values of m and use the result to study the quantum anharmonic oscillator problem in the heisenberg approach in particular we derive generalized expressions for energy eigenvalues and frequency shifts for the hamiltonian h x2 2 2 2 m xm we also derive a closed form first order multiscale perturbation theoretic operator solution of this hamiltonian with a view to generalize some recent results of bender and bettencourt 1996 phys rev lett 77 4114 1996 phys rev d 54 1710

penalized bregman divergence for large dimensional regression and classification regularization methods are characterized by loss functions measuring data fits and penalty terms constraining model parameters the commonly used quadratic loss is not suitable for classification with binary responses whereas the loglikelihood function is not readily applicable to models where the exact distribution of observations is unknown or not fully specified we introduce the penalized bregman divergence by replacing the negative loglikelihood in the conventional penalized likelihood with bregman divergence which encompasses many commonly used loss functions in the regression analysis classification procedures and machine learning literature we investigate new statistical properties of the resulting class of estimators with the number pn of parameters either diverging with the sample size n or even nearly comparable with n and develop statistical inference tools it is shown that the resulting penalized estimator combined with appropriate penalties achieves the same oracle property as the penalized likelihood estimator but asymptotically does not rely on the complete specification of the underlying distribution furthermore the choice of loss function in the penalized classifiers has an asymptotically relatively negligible impact on classification performance we illustrate the proposed method for quasilikelihood regression and binary classification with simulation evaluation and real data application

the caldeira leggett quantum master equation in wigner phase space continued fraction solution and application to brownian motion in periodic potentials the continued fraction method of solving classical fokker planck equations has been adapted to tackle quantum master equations of the caldeira leggett type this was done taking advantage of the phase space wigner representation of the quantum density matrix the approach differs from those in which some continued fraction expression is found for a certain quantity in that the full solution of the master equation is obtained by continued fraction methods this allows us to study in detail the effects of the environment fluctuations and dissipation on several classes of nonlinear quantum systems we apply the method to the canonical problem of quantum brownian motion in periodic potentials both for cosine and ratchet potentials lacking inversion symmetry

quantum brownian motion and generalizations of hermite polynomials abstract we introduce new families of orthogonal polynomials h d n motivated by the non equilibrium evolution of a quantum brownian particle qbp the h d n s generalize non trivially the standard hermite polynomials employed for classical brownian motion we treat several models labelled by d for a non equilibrium qbp by means of the wigner function w in the presence of a heat bath at thermal equilibrium with and without ab initio friction for long times for a suitable class of initial conditions the non equilibrium wigner function w should approach in some sense the time independent equilibrium wigner function w e q d which describes the thermal equilibrium of the qbp with the heat bath and plays a central role w e q d is chosen to be the weight function which orthogonalizes the h d n s new results on w e q d and on the h d n s are reported we justify the key role of the h d n s as follows using the h d n s moments w e q d n and w n are introduced for w e q d and w respectively at equilibrium all moments w e q d n except the lowest one w e q d 0 vanish identically off equilibrium one expects that for long times for suitable initial conditions i all non equilibrium moments w n except the lowest moment w 0 will approach zero while ii the lowest non equilibrium moment w 0 will tend to w e q d 0 0 to complete the justification we outline how the approximate long time non equilibrium theories determined by w 0 for the different models d yield smoluchowski equations and irreversible evolutions of the qbp towards thermal equilibrium

a strong version of the redundancy capacity theorem of universal coding the capacity of the channel induced by a given class of sources is well known to be an attainable lower bound on the redundancy of universal codes with respect to this class both in the minimax sense and in the bayesian maximin sense we show that this capacity is essentially a lower bound also in a stronger sense that is for most sources in the class this result extends rissanen s 1984 1986 lower bound for parametric families we demonstrate the applicability of this result in several examples e g parametric families with growing dimensionality piecewise fixed sources arbitrarily varying sources and noisy samples of learnable functions finally we discuss implications of our results to statistical inference

conditional entropy and data processing an axiomatic approach based on core concavity this work presents an axiomatization for entropy based on an extension of concavity called core concavity we show that core concavity characterizes the largest class of functions for which the data processing inequality holds under the assumption that conditional entropy is defined as a generalized average also under the same assumption we show that data processing and conditioning reduces entropy properties are equivalent we prove several properties of core concave functions including generalization of perfect secrecy and of fano s inequality we also show that definitions of conditional entropy based on worst case can be retrieved as limit cases of generalized averages a connection between statistical decision making and this axiomatic approach is also presented

general entropy criteria for inverse problems with applications to data compression pattern classification and cluster analysis minimum distance approaches are considered for the reconstruction of a real function from finitely many linear functional values an optimal class of distances satisfying an orthogonality condition analogous to that enjoyed by linear projections in hilbert space is derived these optimal distances are related to measures of distances between probability distributions recently introduced by c r rao and t k nayak 1985 and possess the geometric properties of cross entropy useful in speech and image compression pattern classification and cluster analysis several examples from spectrum estimation and image processing are discussed

an algorithm for calculating γ minimax decision rules under generalized moment conditions we present an algorithm for calculating a minimax decision rule when r is given by a finite number of generalized moment conditions such a decision rule minimizes the maximum of the integrals of the risk function with respect to all distributions in r the inner maximization problem is approximated by a sequence of linear programs this approximation is combined with an elimination technique which quickly reduces the domain of the variables of the outer minimization problem to test for convergence in a final step the inner maximization problem has to be completely solved once for the candidate of the r minimax rule found by the algorithm for an infinite compact parameter space this is done by semi infinite programming the algorithm is applied to calculate robustified bayesian designs in a logistic regression model and r minimax tests in monotone decision problems

relaxation dynamics of a particle in the presence of an external potential exact solution in terms of matrix continued fractions exact expressions for the laplace transform of the after effect function arising from the solution of the langevin or underlying fokker planck equation for brownian motion in an external potential are obtained as a sum of products of infinite matrix continued fractions this is accomplished by reducing the scalar multiterm recurrence relations associated with the fokker planck equation to matrix three term recurrence relations the solution is illustrated by considering the problem of dielectric relaxation of a single axis rotator subjected to both constant and crystalline anisotropy fields

quantum smoluchowski equation for driven systems we consider a driven quantum harmonic oscillator strongly coupled to a heat bath starting from the exact quantum langevin equation we use a green s function approach to determine the corresponding semiclassical equation for the wigner phase space distribution in the limit of high friction and high temperature we apply brinkman s method to derive the quantum smoluchowski equation for the probability distribution in position space we further determine the range of validity of the equation and discuss the special case of a brownian parametric oscillator

time scales and diffusion mechanisms in the kramers equation with periodic potentials i abstract noise assisted diffusion on a periodic substrate is studied solving the klein kramers equation kke by the matrix continued fraction method assuming a periodic cosine potential and a homogeneous friction the one particle dynamics is completely characterized both at short and long times calculating up to large momentum transfer the dynamic structure factor its full width at half maximum the velocity self correlation spectrum and the mean square displacement four different dynamical regimes are found as the potential strength or the friction vary at high potential barriers diffusion proceeds by single or multiple jumps at high and low friction respectively while two different kinds of continuous unactivated diffusion are found at lower barriers these dynamical features are discussed in terms of three characteristic time scales whose relations determine the diffusion mechanisms

maximum entropy versus minimum risk and applications to some classical discrete distributions the game which can be taken to lie behind the maximum entropy principle is studied refining previous techniques new theoretical results are obtained these results are illustrated by concrete examples pertaining to well known classical models

path integral approach to quantum brownian motion abstract we apply the influence functional method of feynman and vernon to the study of brownian motion at arbitrary temperature by choosing a specific model for the dissipative interaction of the system of interest with its environment we are able to evaluate the influence functional in closed form and express it in terms of a few parameters such as the phenomenological viscosity coefficient we show that in the limit h 0 the results obtained from the influence functional formalism reduce to the classical fokker planck equation in the case of a simple harmonic oscillator with arbitrarily strong damping and at arbitrary temperature we obtain an explicit expression for the time evolution of the complete density matrix x x t when the system starts in a particular kind of pure state we compare our results with those of other approaches to the problem of dissipation in quantum mechanics

on σ supersoluble groups and one generalization of clt groups abstract let g be a finite group and i i i be a partition of the set of all primes p a chief factor h k of g is said to be central in g if the semidirect product h k g c g h k is a i group for some i i the group g is said to be nilpotent if either g 1 or every chief factor of g is central let g n be the nilpotent residual of g that is the intersection of all normal subgroups n of g with nilpotent quotient g n then we say that g is supersoluble if each chief factor of g below g n is cyclic in this paper we study properties of supersoluble groups and also consider some applications of such groups in the theory of generalized clt groups

mixability is bayes risk curvature relative to log loss mixability of a loss characterizes fast rates in the online learning setting of prediction with expert advice the determination of the mixability constant for binary losses is straight forward but opaque in the binary case we make this transparent and simpler by characterising mixability in terms of the second derivative of the bayes risk of proper losses we then extend this result to multiclass proper losses where there are few existing results we show that mixability is governed by the maximum eigenvalue of the hessian of the bayes risk relative to the hessian of the bayes risk for log loss we conclude by comparing our result to other work that bounds prediction performance in terms of the geometry of the bayes risk although all calculations are for proper losses we also show how to carry the results across to improper losses

fast rates in statistical and online learning the speed with which a learning algorithm converges as it is presented with more data is a central problem in machine learning a fast rate of convergence means less data is needed for the same level of performance the pursuit of fast rates in online and statistical learning has led to the discovery of many conditions in learning theory under which fast learning is possible we show that most of these conditions are special cases of a single unifying condition that comes in two forms the central condition for proper learning algorithms that always output a hypothesis in the given model and stochastic mixability for online algorithms that may make predictions outside of the model we show that under surprisingly weak assumptions both conditions are in a certain sense equivalent the central condition has a re interpretation in terms of convexity of a set of pseudoprobabilities linking it to density estimation under misspecification for bounded losses we show how the central condition enables a direct proof of fast rates and we prove its equivalence to the bernstein condition itself a generalization of the tsybakov margin condition both of which have played a central role in obtaining fast rates in statistical learning yet while the bernstein condition is two sided the central condition is one sided making it more suitable to deal with unbounded losses in its stochastic mixability form our condition generalizes both a stochastic exp concavity condition identified by juditsky rigollet and tsybakov and vovk s notion of mixability our unifying conditions thus provide a substantial step towards a characterization of fast rates in statistical learning similar to how classical mixability characterizes constant regret in the sequential prediction with expert advice setting

bregman divergences and surrogates for learning bartlett et al 2006 recently proved that a ground condition for surrogates classification calibration ties up their consistent minimization to that of the classification risk and left as an important problem the algorithmic questions about their minimization in this paper we address this problem for a wide set which lies at the intersection of classification calibrated surrogates and those of murata et al 2004 this set coincides with those satisfying three common assumptions about surrogates equivalent expressions for the members sometimes well known follow for convex and concave surrogates frequently used in the induction of linear separators and decision trees most notably they share remarkable algorithmic features for each of these two types of classifiers we give a minimization algorithm provably converging to the minimum of any such surrogate while seemingly different we show that these algorithms are offshoots of the same ldquomasterrdquo algorithm this provides a new and broad unified account of different popular algorithms including additive regression with the squared loss the logistic loss and the top down induction performed in cart c4 5 moreover we show that the induction enjoys the most popular boosting features regardless of the surrogate experiments are provided on 40 readily available domains

semiclassical klein kramers and smoluchowski equations for the brownian motion of a particle in an external potential the quantum brownian motion of a particle in an external potential v x is treated using the master equation for the wigner distribution function w x p t in phase space x p a heuristic method of determination of diffusion coefficients in the master equation is proposed the time evolution equation so obtained contains explicit quantum correction terms up to o 4 and in the classical limit 0 reduces to the klein kramers equation for a quantum oscillator the method yields an evolution equation for w x p t coinciding with that of agarwal 1971 phys rev a 4 739 in the non inertial regime by applying the brinkman expansion of the momentum distribution in weber functions brinkman 1956 physica 22 29 the corresponding semiclassical smoluchowski equation is derived

economic interpretation of probabilities estimated by maximum likelihood or score the conventional method of estimating a probability prediction model by maximum likelihood mle is a form of maximum score estimation with economic meaning of all the probabilities that a given model might have produced those obtained by mle yield maximum in sample betting return to a log utility investor recognition of this affinity between mle and log utility begs the wider methodological question of whether different decision makers benefit in different degrees from different probabilities probabilities produced by mle can be either too conservative or too bold relative to those found by maximizing utility under more risk tolerant or risk averse score functions a very not very risk averse user who bets characteristically small large fractions of wealth based on a conservative forecast is bound to make a rapidly slowly increasing bet as the forecast probability becomes progressively bolder or more distant from the market probability the effect of this interaction between risk aversion and forecast is that a highly risk averse user may need a much bolder forecast to obtain the same certainty equivalent as a more risk tolerant investor it follows more broadly that professional forecasters should anticipate how a client with given risk aversion expects to gain from any given forecast or forecast revision before committing resources toward making a better informed but still honest forecast this paper was accepted by peter wakker decision analysis

composite binary losses we study losses for binary classification and class probability estimation and extend the understanding of them from margin losses to general composite losses which are the composition of a proper loss with a link function we characterise when margin losses can be proper composite losses explicitly show how to determine a symmetric loss in full from half of one of its partial losses introduce an intrinsic parametrisation of composite binary losses and give a complete characterisation of the relationship between proper losses and classification calibrated losses we also consider the question of the best surrogate binary loss we introduce a precise notion of best and show there exist situations where two convex surrogate losses are incommensurable we provide a complete explicit characterisation of the convexity of composite binary losses in terms of the link function and the weight function associated with the proper loss which make up the composite loss this characterisation suggests new ways of surrogate tuning as well as providing an explicit characterisation of when bregman divergences on the unit interval are convex in their second argument finally in an appendix we present some new algorithm independent results on the relationship between properness convexity and robustness to misclassification noise for binary losses and show that all convex proper losses are non robust to misclassification noise

analytical inversion of general tridiagonal matrices in this paper we give a complete analysis for general tridiagonal matrix inversion for both non block and block cases and provide some very simple analytical formulae which immediately lead to closed forms for some special cases such as symmetric or toeplitz tridiagonal matrices

differential privacy for bayesian inference through posterior sampling differential privacy formalises privacy preserving mechanisms that provide access to a database can bayesian inference be used directly to provide private access to data the answer is yes under certain conditions on the prior sampling from the posterior distribution can lead to a desired level of privacy and utility for a uniform treatment we define differential privacy over arbitrary data set metrics outcome spaces and distribution families this allows us to also deal with non i i d or non tabular data sets we then prove bounds on the sensitivity of the posterior to the data which delivers a measure of robustness we also show how to use posterior sampling to provide differentially private responses to queries within a decision theoretic framework finally we provide bounds on the utility of answers to queries and on the ability of an adversary to distinguish between data sets the latter are complemented by a novel use of le cam s method to obtain lower bounds on distinguishability our results hold for arbitrary metrics including those for the common definition of differential privacy for specific choices of the metric we give a number of examples satisfying our assumptions

notions of the ergodic hierarchy for curved statistical manifolds abstract we present an extension of the ergodic mixing and bernoulli levels of the ergodic hierarchy for statistical models on curved manifolds making use of elements of the information geometry this extension focuses on the notion of statistical independence between the microscopical variables of the system moreover we establish an intimately relationship between statistical models and families of probability distributions belonging to the canonical ensemble which for the case of the quadratic hamiltonian systems provides a closed form for the correlations between the microvariables in terms of the temperature of the heat bath as a power law from this we obtain an information geometric method for studying hamiltonian dynamics in the canonical ensemble we illustrate the results with two examples a pair of interacting harmonic oscillators presenting phase transitions and the 2 2 gaussian ensembles in both examples the scalar curvature results a global indicator of the dynamics

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

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

maximum likelihood estimation in log linear models we study maximum likelihood estimation in log linear models under conditional poisson sampling schemes we derive necessary and sufficient conditions for existence of the maximum likelihood estimator mle of the model parameters and investigate estimability of the natural and mean value parameters under a nonexistent mle our conditions focus on the role of sampling zeros in the observed table we situate our results within the framework of extended exponential families and we exploit the geometric properties of log linear models we propose algorithms for extended maximum likelihood estimation that improve and correct the existing algorithms for log linear model analysis

on formations of finite groups with the generalised wielandt property for residuals abstract a formation f of finite groups has the generalised wielandt property for residuals or f is a gwp formation if the f residual of a group generated by two f subnormal subgroups is the subgroup generated by their f residuals we prove that every gwp formation is saturated this is one of the crucial steps in the hunt for a solution of the classification problem

onf subnormal subgroups andf residuals of finite soluble groups all groups that we consider are finite and soluble recall that a formation is a class of groups which is closed under homomorphic images and subdirect products hence if f is a formation and g is a group which is a direct product of the subgroups a and b then g is in f if and only if a and b lie in f more generally doerk and w x hawkes 4 iv 1 18 proved that if g is a group such that g s a b then g s a b where g is the f residual of g that is the smallest normal subgroup of g with quotient in f the main purpose of this paper is the development of this result by means of the concept of f subnormal subgroup suppose that f is a saturated formation a maximal subgroup m of a group g is called f normal in g if grcore m g f a subgroup h of g g is called f subnormal in g if either h s g or there exists a chain h s h f h f f h s g such that h is an f normal maximal 0 1 n i subgroup of h for 0 f i n it is clear that if f s n the saturated iq1 formation of all nilpotent groups the f subnormal subgroups of g are exactly the subnormal subgroups of g let f be a subgroup closed saturated formation containing n it is rather easy to see that if f is closed under the product of normal subgroups then g s a b f for every pair of subnormal subgroups a and b such that g s ab this result does not remain true if a and b are

on sublattices of the subgroup lattice defined by formation fitting sets abstract throughout this paper g always denotes a group and l g is the lattice of all subgroups of g if k h g then h k is called a section of g such a section is called normal if k h g we call any set of normal sections of g a stratification of g provided i is g closed that is h k whenever h k g t l and ii l k h l for each triple k l h where h k and l g now let be any stratification of g then we write l g to denote the set of all subgroups a of g such that a g a g we say that a stratification of g is a formation fitting set of g provided i h k n for every two sections h k h n and ii h v k for every two sections h k v k we prove that the set l g forms a sublattice of l g for every formation fitting set of g and we also discuss some applications of sublattices of this kind

scoring rules generalized entropy and utility maximization information measures arise in many disciplines including forecasting where scoring rules are used to provide incentives for probability estimation signal processing where information gain is measured in physical units of relative entropy decision analysis where new information can lead to improved decisions and finance where investors optimize portfolios based on their private information and risk preferences in this paper we generalize the two most commonly used parametric families of scoring rules and demonstrate their relation to well known generalized entropies and utility functions shedding new light on the characteristics of alternative scoring rules as well as duality relationships between utility maximization and entropy minimization in particular we show that weighted forms of the pseudospherical and power scoring rules correspond exactly to measures of relative entropy divergence with convenient properties and they also correspond exactly to the solutions of expected utility

brownian motion quantum corrections and a generalization of the hermite polynomials the nonequilibrium evolution of a brownian particle in the presence of a heat bath at thermal equilibrium without imposing any friction mechanism from the outset is considered using a suitable family of orthogonal polynomials moments of the nonequilibrium probability distribution for the brownian particle are introduced which fulfill a recurrence relation we review the case of classical brownian motion in which the orthogonal polynomials are the hermite ones and the recurrence relation is a three term one after having performed a long time approximation in the recurrence relation the approximate nonequilibrium theory yields irreversible evolution of the brownian particle towards thermal equilibrium with the heat bath for quantum brownian motion which is the main subject of the present work we restrict ourselves to include the first quantum correction this leads us to introduce a new family of orthogonal polynomials which generalize the hermite ones some general properties of the new family are established the recurrence relation for the new moments of the nonequilibrium distribution including the first quantum correction turns out to be also a three term one which justifies the new family of polynomials a long time approximation on the new three term recurrence relation describes irreversible evolution towards equilibrium for the new moment of lowest order the standard smoluchowski equations for the lowest order moments are recovered consistently both classically and quantum mechanically

a general minimax result for relative entropy suppose nature picks a probability measure p sub spl theta on a complete separable metric space x at random from a measurable set p sub spl theta p spl theta spl theta spl isin spl theta then without knowing spl theta a statistician picks a measure q on s finally the statistician suffers a loss d p sub 0 spl par q the relative entropy between p sub spl theta and q we show that the minimax and maximin values of this game are always equal and there is always a minimax strategy in the closure of the set of all bayes strategies this generalizes previous results of gallager 1979 and davisson and leon garcia 1980

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

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

how to calculate the wigner function from phase space we present a method for the direct computation of the wigner function by solving a coupled system of linear partial differential equations in phase space our modified spectral method relies on chebyshev polynomials since this approach allows us to include arbitrary high orders of partial derivatives our procedure is applicable to arbitrary binding potentials we apply our scheme to wigner functions of the harmonic oscillator the morse oscillator and an asymmetric double well potential

maximum entropy reconstruction using derivative information part 1 fisher information and convex duality maximum entropy spectral density estimation is a technique for reconstructing an unknown density function from some known measurements by maximizing a given measure of entropy of the estimate here we present a variety of new entropy measures which attempt to control derivative values of the densities our models apply among others to the inference problem based on the averaged fisher information measure the duality theory we develop resembles models used in convex optimal control problems we present a variety of examples including relaxed moment matching with fisher information and best interpolation on a strip

minimax robust a priori information aware channel equalization an a priori information aware channel equalizer accepts some prior information about the unknown transmit symbols in addition to the knowledge of channel coefficients and noise covariance and therefore offers a superior performance as compared to an equalizer that lacks this additional information such an equalizer is employed for instance in turbo equalization systems where it is placed in conjunction with a channel decoder that provides the necessary a priori information in this paper we consider the design of such an a priori information aware equalizer under an imperfect knowledge of channel parameters precisely speaking we pursue a minimax optimization procedure to robustify the design of the equalizer in presence of an uncertainty about the channel coefficients as well as the interference plus noise covariance to this end we employ a kullback leibler divergence based and a norm based uncertainty class obtain closed form expressions for the worst case uncertainties and finally arrive at the optimal mean square error mse equalizers that offer the best worst case performance under uncertainty finally we show that the proposed minimax robust equalizer achieves significant performance gains in comparison to the conventional nonrobust equalizer both as a standalone entity and also as part of an iterative detection system

continued fraction as a discrete nonlinear transform the connection between a taylor series and a continued fraction involves a nonlinear relation between the taylor coefficients an and the continued fraction coefficients bn in many instances it turns out that this nonlinear relation transforms a complicated sequence an into a very simple one bn this simplification is illustrated in the context of graph combinatorics

some permutability properties related to f hypercentrally embedded subgroups of finite groups let f denote a saturated formation in this paper we study some properties of f hypercentrally embedded subgroups i e those subgroups t of a finite group g such that every chief factor of g between its core and its normal closure is f central in g we prove that these subgroups form a sublattice of the lattice of all subgroups of g i ff is subgroup closed the main result of the paper is the following if f contains the class of nilpotent groups and g is a soluble group a subgroup t which permutes with all sylow subgroups of g is f hypercentrally embedded in g if and only if t permutes with some f normalizer of g 2003 elsevier science usa all rights reserved

generalised entropies and asymptotic complexities of languages the paper explores connections between asymptotic complexity and generalised entropy asymptotic complexity of a language a language is a set of finite or infinite strings is a way of formalising the complexity of predicting the next element in a sequence it is the loss per element of a strategy asymptotically optimal for that language generalised entropy extends shannon entropy to arbitrary loss functions it is the optimal expected loss given a distribution on possible outcomes it turns out that the set of tuples of asymptotic complexities of a language w r t different loss functions can be described by means of the generalised entropies corresponding to the loss functions

compatible weighted proper scoring rules many proper scoring rules such as the brier and log scoring rules implicitly reward a probability forecaster relative to a uniform baseline distribution recent work has motivated weighted proper scoring rules which have an additional baseline parameter to date two families of weighted proper scoring rules have been introduced the weighted power and pseudospherical scoring families these families are compatible with the log scoring rule when the baseline maximizes the log scoring rule over some set of distributions the baseline also maximizes the weighted power and pseudospherical scoring rules over the same set we characterize all weighted proper scoring families and prove a general property every proper scoring rule is compatible with some weighted scoring family and every weighted scoring family is compatible with some proper scoring rule

information theoretic asymptotics of bayes methods in the absence of knowledge of the true density function bayesian models take the joint density function for a sequence of n random variables to be an average of densities with respect to a prior the authors examine the relative entropy distance d sub n between the true density and the bayesian density and show that the asymptotic distance is d 2 log n c where d is the dimension of the parameter vector therefore the relative entropy rate d sub n n converges to zero at rate log n n the constant c which the authors explicitly identify depends only on the prior density function and the fisher information matrix evaluated at the true parameter value consequences are given for density estimation universal data compression composite hypothesis testing and stock market portfolio selection

composite multiclass losses we consider loss functions for multiclass prediction problems we show when a multiclass loss can be expressed as a proper composite loss which is the composition of a proper loss and a link function we extend existing results for binary losses to multiclass losses we subsume results on classification calibration by relating it to properness we determine the stationarity condition bregman representation order sensitivity and quasi convexity of multiclass proper losses we then characterise the existence and uniqueness of the composite representation for multiclass losses we show how the composite representation is related to other core properties of a loss mixability admissibility and strong convexity of multiclass losses which we characterise in terms of the hessian of the bayes risk we show that the simple integral representation for binary proper losses can not be extended to multiclass losses but offer concrete guidance regarding how to design different loss functions the conclusion drawn from these results is that the proper composite representation is a natural and convenient tool for the design of multiclass loss functions