stochastic mechanics of reciprocal diffusions the dynamics and kinematics of reciprocal diffusions were examined in a previous paper j math phys 34 1846 1993 where it was shown that reciprocal diffusions admit a chain of conservation laws which close after the first two laws for two disjoint subclasses of reciprocal diffusions the markov and quantum diffusions for the case of quantum diffusions the conservation laws are equivalent to schrodinger s equation the markov diffusions were employed by schrodinger sitzungsber preuss akad wiss phys math kl 144 1931 ann inst h poincare 2 269 1932 nelson dynamical theories of brownian motion princeton university princeton nj 1967 quantum fluctuations princeton university princeton nj 1985 and other researchers to develop stochastic formulations of quantum mechanics called stochastic mechanics we propose here an alternative version of stochastic mechanics based on quantum diffusions a procedure is presented for constructing the quantum diffusion associated to a given wave function it is shown that quantum diffusions satisfy the uncertainty principle and have a locality property whereby given two dynamically uncoupled but statistically correlated particles the marginal statistics of each particle depend only on the local fields to which the particle is subjected however like wigner s joint probability distribution for the position and momentum of a particle the finite joint probability densities of quantum diffusions may take negative values

discrete time classical and quantum markovian evolutions maximum entropy problems on path space the theory of schrodinger bridges for diffusion processes is extended to classical and quantum discrete time markovian evolutions the solution of the path space maximum entropy problems is obtained from the a priori model in both cases via a suitable multiplicative functional transformation in the quantum case nonequilibrium time reversal of quantum channels is discussed and space time harmonic processes are introduced

estimating a population of parameter values using bayes and empirical bayes methods abstract in standard bayes and empirical bayes component decision problems estimating inidividual parameters is the primary goal in multiple comparison problems and in comparisons of histograms of estimates however the primary goal is to produce parameter estimates that can be considered as an ensemble for example the histogram of estimates should be a good estimate of the histogram of parameters standard methods of estimating by the posterior expectation do minimize symmetric componentwise losses such as squared error but they produce ensembles of estimates with a sample variance smaller than the posterior expected sample variance for parameters in this article we propose new bayes and empirical bayes estimates that minimize a distance function between the empirical cdf of the estimates and the true parameters these estimators are weighted averages of the prior mean and the data with weight on the data being approximately the square root of that for the posterior expectation we give theoreti

variational processes and stochastic versions of mechanics the dynamical structure of any reasonable stochastic version of classical mechanics is investigated including the version created by nelson e nelson quantum fluctuations princeton u p princeton nj 1985 phys rev 150 1079 1966 for the description of quantum phenomena two different theories result from this common structure one of them is the imaginary time version of nelson s theory whose existence was unknown and yields a radically new probabilistic interpretation of the heat equation the existence and uniqueness of all the involved stochastic processes is shown under conditions suggested by the variational approach of yasue k yasue j math phys 22 1010 1981

random removal of edges from scale free graphs abstract it has been discovered that many naturally occurring networks the internet the power grid of the western us various biological networks etc satisfy a power law degree distribution such scale free networks have many interesting properties one of which is robustness to random damage this problem has been analyzed from the point of view of node deletion and connectedness recently it has also been considered from the point of view of node deletion and scale preservation in this paper we consider the problem from the point of view of edge deletion and scale preservation in agreement with the work on node deletion and scale preservation we show that a scale free graph should not be expected to remain scale free when edges are removed at random

empirical bayes estimates of age standardized relative risks for use in disease mapping there have been many attempts in recent years to map incidence and mortality from diseases such as cancer such maps usually display either relative rates in each district as measured by a standardized mortality ratio smr or some similar index or the statistical significance level for a test of the difference between the rates in that district and elsewhere neither of these approaches is fully satisfactory and we propose a new approach using empirical bayes estimation the resulting estimators represent a weighted compromise between the smr the overall mean relative rate and a local mean of the relative rate in nearby areas the compromise solution depends on the reliability of each individual smr and on estimates of the overall amount of dispersion of relative rates over different districts

constrained bayes estimation with applications abstract bayesian techniques are widely used in these days for simultaneous estimation of several parameters in compound decision problems often however the main objective is to produce an ensemble of parameter estimates whose histogram is in some sense close to the histogram of population parameters this is for example the situation in subgroup analysis where the problem is not only to estimate the different components of a parameter vector but also to identify the parameters that are above and the others that are below a certain specified cutoff point we have proposed in this paper bayes estimates in a very general context that meet this need these estimates are obtained by matching the first two moments of the histogram of the estimates and the posterior expectations of the first two moments of the histogram of the parameters and minimizing subject to these conditions the posterior expectation of the euclidean distance between the estimates and the parameters several applications of the m

remarks on linear and nonlinear filtering this communication tries to give some insight into relationships existing between viterbi and the forward backward algorithm used in the context of hidden markov models on the one hand and kalman filtering and rauch tung striebel smoothing on the other we give a unifying view which shows how those algorithms are related and give an example of a nonlinear hybrid system that can be filtered through a mixed algorithm

disease mapping using empirical bayes and bayes methods on mortality statistics in the netherlands data from the 64 public health service districts in the netherlands describing the health status of the dutch population were used to construct maps for several causes of deaths the choice and estimation of the relative risk measure is described the prior expected number of deaths was modelled using a poisson regression approach based on a model with main effects of district and age desirable properties of risk parameters for disease mapping are that they both reflect the level of risk and cope with the instability in the observed measure caused by the numbers at risk in each district different techniques for estimation of parameters were applied empirical bayes estimation eb using a non parametric prior and a gamma prior and a bayesian approach b with a uniform prior for the parametric eb also a constrained estimator was used the eb techniques studied in this paper shift or smooth the values of the risk parameter towards a global mean in the bayesian method applied here spatial dependence among districts can be modelled that is the estimates are smoothed towards a local mean the three eb estimates gave by and large similar results although the constrained eb estimate smoothed less as was expected the bayesian estimates smoothed the estimates more or less similarly to the constrained eb

stochastic relaxation gibbs distributions and the bayesian restoration of images we make an analogy between images and statistical mechanics systems pixel gray levels and the presence and orientation of edges are viewed as states of atoms or molecules in a lattice like physical system the assignment of an energy function in the physical system determines its gibbs distribution because of the gibbs distribution markov random field mrf equivalence this assignment also determines an mrf image model the energy function is a more convenient and natural mechanism for embodying picture attributes than are the local characteristics of the mrf for a range of degradation mechanisms including blurring nonlinear deformations and multiplicative or additive noise the posterior distribution is an mrf with a structure akin to the image model by the analogy the posterior distribution defines another imaginary physical system gradual temperature reduction in the physical system isolates low energy states annealing or what is the same thing the most probable states under the gibbs distribution the analogous operation under the posterior distribution yields the maximum a posteriori map estimate of the image given the degraded observations the result is a highly parallel relaxation algorithm for map estimation we establish convergence properties of the algorithm and we experiment with some simple pictures for which good restorations are obtained at low signal to noise ratios

high level primitives for recursive maximum likelihood estimation this paper proposes a high level language constituted of a small number of primitives and macros for describing recursive maximum likelihood ml estimation algorithms this language is applicable to estimation problems involving linear gaussian models or processes taking values in a finite set the use of high level primitives allows the development of highly modular ml estimation algorithms based on simple numerical building blocks the primitives which correspond to the combination of different measurements the extraction of sufficient statistics and the conversion of the status of a variable from unknown to observed or vice versa are first defined for linear gaussian relations specifying mixed deterministic stochastic information about the system variables these primitives are used to define other macros and are illustrated by deriving new filtering and smoothing algorithms for linear descriptor systems the primitives are then extended to finite state processes and used to implement the viterbi ml state sequence estimator for a hidden markov model

reciprocal diffusions in flat space we describe the theory of reciprocal diffusions in flat space a reciprocal process is a markov random field on a one dimensional parameter space every markov process is reciprocal but not vice versa we descibe the first and second order mean differential characteristics of reciprocal diffusions this includes a new definition of stochastic acceleration we show that reciprocal diffusions satisfy stochastic differential equations of second order associated to a reciprocal diffusion is a sequence of conservation laws the first two of which are the familiar continuity and euler equations there are two cases where these laws can be closed after the first two they are the mutually exclusive subclasses of markov and quantum diffusions the latter corresponds to solutions of the schrodinger equation and may be part of a stochastic description of quantum mechanics

a second order equation for schrodinger bridges with applications to the hot gas experiment and entropic transportation cost the schrodinger problem is obtained by replacing the mean square distance with the relative entropy in the monge kantorovich problem it was first addressed by schrodinger as the problem of describing the most likely evolution of a large number of brownian particles conditioned to reach an unexpected configuration its optimal value the entropic transportation cost and its optimal solution the schrodinger bridge stand as the natural probabilistic counterparts to the transportation cost and displacement interpolation moreover they provide a natural way of lifting from the point to the measure setting the concept of brownian bridge in this article we prove that the schrodinger bridge solves a second order equation in the riemannian structure of optimal transport roughly speaking the equation says that its acceleration is the gradient of the fisher information using this result we obtain a fine quantitative description of the dynamics and a new functional inequality for the entropic transportation cost that generalize talagrand s transportation inequality finally we study the convexity of the fisher information along schrodigner bridges under the hypothesis that the associated reciprocal characteristic is convex the techniques developed in this article are also well suited to study the feynman kac penalisations of brownian motion

hidden markov processes an overview of statistical and information theoretic aspects of hidden markov processes hmps is presented an hmp is a discrete time finite state homogeneous markov chain observed through a discrete time memoryless invariant channel in recent years the work of baum and petrie 1966 on finite state finite alphabet hmps was expanded to hmps with finite as well as continuous state spaces and a general alphabet in particular statistical properties and ergodic theorems for relative entropy densities of hmps were developed consistency and asymptotic normality of the maximum likelihood ml parameter estimator were proved under some mild conditions similar results were established for switching autoregressive processes these processes generalize hmps new algorithms were developed for estimating the state parameter and order of an hmp for universal coding and classification of hmps and for universal decoding of hidden markov channels these and other related topics are reviewed

empirical bayes estimates of cancer mortality rates using spatial models we give results of empirical bayes eb estimation of mortality rates designed to smooth observed smr when random fluctuation of the observed deaths is important we have specially studied the case where the prior distributions of the eb method have a spatial structure the need for spatial modelling of cancer mortality rates in france is first shown with testing autocorrelation and fitting autoregressive spatial models conditional car or simultaneous sar a positive autocorrelation of the rates is shown for most cancer sites studied as expected eb estimates of mortality rates for common tumours are similar to smrs for rare tumours the eb method identifies the extreme rates more clearly than smrs by smoothing the smrs with large variances car or sar models are adequate prior distributions for autocorrelated rates and produce quite similar rate estimates

local kinetic interpretation of entropy production through reversed diffusion the time reversal of stochastic diffusion processes is revisited with emphasis on the physical meaning of the time reversed drift and the noise prescription in the case of multiplicative noise the local kinematics and mechanics of free diffusion are linked to the hydrodynamic description these properties also provide an interpretation of the pope ching formula for the steady state probability density function along with a geometric interpretation of the fluctuation dissipation relation finally the statistics of the local entropy production rate of diffusion are discussed in the light of local diffusion properties and a stochastic differential equation for entropy production is obtained using the girsanov theorem for reversed diffusion the results are illustrated for the ornstein uhlenbeck process

approximate inference in generalized linear mixed models statistical approaches to overdispersion correlated errors shrinkage estimation and smoothing of regression relationships may be encompassed within the framework of the generalized linear mixed model glmm given an unobserved vector of random effects observations are assumed to be conditionally independent with means that depend on the linear predictor through a specified link function and conditional variances that are specified by a variance function known prior weights and a scale factor the random effects are assumed to be normally distributed with mean zero and dispersion matrix depending on unknown variance components for problems involving time series spatial aggregation and smoothing the dispersion may be specified in terms of a rank deficient inverse covariance matrix approximation of the marginal quasi likelihood using laplace s method leads eventually to estimating equations based on penalized quasilikelihood or pql for the mean parameters and pseudo likelihood for the variances im

optimal smoothing for finite state hidden reciprocal processes this technical note addresses modelling and estimation of a class of finite state random processes called hidden reciprocal chains hrc a hidden reciprocal chain consists of a finite state reciprocal process together with an observation process conditioned on the reciprocal process much as in the case of a hidden markov model hmm the key difference between markov models and reciprocal models is that reciprocal models are non causal the technical note presents a characterization of a hrc by a finite set of hidden markov bridges which are hmms with the final state fixed the technical note then uses this characterization to derive the optimal fixed interval smoother for a hrc performance of linear and optimal smoothers derived for both hmm and hrc are compared using simulations for a class of hrc derived from underlying markov transitions these experiments suggest that not surprisingly the performance of the optimal hmm and hrc smoothers are signifcantly better than their linear counterparts and that some performance improvement is obtained using the hrc smoothers compared to the hmm smoothers the technical note concludes by mentioning some ongoing and future work which exploits this new markov bridge characterization of a hrc

modeling and estimation of discrete time gaussian reciprocal processes discrete time gaussian reciprocal processes are characterized in terms of a second order two point boundary value nearest neighbor model driven by a locally correlated noise whose correlation is specified by the model dynamics this second order model is the analog for reciprocal processes of the standard first order state space models for markov processes the model is used to obtain a solution to the smoothing problem for reciprocal processes the resulting smoother obeys second order equations whose structure is similar to that of the kalman filter for gauss markov processes it is shown that the smoothing error is itself a reciprocal process

effect of edge removal on topological and functional robustness of complex networks abstract we study the robustness of several network models subject to edge removal the robustness is measured by the statistics of network breakdowns where a breakdown is defined as the destroying of the total connectedness of a network rather than the disappearance of the giant component we introduce a simple traffic dynamics as the function of a network topology and the total connectedness can be destroyed in the sense of either the topology or the function the overall effect of the topological breakdown and the functional breakdown as well as the relative importance of the topological robustness and the functional robustness are studied under two edge removal strategies

modelling and estimation for finite state reciprocal processes a reciprocal equation is a kind of descriptor linear discrete index stochastic system which is well known be satisfied pathwise by all gaussian reciprocal processes from a system theoretic point of view it is a kind of noncausal linear system in the sense that the solution of it cannot be determined by only an initial condition indeed requiring the terminal state as well besides all the input function between initial and terminal states also nice properties are known of a reciprocal equation such as the equivalence of it with a couple of ordinary causal dynamic systems running in opposite directions for these reasons here we assume a reciprocal equation as the target of stochastic realization for the class of finite state reciprocal processes also named reciprocal chains the central result of the present paper is showing that any canonical reciprocal chain i e valued in the canonical base of realrn n being the cardinality of the set of chain s states satisfies pathwise a reciprocal equation in a n2 dimensional canonical variable or in other word a quadratic reciprocal equation named augmented state reciprocal model asrm also for a partially observed reciprocal chain a linear optimal smoother is derived all the results here presented are based upon the idea that a reciprocal chain is a combination of markov bridges to this purpose other forms besides the asrm are presented in order to make clear the meaning of this combination as well as to prove that the linear smoother can be actually implemented as n smoothers all operating independently on each markov bridge component

effect of random edge failure on the average path length we study the effect of random removal of edges on the average path length apl in a large class of uncorrelated random networks in which vertices are characterized by hidden variables controlling the attachment of edges between pairs of vertices a formula for approximating the apl of networks suffering random edge removal is derived first then the formula is confirmed by simulations for classical er erdos and renyi random graphs ba barabasi and albert networks networks with exponential degree distributions as well as random networks with asymptotic power law degree distributions with exponent 2

dynamics and kinematics of reciprocal diffusions this paper examines the dynamics and kinematics of reciprocal diffusions reciprocal processes were introduced by bernstein in 1932 and were later studied in detail by jamison the reciprocal diffusions are constructed here by specifying their finite joint densities in terms of the green s function of a general heat operator and an end point density a path integral interpretation of the heat operator green s function is provided which is used to derive a stochastic form of newton s law as well as a conditional distribution for the velocity of a diffusing particle given its position these results are then employed to derive two conservation laws expressing the conservation of mass and momentum the conservation laws do not form a closed system of equations in general except for two subclasses of reciprocal diffusions the markov and quantum diffusions

on the convergence properties of the em algorithm two convergence aspects of the em algorithm are studied i does the em algorithm find a local maximum or a stationary value of the incomplete data likelihood function ii does the sequence of parameter estimates generated by em converge several convergence results are obtained under conditions that are applicable to many practical situations two useful special cases are a if the unobserved complete data specification can be described by a curved exponential family with compact parameter space all the limit points of any em sequence are stationary points of the likelihood function b if the likelihood function is unimodal and a certain differentiability condition is satisfied then any em sequence converges to the unique maximum likelihood estimate a list of key properties of the algorithm is included

hidden gauss markov models for signal classification continuous state hidden markov models cs hmms are developed as a tool for signal classification analogs of the baum 1972 viterbi 1962 and baum welch algorithms are formulated for this class of models the cs hmm algorithms are then specialized to hidden gauss markov models hgmms with linear gaussian state transition and output densities a new gaussian refactorization lemma is used to show that the baum and viterbi algorithms for hgmms are implemented by two different formulations of the fixed interval kalman smoother the measurement likelihoods obtained from the forward pass of the hgmm baum algorithm and from the kalman filter innovation sequence are shown to be equal a direct link between the baum welch training algorithm and an existing expectation maximization em algorithm for gaussian models is demonstrated a new expression for the cross covariance between time adjacent states in hgmms is derived from the off diagonal block of the conditional joint covariance matrix a parameter invariance structure is noted for the hgmm likelihood function cs hmms and hgmms are extended to incorporate mixture densities for the a priori density of the initial state application of hgmms to signal classification is demonstrated with a three class test simulation

using empirical bayes methods in biopharmaceutical research a compound sampling model where a unit specific parameter is sampled from a prior distribution and then observed are generated by a sampling distribution depending on the parameter underlies a wide variety of biopharmaceutical data for example in a multi centre clinical trial the true treatment effect varies from centre to centre observed treatment effects deviate from these true effects through sampling variation knowledge of the prior distribution allows use of bayesian analysis to compute the posterior distribution of clinic specific treatment effects frequently summarized by the posterior mean and variance more commonly with the prior not completely specified observed data can be used to estimate the prior and use it to produce the posterior distribution an empirical bayes or variance component analysis in the empirical bayes model the estimated prior mean gives the typical treatment effect and the estimated prior standard deviation indicates the heterogeneity of treatment effects in both the bayes and empirical bayes approaches estimated clinic effects are shrunken towards a common value from estimates based on single clinics this shrinkage produces more efficient estimates in addition the compound model helps structure approaches to ranking and selection provides adjustments for multiplicity allows estimation of the histogram of clinic specific effects and structures incorporation of external information this paper outlines the empirical bayes approach coverage will include development and comparison of approaches based on parametric priors for example a gaussian prior with unknown mean and variance and non parametric priors discussion of the importance of accounting for uncertainty in the estimated prior comparison of the output and interpretation of fixed and random effects approaches to estimating population values estimating histograms and identification of key considerations in the use and interpretation of empirical bayes methods

2 d recursive modelling of homogeneous discrete gaussian markov fields a 2 d stochastic realization is obtained for an homogeneous discrete gaussian markov field in two dimensions in specific it is shown that such markov fields can be represented as the unique pathwise solution of a 2 d recursive system of equations with a locally correlated forcing noise the idea of stochastic dynamic system in two dimensions gets realized this way two types of domain are considered spheric and rectangular

transformations of diffusion and schrodinger processes a transformation by means of a new type of multiplicative functionals is given which is a generalization of doob s space time harmonic transformation in the case of arbitrary non harmonic function t x which may vanish on a subset of a b x d the transformation induces an additional singular drift term like in the case of doob s space time harmonic transformation to handle the transformation an integral equation of singular perturbations and a diffusion equation with singular potentials are discussed and the feynman kac theorem is established for a class of singular potentials the transformation is applied to schrodinger processes which are defined following an idea of e schrodinger 1931

a simplified variational characterization of schrodinger processes the variational problem e 1 2 t0 t a t xt 2 dt t0c t xt dt min where t is the drift process of a diffusion process with unit diffusion coefficient and given initial and final distributions a is a given vector field and c is a given scalar field is considered it is shown that the solution is given by a certain markovian diffusion process which in the case a 0 c 0 first was investigated by schrodinger sitzungsber preuss akad wiss phys math kl 1931 144

extremal flows in wasserstein space we develop an intrinsic geometric approach to the calculus of variations in the wasserstein space we show that the flows associated with the schrodinger bridge with general prior with optimal mass transport and with the madelung fluid can all be characterized as annihilating the first variation of a suitable action we then discuss the implications of this unified framework for stochastic mechanics it entails in particular a sort of fluid dynamic reconciliation between bohm s and nelson s stochastic mechanics we develop an intrinsic geometric approach to the calculus of variations in the wasserstein space we show that the flows associated with the schrodinger bridge with general prior with optimal mass transport and with the madelung fluid can all be characterized as annihilating the first variation of a suitable action we then discuss the implications of this unified framework for stochastic mechanics it entails in particular a sort of fluid dynamic reconciliation between bohm s and nelson s stochastic mechanics