approximate likelihood for large irregularly spaced spatial data likelihood approaches for large irregularly spaced spatial datasets are often very difficult if not infeasible to implement due to computational limitations even when we can assume normality exact calculations of the likelihood for a gaussian spatial process observed at n locations requires o n3 operations we present a version of whittle s approximation to the gaussian log likelihood for spatial regular lattices with missing values and for irregularly spaced datasets this method requires o nlog2n operations and does not involve calculating determinants we present simulations and theoretical results to show the benefits and the performance of the spatial likelihood approximation method presented here for spatial irregularly spaced datasets and lattices with missing values we apply these methods to estimate the spatial structure of sea surface temperatures using satellite data with missing values

on conditional and intrinsic autoregressions summary gaussian conditional autoregressions have been widely used in spatial statistics and bayesian image analysis where they are intended to describe interactions between random variables at fixed sites in euclidean space the main appeal of these distributions is in the markovian interpretation of their full conditionals intrinsic autoregressions are limiting forms that retain the markov property despite being improper they can have advantages over the standard autoregressions both conceptually and in practice for example they often avoid difficulties in parameter estimation without apparent loss or exhibit appealing invariances as in texture analysis however on small arrays and in nonlattice applications both forms of autoregression can lead to undesirable second order characteristics either in the variables themselves or in contrasts among them this paper discusses standard and intrinsic autoregressions and describes how the problems that arise can be alleviated using dempster s 1972 algorithm or an appropriate modification the approach represents a partial synthesis of standard geostatistical and gaussian markov random field formulations some nonspatial applications are also mentioned

bayesian sparse multiple regression for simultaneous rank reduction and variable selection we develop a bayesian methodology aimed at simultaneously estimating low rank and row sparse matrices in a high dimensional multiple response linear regression model we consider a carefully devised shrinkage prior on the matrix of regression coefficients which obviates the need to specify a prior on the rank and shrinks the regression matrix towards low rank and row sparse structures we provide theoretical support to the proposed methodology by proving minimax optimality of the posterior mean under the prediction risk in ultra high dimensional settings where the number of predictors can grow sub exponentially relative to the sample size a one step post processing scheme induced by group lasso penalties on the rows of the estimated coefficient matrix is proposed for variable selection with default choices of tuning parameters we additionally provide an estimate of the rank using a novel optimization function achieving dimension reduction in the covariate space we exhibit the performance of the proposed methodology in an extensive simulation study and a real data example

isoperimetry for wreath products of markov chains and multiplicity of selfintersections of random walks we prove an isoperimetric inequality for wreath products of markov chains with variable fibers we use isoperimetric inequalities for wreath products to estimate the return probability of random walks on infinite groups and graphs drift of random loops the expected value e exp trn where rn is the number of distinct sites visited up to the moment n and more generally open image in new window where lz n is the number of visits of z up to the moment n and f x y is some non negative function

propagation of probabilities means and variances in mixed graphical association models abstract a scheme is presented for modeling and local computation of exact probabilities means and variances for mixed qualitative and quantitative variables the models assume that the conditional distribution of the quantitative variables given the qualitative is multivariate gaussian the computational architecture is set up by forming a tree of belief universes and the calculations are then performed by local message passing between universes the asymmetry between the quantitative and qualitative variables sets some additional limitations for the specification and propagation structure approximate methods when these are not appropriately fulfilled are sketched it has earlier been shown how to exploit the local structure in the specification of a discrete probability model for fast and efficient computation thereby paving the way for exploiting probability based models as parts of realistic systems for planning and decision support the purpose of this article is to extend this computational s

modelling spatially correlated data via mixtures a bayesian approach the paper develops mixture models for spatially indexed data we confine attention to the case of finite typically irregular patterns of points or regions with prescribed spatial relationships and to problems where it is only the weights in the mixture that vary from one location to another our specific focus is on poisson distributed data and applications in disease mapping we work in a bayesian framework with the poisson parameters drawn from gamma priors and an unknown number of components we propose two alternative models for spatially dependent weights based on transformations of autoregressive gaussian processes in one the logistic normal model the mixture component labels are exchangeable in the other the grouped continuous model they are ordered reversible jump markov chain monte carlo algorithms for posterior inference are developed finally the performances of both of these formulations are examined on synthetic data and real data on mortality from a rare disease

fast and exact simulation of stationary gaussian processes through circulant embedding of the covariance matrix geostatistical simulations often require the generation of numerous realizations of a stationary gaussian process over a regularly meshed sample grid omega this paper shows that for many important correlation functions in geostatistics realizations of the associated process over m 1 equispaced points on a line can be produced at the cost of an initial fft of length 2m with each new realization requiring an additional fft of the same length in particular the paper first notes that if an m 1 times m 1 toeplitz correlation matrix r can be embedded in a nonnegative definite 2m times2m circulant matrix s exact realizations of the normal multivariate y sim cal n 0 r can be generated via ffts of length 2m theoretical results are then presented to demonstrate that for many commonly used correlation structures the minimal embedding in which m m is nonnegative definite extensions to simulations of stationary fields in higher dimensions are also provided and illustrated

efficient gaussian sampling for solving large scale inverse problems using mcmc the resolution of many large scale inverse problems using mcmc methods requires a step of drawing samples from a high dimensional gaussian distribution while direct gaussian sampling techniques such as those based on cholesky factorization induce an excessive numerical complexity and memory requirement sequential coordinate sampling methods present a low rate of convergence based on the reversible jump markov chain framework this paper proposes an efficient gaussian sampling algorithm having a reduced computation cost and memory usage while maintaining the theoretical convergence of the sampler the main feature of the algorithm is to perform an approximate resolution of a linear system with a truncation level adjusted using a self tuning adaptive scheme allowing to achieve the minimal computation cost per effective sample the connection between this algorithm and some existing strategies is given and its performance is illustrated on a linear inverse problem of image resolution enhancement

on drift and entropy growth for random walks on groups in this paper we consider drift and entropy growth for symmetric finitary random walks on finitely generated groups we construct examples of various intermediate asymptotics of the drift for such random walks we establish general inequalities which connect drift entropy and exponential growth rate of groups then we apply these inequalities to get estimates for entropy in particular examples

likelihood analysis of non gaussian measurement time series summary in this paper we provide methods for estimating non gaussian time series models these techniques rely on markov chain monte carlo to carry out simulation smoothing and bayesian posterior analysis of parameters and on importance sampling to estimate the likelihood function for classical inference the time series structure of the models is used to ensure that our simulation algorithms are efficient

a multiresolution stochastic process model for predicting basketball possession outcomes abstractbasketball games evolve continuously in space and time as players constantly interact with their teammates the opposing team and the ball however current analyses of basketball outcomes rely on discretized summaries of the game that reduce such interactions to tallies of points assists and similar events in this article we propose a framework for using optical player tracking data to estimate in real time the expected number of points obtained by the end of a possession this quantity called expected possession value epv derives from a stochastic process model for the evolution of a basketball possession we model this process at multiple levels of resolution differentiating between continuous infinitesimal movements of players and discrete events such as shot attempts and turnovers transition kernels are estimated using hierarchical spatiotemporal models that share information across players while remaining computationally tractable on very large data sets in addition to estima

circuit theory and model based inference for landscape connectivity circuit theory has seen extensive recent use in the field of ecology where it is often applied to study functional connectivity the landscape is typically represented by a network of nodes and resistors with the resistance between nodes a function of landscape characteristics the effective distance between two locations on a landscape is represented by the resistance distance between the nodes in the network circuit theory has been applied to many other scientific fields for exploratory analyses but parametric models for circuits are not common in the scientific literature to model circuits explicitly we demonstrate a link between gaussian markov random fields and contemporary circuit theory using a covariance structure that induces the necessary resistance distance this provides a parametric model for second order observations from such a system in the landscape ecology setting the proposed model provides a simple framework where inference can be obtained for effects that landscape features ha

latent variable models with nonparametric interaction effects of latent variables renal disease is one of the common complications of diabetes especially for asian populations moreover cardiovascular and renal diseases share common risk factors this paper proposes a latent variable model with nonparametric interaction effects of latent variables for a study based on the hong kong diabetes registry which was established in 1995 as part of a continuous quality improvement program at the prince of wales hospital in hong kong renal outcome outcome latent variable is regressed in terms of cardiac function and diabetes explanatory latent variables through an additive structural equation formulated using a series of unspecified univariate and bivariate smooth functions the bayesian p splines approach along with a markov chain monte carlo algorithm is proposed to estimate smooth functions unknown parameters and latent variables in the model the performance of the developed methodology is demonstrated via a simulation study the effect of the nonparametric interaction of cardiac function and diabetes on renal outcome is investigated using the proposed methodology copyright 2013 john wiley sons ltd

on gibbs sampling for state space models summary we show how to use the gibbs sampler to carry out bayesian inference on a linear state space model with errors that are a mixture of normals and coefficients that can switch over time our approach simultaneously generates the whole of the state vector given the mixture and coefficient indicator variables and simultaneously generates all the indicator variables conditional on the state vectors the states are generated efficiently using the kalman filter we illustrate our approach by several examples and empirically compare its performance to another gibbs sampler where the states are generated one at a time the empirical results suggest that our approach is both practical to implement and dominates the gibbs sampler that generates the states one at a time

updating schemes correlation structure blocking and parameterization for the gibbs sampler in this paper many convergence issues concerning the implementation of the gibbs sampler are investigated exact computable rates of convergence for gaussian target distributions are obtained different random and non random updating strategies and blocking combinations are compared using the rates the effect of dimensionality and correlation structure on the convergence rates are studied some examples are considered to demonstrate the results for a gaussian image analysis problem several updating strategies are described and compared for problems in bayesian linear models several possible parameterizations are analysed in terms of their convergence rates characterizing the optimal choice

an explicit link between gaussian fields and gaussian markov random fields the stochastic partial differential equation approach continuously indexed gaussian fields gfs are the most important ingredient in spatial statistical modelling and geostatistics the specification through the covariance function gives an intuitive interpretation of the field properties on the computational side gfs are hampered with the big n problem since the cost of factorizing dense matrices is cubic in the dimension although computational power today is at an all time high this fact seems still to be a computational bottleneck in many applications along with gfs there is the class of gaussian markov random fields gmrfs which are discretely indexed the markov property makes the precision matrix involved sparse which enables the use of numerical algorithms for sparse matrices that for fields in r 2 only use the square root of the time required by general algorithms the specification of a gmrf is through its full conditional distributions but its marginal properties are not transparent in such a parameterization we show that using an approximate stochastic weak solution to linear stochastic partial differential equations we can for some gfs in the matern class provide an explicit link for any triangulation of r d between gfs and gmrfs formulated as a basis function representation the consequence is that we can take the best from the two worlds and do the modelling by using gfs but do the computations by using gmrfs perhaps more importantly our approach generalizes to other covariance functions generated by spdes including oscillating and non stationary gfs as well as gfs on manifolds we illustrate our approach by analysing global temperature data with a non stationary model defined on a sphere less

liouville property for groups and manifolds we introduce a method to estimate the entropy of random walks on groups we apply this method to exhibit the existence of compact manifolds with amenable fundamental groups such that the universal cover is not liouville we also use the criterion to prove that a finitely generated solvable group admits a symmetric measure with non trivial poisson boundary if and only if this group is not virtually nilpotent this in particular shows that any polycyclic group admits a symmetric measure such that its boundary does not readily interprete in terms of the ambient lie group as another application we get a series of examples of amenable groups such that any finite entropy non degenerate measure on them has non trivial poisson boundary since the groups in question are amenable they do admit measures such that the corresponding random walks have trivial boundary the above shows that such measures on these groups have infinite entropy

the hessian method highly efficient simulation smoothing in a nutshell abstract i introduce the hessian highly efficient simulation smoothing in a nutshell method for numerically efficient simulation smoothing in state space models with univariate states given a vector of parameters the vector of states 1 n is gaussian and the observed vector y y 1 y n need not be i describe a procedure to construct a close approximation q y to the target density p y it requires code to compute five derivatives of log p y t t with respect to t t 1 n and is not otherwise model specific since q y is proper fully normalised and simulable it can be used as an importance density for importance sampling is or as a proposal density for markov chain monte carlo mcmc hessian is an acronym but it also refers to the sparse hessian matrix of log p y with respect to the hessian method is based on sparse matrix operations rather than the kalman filter i construct q y and a related approximation q y of p y for two stochastic volatility models two stochastic count models and a stochastic duration model i illustrate their use for numerical approximation of likelihood function values and marginal likelihoods using is and for posterior inference using is and mcmc compared with other simulation smoothing methods the hessian method is highly numerically efficient in an is application featuring a student s t stochastic volatility model and n 8851 daily log returns the efficiency of is for numerical approximation of the elements of the posterior mean e y is between 80 and 100

fast sampling with gaussian scale mixture priors in high dimensional regression we propose an efficient way to sample from a class of structured multivariate gaussian distributions the proposed algorithm only requires matrix multiplications and linear system solutions its computational complexity grows linearly with the dimension unlike existing algorithms that rely on cholesky factorizations with cubic complexity the algorithm is broadly applicable in settings where gaussian scale mixture priors are used on high dimensional parameters its effectiveness is illustrated through a high dimensional regression problem with a horseshoe prior on the regression coefficients other potential applications are outlined

on random walks on wreath products wreath products are a type of semidirect product they play an important role in group theory this paper studies the basic behavior of simple random walks on such groups and shows that these walks have interesting somewhat exotic behaviors the crucial fact is that the probability of return to the starting point of certain walks on wreath products is closely related to some functionals of the local times of a walk taking place on a simpler factor group

approximating hidden gaussian markov random fields summary gaussian markov random field gmrf models are frequently used in a wide variety of applications in most cases parts of the gmrf are observed through mutually independent data hence the full conditional of the gmrf a hidden gmrf hgmrf is of interest we are concerned with the case where the likelihood is non gaussian leading to non gaussian hgmrf models several researchers have constructed block sampling markov chain monte carlo schemes based on approximations of the hgmrf by a gmrf using a second order expansion of the log density at or near the mode this is possible as the gmrf approximation can be sampled exactly with a known normalizing constant the markov property of the gmrf approximation yields computational efficiency the main contribution in the paper is to go beyond the gmrf approximation and to construct a class of non gaussian approximations which adapt automatically to the particular hgmrf that is under study the accuracy can be tuned by intuitive parameters to nearly any precision these non gaussian approximations share the same computational complexity as those which are based on gmrfs and can be sampled exactly with computable normalizing constants we apply our approximations in spatial disease mapping and model based geostatistical models with different likelihoods obtain procedures for block updating and construct metropolized independence samplers

a bayesian semiparametric latent variable model for mixed responses in this paper we introduce a latent variable model lvm for mixed ordinal and continuous responses where covariate effects on the continuous latent variables are modelled through a flexible semiparametric gaussian regression model we extend existing lvms with the usual linear covariate effects by including nonparametric components for nonlinear effects of continuous covariates and interactions with other covariates as well as spatial effects full bayesian modelling is based on penalized spline and markov random field priors and is performed by computationally efficient markov chain monte carlo mcmc methods we apply our approach to a german social science survey which motivated our methodological development

sampling from gaussian markov random fields using stationary and non stationary subgraph perturbations gaussian markov random fields gmrfs or gaussian graphical models have been widely used in many applications efficiently drawing samples from gmrfs has been an important research problem in this paper we introduce the subgraph perturbation sampling algorithm which makes use of any pre existing tractable inference algorithm for a subgraph by perturbing this algorithm so as to yield asymptotically exact samples for the intended distribution we study the stationary version where a single fixed subgraph is used in all iterations as well as the non stationary version where tractable subgraphs are adaptively selected the subgraphs used can have any structure for which efficient inference algorithms exist for example tree structured low tree width or having a small feedback vertex set we present new theoretical results that give convergence guarantees for both stationary and non stationary graphical splittings our experiments using both simulated models and large scale real models demonstrate that this subgraph perturbation algorithm efficiently yields accurate samples for many graph topologies

modeling spatial covariance using the limiting distribution of spatio temporal random walks abstractwe present an approach for modeling areal spatial covariance in observed genetic allele data by considering the stationary limiting distribution of a spatio temporal markov random walk model for gene flow this stationary distribution corresponds to an intrinsic simultaneous autoregressive sar model for spatial correlation and provides a principled approach to specifying areal spatial models when a spatio temporal generating process can be assumed we apply the approach to a study of spatial genetic variation of trout in a stream network in connecticut usa

random walks on braid groups brownian bridges complexity and statistics we investigate the limit behaviour of random walks on some non commutative discrete groups related to knot theory namely we study the connection between the limit behaviour of the lyapunov exponent of products of non commutative random matrices generators of the braid group and the asymptotics of powers of the algebraic invariants of randomly generated knots we turn the simplest problems of knot statistics into the context of random walks on hyperbolic groups we also consider the limit distribution of brownian bridges on so called locally non commutative groups

fast sampling of gaussian markov random fields this paper demonstrates how gaussian markov random fields conditional autoregressions can be sampled quickly by using numerical techniques for sparse matrices the algorithm is general and efficient and expands easily to various forms for conditional simulation and evaluation of normalization constants we demonstrate its use by constructing efficient block updates in markov chain monte carlo algorithms for disease mapping

bicoprime factorizations of the plant and their relation to right and left coprime factorizations in a general algebraic framework starting with a bicoprime factorization p n sub pr d sup 1 n sub pl a right coprime factorization n sub p d sub p sup 1 a left coprime factorization d sup 1 sub p n sub p and the generalized bezout identities associated with the pairs n sub p d sub p and d sub p n sub p are obtained the set of all h stabilizing compensators for p in the unity feedback configuration s p c are expressed in terms of n sub pr d n sub pt and the elements of the bezout identity the state space representation p c si a sup 1 b is included as an example

maximum likelihood estimation for spatial models by markov chain monte carlo stochastic approximation we propose a two stage algorithm for computing maximum likelihood estimates for a class of spatial models the algorithm combines markov chain monte carlo methods such as the metropolis hastings green algorithm and the gibbs sampler and stochastic approximation methods such as the off line average and adaptive search direction a new criterion is built into the algorithm so stopping is automatic once the desired precision has been set simulation studies and applications to some real data sets have been conducted with three spatial models we compared the algorithm proposed with a direct application of the classical robbins monro algorithm using wiebe s wheat data and found that our procedure is at least 15 times faster

bayesian analysis of agricultural field experiments the paper describes bayesian analysis for agricultural field experiments a topic that has received very little previous attention despite a vast frequentist literature adoption of the bayesian paradigm simplifies the interpretation of the results especially in ranking and selection also complex formulations can be analysed with comparative ease by using markov chain monte carlo methods a key ingredient in the approach is the need for spatial representations of the unobserved fertility patterns this is discussed in detail problems caused by outliers and by jumps in fertility are tackled via hierarchical f formulations that may find use in other contexts the paper includes three analyses of variety trials for yield and one example involving binary data none is entirely straightforward some numerical comparisons with frequentist analyses are made

bayesian extrapolation of space time trends in cancer registry data nous avons utilise un modele bayesien sur des donnees de mortalite par cancer de l estomac en allemagne de l ouest les donnees sont stratifiees par groupe d ge annee et district en utilisant sur cesdonnees un modele age periode cohorte avec une composante spatiale nous avons cherche a mettre en evidence des interactions spatic temporelles nous avons de plus etudie si des interactions periode composante spatiale ou cohorte composante spatiale etaient plus appropriees pour faire de la prediction sur les taux de mortalite pour chacun des elements du modele bayesien nous avons utilise des chaines de markov avec marche aleatoire gaussienne comme a priori les inferences statistiques ont ete obtenues a partir d un algorithme d echantillonnage par bloc recemment propose dans la litterature pour ce genre de modele

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

disease mapping of stage specific cancer incidence data we propose two approaches for the spatial analysis of cancer incidence data with additional information on the stage of the disease at time of diagnosis the two formulations are extensions of commonly used models for multicategorical response data on an ordinal scale we include spatial and age group effects in both formulations which we estimate in a nonparametric smooth way more specifically we adopt a fully bayesian approach based on gaussian pairwise difference priors where additional smoothing parameters are treated as unknown as well we apply our methods to data on cervical cancer in the former german democratic republic the results suggest that there are large spatial differences in the stage proportions which indicates spatial variability with respect to the introduction and effectiveness of screening programs

analytic properties and covariance functions for a new class of generalized gibbs random fields spartan spatial random fields ssrfs are generalized gibbs random fields equipped with a coarse graining kernel that acts as a low pass filter for the fluctuations ssrfs are defined by means of physically motivated spatial interactions and a small set of free parameters interaction couplings this paper focuses on the fluctuation gradient curvature fgc ssrf model henceforth referred to as fgc ssrf this model is defined on the euclidean space r by means of interactions proportional to the squares of the field realizations as well as their gradient and curvature the permissibility criteria of fgc ssrfs are extended by considering the impact of a finite bandwidth kernel it is proved that the fgc ssrfs are almost surely differentiable in the case of finite bandwidth asymptotic explicit expressions for the spartan covariance function are derived for d 1 and d 3 both known and new covariance functions are obtained depending on the value of the fgc ssrf shape parameter nonlinear dependence of the covariance integral scale on the fgc ssrf characteristic length is established and it is shown that the relation becomes linear asymptotically the results presented in this paper are useful in random field parameter inference and in spatial interpolation of irregularly spaced samples

rapid spatially coupled avo inversion in the fourier domain spatial coupling of the model parameters in an inversion problem provides lateral consistency and robust solutions we have defined the inversion problem in a bayesian framework where the solution is represented by a posterior distribution obtained from a prior distribution and a likelihood model for the recorded data the spatial coupling of the model parameters is imposed via the prior distribution by a spatial correlation function in the fourier domain the spatially correlated model parameters can be decoupled and the inversion problem can be solved independently for each frequency component for a spatial model parameter represented on n grid nodes the computing time for the inversion in the fourier domain follows a linear function of the number of grid nodes while the computing time for the fast fourier transform follows an n log n function we have developed a 3d linearized amplitude variation with offset avo inversion method with spatially coupled model parameters where the objective is to

bayesian hemodynamic parameter estimation by bolus tracking perfusion weighted imaging a delay insensitive probabilistic method for estimating hemodynamic parameters delays theoretical residue functions and concentration time curves by computed tomography ct and magnetic resonance mr perfusion weighted imaging is presented only a mild stationarity hypothesis is made beyond the standard perfusion model new microvascular parameters with simple hemodynamic interpretation are naturally introduced simulations on standard digital phantoms show that the method outperforms the oscillating singular value decomposition osvd method in terms of goodness of fit linearity statistical and systematic errors on all parameters especially at low signal to noise ratios snrs delay is always estimated sharply with user supplied resolution and is purely arterial by contrast to osvd time to maximum tmax that is very noisy and biased by mean transit time mtt blood volume and snr residue functions and signals estimates do not suffer overfitting anymore one ct acute stroke case confirms simulation results and highlights the ability of the method to reliably estimate mtt when snr is low delays look promising for delineating the arterial occlusion territory and collateral circulation

bayesian estimation for optimized structured illumination microscopy structured illumination microscopy is a recent imaging technique that aims at going beyond the classical optical resolution by reconstructing high resolution hr images from low resolution lr images acquired through modulation of the transfer function of the microscope the classical implementation has a number of drawbacks such as requiring a large number of images to be acquired and parameters to be manually set in an ad hoc manner that have until now hampered its wide dissemination here we present a new framework based on a bayesian inverse problem formulation approach that enables the computation of one hr image from a reduced number of lr images and has no specific constraints on the modulation moreover it permits to automatically estimate the optimal reconstruction hyperparameters and to compute an uncertainty bound on the estimated values we demonstrate through numerical evaluations on simulated data and examples on real microscopy data that our approach represents a decisive advance for a wider use of hr microscopy through structured illumination

rate of escape of random walks on wreath products and related groups this article examines the rate of escape for a random walk on g z and proves laws of the iterated logarithm for both the inner and outer radius of escape the class of g for which these results hold includes finite g as well as groups of the form h z so the construction can be iterated laws of the iterated logarithm are also found for random walk on baumslag solitar groups and a discrete version of the sol geometry

random walks on the lamplighter group kaimanovich and vershik described certain finitely generated groups of exponential growth such that simple random walk on their cayley graph escapes from the identity at a sublinear rate or equivalently all bounded harmonic functions on the cayley graph are constant here we focus on a key example called g 1 by kaimanovich and vershik and show that inward biased random walks on g 1 move outward faster than simple random walk indeed they escape from the identity at a linear rate provided that the bias parameter is smaller than the growth rate of g 1 these walks can be viewed as random walks interacting with a dynamical environment on mathbb z the proof uses potential theory to analyze a stationary environment as seen from the moving particle

covariance structure of the gibbs sampler with applications to the comparisons of estimators and augmentation schemes summary we study the covariance structure of a markov chain generated by the gibbs sampler with emphasis on data augmentation when applied to a bayesian missing data problem the gibbs sampler produces two natural approximations for the posterior distribution of the parameter vector the empirical distribution based on the sampled values of the parameter vector and a mixture of complete data posteriors we prove that rao blackwellization causes a one lag delay for the autocovariances among dependent samples obtained from data augmentation and consequently the mixture approximation produces estimates with smaller variances than the empirical approximation the covariance structure results are used to compare different augmentation schemes it is shown that collapsing and grouping random components in a gibbs sampler with two or three components usually result in more efficient sampling schemes

sampling gaussian distributions in krylov spaces with conjugate gradients this paper introduces a conjugate gradient sampler that is a simple extension of the method of conjugate gradients cg for solving linear systems the cg sampler iteratively generates samples from a gaussian probability density using either a symmetric positive definite covariance or precision matrix whichever is more convenient to model similar to how the lanczos method solves an eigenvalue problem the cg sampler approximates the covariance or precision matrix in a small dimensional krylov space as with any iterative method the cg sampler is efficient for high dimensional problems where forming the covariance or precision matrix is impractical but operating by the matrix is feasible in exact arithmetic the sampler generates gaussian samples with a realized covariance that converges to the covariance of interest in finite precision the sampler produces a gaussian sample with a realized covariance that is the best approximation to the desired covariance in the smaller dimensional krylov space