auxiliary variable methods for markov chain monte carlo with applications abstract suppose that one wishes to sample from the density x using markov chain monte carlo mcmc an auxiliary variable u and its conditional distribution u x can be defined giving the joint distribution x u x u x a mcmc scheme that samples over this joint distribution can lead to substantial gains in efficiency compared to standard approaches the revolutionary algorithm of swendsen and wang is one such example besides reviewing the swendsen wang algorithm and its generalizations this article introduces a new auxiliary variable method called partial decoupling two applications in bayesian image analysis are considered a binary classification problem in which partial decoupling out performs swendsen wang and single site metropolis methods and a positron emission tomography pet reconstruction that uses the gray level prior of geman and mcclure a generalized swendsen wang algorithm is developed for this problem which reduces the computing time to the point where mcmc is a viab

stochastic inversion for electromagnetic geophysics practical challenges and improving convergence efficiency traditional deterministic geophysical inversion algorithms are not designed to provide a robust evaluation of uncertainty that reflects the limitations of the geophysical technique stochastic inversions which do provide a sampling based measure of uncertainty are computationally expensive and not straightforward to implement for nonexperts nonstatisticians our results include stochastic inversion for magnetotelluric and controlled source electromagnetic data two markov chain sampling algorithms metropolis hastings and slice sampler can significantly decrease the computational expense compared to using either sampler alone the statistics of the stochastic inversion allow for 1 variances that better reveal the measurement sensitivities of the two different electromagnetic techniques than traditional techniques and 2 models defined by the median and modes of parameter probability density functions which produce amplitude and phase data that are consistent with the observed data in general pa

gibbs sampling for bayesian non conjugate and hierarchical models by using auxiliary variables summary we demonstrate the use of auxiliary or latent variables for sampling non standard densities which arise in the context of the bayesian analysis of non conjugate and hierarchical models by using a gibbs sampler their strategic use can result in a gibbs sampler having easily sampled full conditionals we propose such a procedure to simplify or speed up the markov chain monte carlo algorithm the strength of this approach lies in its generality and its ease of implementation the aim of the paper therefore is to provide an alternative sampling algorithm to rejection based methods and other sampling approaches such as the metropolis hastings algorithm

a version of the swendsen wang algorithm for restoration of images degraded by poisson noise abstract an algorithm for restoration of images degraded by poisson noise is proposed the algorithm belongs to the family of markov chain monte carlo methods with auxiliary variables we explicitly use the fact that medical images consist of finitely many often relatively few grey levels the continuous scale of grey levels is discretized in an adaptive way so that a straightforward application of the swendsen wang phys rev lett 58 1987 86 algorithm becomes possible partial decoupling method due to higdon j am statist assoc 93 1998 442 585 is also incorporated into the algorithm simulation results suggest that the algorithm is reliable and efficient

mobility of solitons in one dimensional lattices with the cubic quintic nonlinearity we investigate mobility regimes for localized modes in the discrete nonlinear schrodinger dnls equation with the cubic quintic onsite terms using the variational approximation va the largest soliton s total power admitting progressive motion of kicked discrete solitons is predicted by com paring the effective kinetic energy with the respective peierls nabarro pn potential barrier the prediction is novel for the dnls model with the cubic only nonlinearity too demonstrating a rea sonable agreement with numerical findings small self focusing quintic term quickly suppresses the mobility in the case of the competition between the cubic self focusing and quintic self defocusing terms we identify parameter regions where odd and even fundamental modes exchange their stabil ity involving intermediate asymmetric modes in this case stable solitons can be set in motion by kicking so as to let them pass the pn barrier unstable solitons spontaneously start oscillatory or progressive motion if they are located respectively below or above a mobility threshold collisions between moving discrete solitons at the competing nonlinearities frame are studied too

optimal system invariant solutions and evolution of weak discontinuity for isentropic drift flux model abstract inequivalent classes of similarity solutions for isentropic drift flux two phase flow model are obtained by using one dimensional optimal classifications the optimality of this classification is shown in a constructive manner i e by constructing various invariant functions including conditional and numerical invariants further with the help of invariant solution we discuss the evolution of weak discontinuity which displays the nature of the general solution

markov chain monte carlo 10 years and still running although markov chain monte carlo mcmc techniques can be traced back to the early 1950s and are almost contemporary with independent monte carlo methods the use of these techniques has only really taken off in the past 10 years at least in mainstream statistics with the work of gelfand and smith 1990 and the rediscovery of metropolis s and hastings s papers the impact on the discipline is deep and durable because these methods have opened new horizons in the scale of the problems that one can deal with thus enhancing the position of statistics in most applied fields the mcmc revolution has in particular boosted bayesian statistics to new heights by providing a virtually universal tool for dealing with integration and optimization problems this can be seen in for instance the explosion of papers dealing with complex models hierarchical modelings nonparametric bayesian estimation and spatial statistics this trend has also created new synergies with mathematicians and probabilists as well as econometricians engineers ecologists astronomers and others for theoretical requests and practical implications of mcmc techniques it is impossible to provide here a complete perspective on mcmc methods and applications for this as well as for other references we refer the reader to the books by gilks richardson and spiegelhalter 1995 and robert and casella 1999 here we focus on four particular aspects of mcmc algorithms theoretical foundations practical implementation bayesian inference and prospects the web provides many relevant sites including the mcmc preprint server http www statslab cam ac uk r mcmc which offers a most useful window to the current research on mcmc wilson s site on perfect simulation http dimacs rutgers edu dbwilson exact h which keeps track of papers and applets on this topic and a site on convergence control techniques http www ensae fr crest statistique robert mcdiag even more recent developments allow for on line computations through applets like breyer s http www maths lancs ac uk breyer java html which impressively illustrates the behavior of different samplers for simple problems note also that the bugs and coda software are freely available at http www mrc bsu cam ac uk bu

mri tissue classification using high resolution bayesian hidden markov normal mixture models magnetic resonance imaging mri is used to identify the major tissues within a subject s brain classification is usually based on a single image providing one measurement for each volume element or voxel in a discretization of the brain a simple model views each voxel as homogeneous belonging entirely to one of the three major tissue types gray matter white matter or cerebrospinal fluid the measurements are normally distributed with means and variances depending on the tissue types of their voxels because nearby voxels tend to be of the same tissue type a markov random field model can be used to capture the spatial similarity of voxels a more realistic model takes into account the fact that some voxels are not homogeneous and contain tissues of more than one type our approach to this problem is to construct a higher resolution image in which each voxel is divided into subvoxels and subvoxels are in turn assumed to be homogeneous and to follow a markov random field model in the present work

markov chains for exploring posterior distributions several markov chain methods are available for sampling from a posterior distribution two important examples are the gibbs sampler and the metropolis algorithm in addition several strategies are available for constructing hybrid algorithms this paper outlines some of the basic methods and strategies and discusses some related theoretical and practical issues on the theoretical side results from the theory of general state space markov chains can be used to obtain convergence rates laws of large numbers and central limit theorems for estimates obtained from markov chain methods these theoretical results can be used to guide the construction of more efficient algorithms for the practical use of markov chain methods standard simulation methodology provides several variance reduction techniques and also give guidance on the choice of sample size and allocation

the swift hohenberg equation with a nonlocal nonlinearity it is well known that aspects of the formation of localised states in a one dimensional swift hohenberg equation can be described by ginzburg landau type envelope equations this paper extends these multiple scales analyses to cases where an additional nonlinear integral term in the form of a convolution is present the presence of a kernel function introduces a new lengthscale into the problem and this results in additional complexity in both the derivation of envelope equations and in the bifurcation structure when the kernel is short range weakly nonlinear analysis results in envelope equations of standard type but whose coefficients are modified in complicated ways by the nonlinear nonlocal term nevertheless these computations can be formulated quite generally in terms of properties of the fourier transform of the kernel function when the lengthscale associated with the kernel is longer our method leads naturally to the derivation of two different novel envelope equations that describe aspects of the dynamics in these new regimes the first of these contains additional bifurcations and unexpected loops in the bifurcation diagram the second of these captures the stretched out nature of the homoclinic snaking curves that arises due to the nonlocal term

geometric convergence rates for stochastically ordered markov chains let n be a markov chain on the state space 0 that is stochastically ordered in its initial state that is a stochastically larger initial state produces a stochastically larger chain at all other times examples of such chains include random walks the number of customers in various queueing systems and a plethora of storage processes a large body of recent literature concentrates on establishing geometric ergodicity of n in total variation that is proving the existence of a limiting probability measure and a number r 1 such that lim n to infty r n sup a in cal b 0 infty vert p x phi n in a pia vert 0 for every deterministic initial state 0 x we seek to identity the largest r that satisfies this relationship a dependent sample path coupling and a foster lyapunov drift inequality are used to derive convergence rate bounds we then show that the bounds obtained are frequently the best possible application of the methods to queues and random walks are included

fixed width output analysis for markov chain monte carlo markov chain monte carlo is a method of producing a correlated sample to estimate features of a target distribution through ergodic averages a fundamental question is when sampling should stop that is at what point the ergodic averages are good estimates of the desired quantities we consider a method that stops the simulation when the width of a confidence interval based on an ergodic average is less than a user specified value hence calculating a monte carlo standard error is a critical step in assessing the simulation output we consider the regenerative simulation and batch means methods of estimating the variance of the asymptotic normal distribution we give sufficient conditions for the strong consistency of both methods and investigate their finite sample properties in various examples

variational approximations to homoclinic snaking in continuous and discrete systems localized structures appear in a wide variety of systems arising from a pinning mechanism due to the presence of a small scale pattern or an imposed grid when there is a separation of length scales the width of the pinning region is exponentially small and beyond the reach of standard asymptotic methods we show how this behavior can be obtained using a variational method for two systems in the case of the quadratic cubic swift hohenberg equation this gives results that are in agreement with recent work using exponential asymptotics in addition the method is applied to a discrete system with cubic quintic nonlinearity giving results that agree well with numerical simulations

variational approximation and the use of collective coordinates we consider propagating spatially localized waves in a class of equations that contain variational and nonvariational terms the dynamics of the waves is analyzed through a collective coordinate approach motivated by the variational approximation we show that there is a natural choice of projection onto collective variables for reducing the governing nonlinear partial differential equation pde to coupled ordinary differential equations odes this projection produces odes whose solutions are exactly the stationary states of the effective lagrangian that would be considered in applying the variational approximation method we illustrate our approach by applying it to a modified fisher equation for a traveling front containing a non constant coefficient nonlinear term we present numerical results that show that our proposed projection captures both the equilibria and the dynamics of the pde much more closely than previously proposed projections

on the applicability of regenerative simulation in markov chain monte carlo we consider the central limit theorem and the calculation of asymptotic standard errors for the ergodic averages constructed in markov chain monte carlo chan geyer 1994 established a central limit theorem for ergodic averages by assuming that the underlying markov chain is geometrically ergodic and that a simple moment condition is satisfied while it is relatively straightforward to check chan geyer s conditions their theorem does not lead to a consistent and easily computed estimate of the variance of the asymptotic normal distribution conversely mykland et al 1995 discuss the use of regeneration to establish an alternative central limit theorem with the advantage that a simple consistent estimator of the asymptotic variance is readily available however their result assumes a pair of unwieldy moment conditions whose verification is difficult in practice in this paper we show that the conditions of chan geyer s theorem are sufficient to establish the central limit theorem of mykland et al this result in conjunction with other recent developments should pave the way for more widespread use of the regenerative method in markov chain monte carlo our results are illustrated in the context of the slice sampler

alternating subspace spanning resampling to accelerate markov chain monte carlo simulation this article provides a simple method to accelerate markov chain monte carlo sampling algorithms such as the data augmentation algorithm and the gibbs sampler via alternating subspace spanning resampling assr the assr algorithm often shares the simplicity of its parent sampler but has dramatically improved efficiency the methodology is illustrated with bayesian estimation for analysis of censored data from fractionated experiments the relationships between assr and existing methods are also discussed

convergence of slice sampler markov chains we analyse theoretical properties of the slice sampler we find that the algorithm has extremely robust geometric ergodicity properties for the case of just one auxiliary variable we demonstrate that the algorithm is stochastically monotone and we deduce analytic bounds on the total variation distance from stationarity of the method by using foster lyapunov drift condition methodology

hierarchical models a current computational perspective 2000 hierarchical models a current computational perspective journal of the american statistical association vol 95 no 452 pp 1312 1316

an analysis of swendsen wang and related sampling methods convergence rates statistical efficiency and sampling costs are studied for the original and extended swendsen wang methods of generating a sample path s j j 1 with equilibrium distribution with r distinct elements on a finite state space x of size n 1 given s j 1 each method uses auxiliary random variables to identify the subset of x from which s j is to be randomly sampled let min and max denote respectively the smallest and largest elements in and let n denote the number of elements in with value max for a single auxiliary variable uniform sampling from the subset and n 1 n r min n r max 1 our results show rapid convergence and high statistical efficiency for large min max or n r n 1 and slow convergence and poor statistical efficiency for small min max and n r n 1 other examples provide additional insight for extended swendsen wang methods with non uniform subset sampling the analysis identifies the properties of a decomposition of x that favour fast convergence and high statistical efficiency in the absence of exploitable special structure subset sampling can be costly regardless of which of these methods is employed

group classification for isothermal drift flux model of two phase flows in this paper a full symmetry group classification for isothermal multiphase drift flux model is presented all invariant functions are developed for the lie algebra which play a vital role in construction of optimal systems further with the help of one dimensional optimal classification group invariant solutions are obtained which describe the asymptotic behavior of general solution

similarity solutions for three dimensional euler equations using lie group analysis a similarity analysis of the nonlinear three dimensional unsteady euler equations of gas dynamics is presented using lie group of transformations with commuting infinitesimal operators symmetry groups admitted by the governing system of partial differential equations pdes are obtained and the complete lie algebra of infinitesimal symmetries is established the symmetry generators are used for constructing similarity variables which lead to a reduced system of equations with one independent variable less at each step and eventually to a system of ordinary differential equations odes in some cases it is possible to solve these equations exactly

maximum likelihood reconstruction for emission tomography previous models for emission tomography et do not distinguish the physics of et from that of transmission tomography we give a more accurate general mathematical model for et where an unknown emission density x y z generates and is to be reconstructed from the number of counts n d in each of d detector units d within the model we give an algorithm for determining an estimate of which maximizes the probability p n of observing the actual detector count data n over all possible densities let independent poisson variables n b with unknown means b b 1 b represent the number of unobserved emissions in each of b boxes pixels partitioning an object containing an emitter suppose each emission in box b is detected in detector unit d with probability p b d d 1 d with p b d a one step transition matrix assumed known we observe the total number n n d of emissions in each detector unit d and want to estimate the unknown b b 1 b for each the observed data n has probability or likelihood p n the em algorithm of mathematical statistics starts with an initial estimate 0 and gives the following simple iterative procedure for obtaining a new estimate new from an old estimate old to obtain k k 1 2 new b old b dd 1 n d p b d old b p b d b 1 b

on linearized coupling conditions for a class of isentropic multiphase drift flux models at pipe to pipe intersections in this paper a general drift flux model describing a subsonic and isentropic multi phase fluid in connected pipes is considered each phase is assumed to be isentropic with its own sonic speed the components are gamma law gases with 1 for such a computational challenge at a junction is the computation of rarefaction waves which do not have a readily available analytical form firstly the well posedness of the riemann problem at the junction is discussed it is suggested that rarefaction waves should be linearized in order to obtain a more efficient numerical method for coupling such multi component flow some computational results on the dynamics of the multi phase gas in the pipes demonstrate the qualitative behavior of this approach

a note on optimal systems for the heat equation a rigorous proof of the optimality of one to five parameter systems for the symmetry group of the one dimensional heat equation is given

geometric convergence and central limit theorems for multidimensional hastings and metropolis algorithms we develop results on geometric ergodicity of markov chains and apply these and other recent results in markov chain theory to multidimensional hastings and metropolis algorithms for those based on random walk candidate distributions we find sufficient conditions for moments and moment generating functions to converge at a geometric rate to a prescribed distribution by phrasing the conditions in terms of the curvature of the densities we show that the results apply to all distributions with positive densities in a large class which encompasses many commonly used statistical forms from these results we develop central limit theorems for the metropolis algorithm converse results showing non geometric convergence rates for chains where the rejection rate is not bounded away from unity are also given these show that the negative definiteness property is not redundant

decomposed slice sampling for factorized distributions abstract slice sampling provides an automatical adjustment to match the characteristics of the distribution although this method has made great success in many situations it becomes limited when the distribution is complex inspired by higdon 1 in this paper we present a decomposed sampling framework based on slice sampling called decomposed slice sampling dss we suppose that the target distribution can be divided into two multipliers so that information in each term can be used respectively the first multiplier is used in the first step of dss to obtain horizontal slices and the last term is used in the second step simulations on four simple distributions indicate the effectiveness of our method compared with slice sampling and hamiltonian monte carlo on gaussian distributions in different dimensions and ten real world datasets the proposed method achieves better performance

preliminary group classification of equations vtt f x vx vxx g x vx a classification is given of equations vtt f v vx vxx g x vx admitting an extension by one of the principal lie algebra of the equation under consideration the paper is one of few applications of a new algebraic approach to the problem of group classification the method of preliminary group classification the result of the work is a wide class of equations summarized in table ii

group invariant solutions of differential equations we introduce the concept of a weak symmetry group of a system of partial differential equations that generalizes the nonclassical method introduced by bluman and cole for finding group invariant

minorization conditions and convergence rates for markov chain monte carlo abstract general methods are provided for analyzing the convergence of discrete time general state space markov chains such as those used in stochastic simulation algorithms including the gibbs sampler the methods provide rigorous a priori bounds on how long these simulations should be run to give satisfactory results results are applied to two models of the gibbs sampler a bivariate normal model and a hierarchical poisson model with gamma conditionals the methods use the notion of minorization conditions for markov chains

lie symmetry analysis optimal systems and exact solutions to the fifth order kdv types of equations abstract in this paper the lie symmetry analysis is performed on the fifth order kdv types of equations which arise in modeling many physical phenomena the similarity reductions and exact solutions are obtained based on the optimal system method then the exact analytic solutions are considered by using the power series method

an efficient markov chain monte carlo method for distributions with intractable normalising constants maximum likelihood parameter estimation and sampling from bayesian posterior distributions are problematic when the probability density for the parameter of interest involves an intractable normalising constant which is also a function of that parameter in this paper an auxiliary variable method is presented which requires only that independent samples can be drawn from the unnormalised density at any particular parameter value the proposal distribution is constructed so that the normalising constant cancels from the metropolis hastings ratio the method is illustrated by producing posterior samples for parameters of the ising model given a particular lattice realisation

parameter expansion for data augmentation abstract viewing the observed data of a statistical model as incomplete and augmenting its missing parts are useful for clarifying concepts and central to the invention of two well known statistical algorithms expectation maximization em and data augmentation recently liu rubin and wu demonstrated that expanding the parameter space along with augmenting the missing data is useful for accelerating iterative computation in an em algorithm the main purpose of this article is to rigorously define a parameter expanded data augmentation px da algorithm and to study its theoretical properties the px da is a special way of using auxiliary variables to accelerate gibbs sampling algorithms and is closely related to reparameterization techniques we obtain theoretical results concerning the convergence rate of the px da algorithm and the choice of prior for the expansion parameter to understand the role of the expansion parameter we establish a new theory for iterative conditional sampling under the tra

batch means and spectral variance estimators in markov chain monte carlo calculating a monte carlo standard error mcse is an important step in the statistical analysis of the simulation output obtained from a markov chain monte carlo experiment an mcse is usually based on an estimate of the variance of the asymptotic normal distribution we consider spectral and batch means methods for estimating this variance in particular we establish conditions which guarantee that these estimators are strongly consistent as the simulation effort increases in addition for the batch means and overlapping batch means methods we establish conditions ensuring consistency in the mean square sense which in turn allows us to calculate the optimal batch size up to a constant of proportionality finally we examine the empirical finite sample properties of spectral variance and batch means estimators and provide recommendations for practitioners

importance sampling for simulation of markovian physical processes transient markov chains are sometimes simulated to estimate rare event probabilities for illustration chains defined by an airborne particle dispersion model are used to estimate the probabilities that released particles reach various locations such estimated probabilities are needed for many purposes including exposure calculations for affected populations and optimization of detector placement by using experimental designs for simulation runs and embedding fitted regression models of output data in importance sampling transition kernels convergence is improved by factors of tens to hundreds

spatial bayesian variable selection with application to functional magnetic resonance imaging we propose a procedure to undertake bayesian variable selection and model averaging for a series of regressions located on a lattice for those regressors that are in common in the regressions we consider using an ising prior to smooth spatially the indicator variables representing whether or not the variable is zero or nonzero in each regression this smooths spatially the probabilities that each independent variable is nonzero in each regression and indirectly smooths spatially the regression coefficients we discuss how single site sampling schemes can be used to evaluate the joint posterior distribution the approach is applied to the problem of functional magnetic resonance imaging in medical statistics where massive datasets arise that require prompt processing here the ising prior with a three dimensional neighborhood structure is used to smooth spatially activation maps from regression models of blood oxygenation the ising prior also has the advantage of allowing incorporation of anatomic prio

positron emission tomography by markov chain monte carlo with auxiliary variables we propose a new algorithm for positron emission tomography pet image reconstruction the algorithm belongs to the family of markov chain monte carlo methods with auxiliary variables the idea is to iteratively generate hidden variables at one step and use them for image restoration at another step the well known model of vardi et al j amer statist assoc 80 1985 8 for pet is combined with the bayesian model of lasota and niemiro pattern recognition 36 2003 931 for the underlying images this latter model takes advantage of the fact that medical images often consist of relatively few grey levels of unknown intensity the algorithm of lasota and niemiro pattern recognition 36 2003 931 is used in the image restoration part of the pet algorithm essentially as a noise filtering and smoothing device it is now equipped with an additional data reconstruction step we include simulation results which suggest that the method is truly reliable we also describe a version of the basic algorithm in which a random simulation step is replaced by computation of expected value similarly as in the em algorithm

a direct algorithm of one dimensional optimal system for the group invariant solutions a direct and systematic algorithm is proposed to find one dimensional optimal system for the group invariant solutions which is attributed to the classification of its corresponding one dimensional lie algebra since the method is based on different values of all the invariants the process itself can both guarantee the comprehensiveness and demonstrate the inequivalence of the optimal system with no further proof to leave the algorithm clear we illustrate each stage with a couple of well known examples the korteweg de vries equation and the heat equation finally we apply our method to the novikov equation and use the found optimal system to investigate the corresponding invariant solutions

bayesian mode regression using mixtures of triangular densities abstract bayesian semiparametric models for mean and median regressions abound but a void for mode regressions exists we fill this gap by nonparametrically modeling the error distribution in such regressions that entails constructing prior distributions on densities which exhibit flexibility while fixing the mode at 0 such priors exist when constraining the mean and median but to our knowledge there is none for the mode our solution with mixtures of triangular distributions results in a conditionally conjugate prior on the space of unimodal untruncated convex densities consistency properties of the resulting modal estimators are studied followed by simulated and real data illustrations

lie symmetries and exact solutions of the barotropic vorticity equation lie group methods are used for the study of various issues related to symmetries and exact solutions of the barotropic vorticity equation the lie symmetries of the barotropic vorticity equations on the f and planes as well as on the sphere in rotating and rest reference frames are determined a symmetry background for reducing the rotating reference frame to the rest frame is presented the one and two dimensional inequivalent subalgebras of the lie invariance algebras of both equations are exhaustively classified and then used to compute invariant solutions of the vorticity equations this provides large classes of exact solutions which include both rossby and rossby haurwitz waves as special cases we also discuss the possibility of partial invariance for the plane equation thereby further extending the family of its exact solutions this is done in a more systematic and complete way than previously available in literature

lie symmetry reductions and exact solutions of a coupled kdv burgers equation in this paper based on classical lie group method with the help of maple software we study a coupled kdv burgers equation and get its infinitesimal generator commutation table of lie algebra symmetry group and similarity reductions in particular solitary wave solutions and similarity solutions of the coupled kdv burgers equation are derived from the reduction equations

graphical gaussian shape models and their application to image segmentation this paper presents a novel approach to shape modeling and a model based image segmentation procedure tailor made for the proposed shape model a common way to represent shape is based on so called key points and leads to shape variables which are invariant with respect to similarity transformations we propose a graphical shape model which relies on a certain conditional independence structure among the shape variables most often it is sufficient to use a sparse underlying graph reflecting both nearby and long distance key point interactions graphical shape models allow for specific shape modeling since e g for the subclass of decomposable graphical gaussian models both model selection procedures and explicit parameter estimates are available a further prerequisite to a successful application of graphical shape models in image analysis is provided by the toolbox of markov chain monte carlo methods offering highly flexible and effective methods for the exploration of a specified distribution for bayesian image segmentation based on a graphical gaussian shape model we suggest applying a hybrid approach composed of the well known gibbs sampler and the more recent slice sampler shape modeling as well as image analysis are demonstrated for the segmentation of vertebrae from two dimensional slices of computer tomography images

a statistical model for positron emission tomography abstract positron emission tomography pet still in its research stages is a technique that promises to open new medical frontiers by enabling physicians to study the metabolic activity of the body in a pictorial manner much as in x ray transmission tomography and other modes of computerized tomography the quality of the reconstructed image in pet is very sensitive to the mathematical algorithm to be used for reconstruction in this article we tailor a mathematical model to the physics of positron emissions and we use the model to describe the basic image reconstruction problem of pet as a standard problem in statistical estimation from incomplete data we describe various estimation procedures such as the maximum likelihood ml method using the em algorithm the method of moments and the least squares method a computer simulation of a pet experiment is then used to demonstrate the ml and the least squares reconstructions the main purposes of this article are to report on what we believe is an

bayesian methods for hidden markov models markov chain monte carlo mcmc sampling strategies can be used to simulate hidden markov model hmm parameters from their posterior distribution given observed data some mcmc methods used in practice for computing likelihood conditional probabilities of hidden states and the most likely sequence of states can be improved by incorporating established recursive algorithms the most important of these is a set of forward backward recursions calculating conditional distributions of the hidden states given observed data and model parameters i show how to use the recursive algorithms in an mcmc context and demonstrate mathematical and empirical results showing a gibbs sampler using the forward backward recursions mixes more rapidly than another sampler often used for hmms iintroduce an augmented variables technique for obtaining unique state labels in hmms and finite mixture models i show how recursive computing allows the statistically efficient use of mcmc output when estimating the hidden states i

fully bayesian estimation of gibbs hyperparameters for emission computed tomography data in recent years many investigators have proposed gibbs prior models to regularize images reconstructed from emission computed tomography data unfortunately hyperparameters used to specify gibbs priors can greatly influence the degree of regularity imposed by such priors and as a result numerous procedures have been proposed to estimate hyperparameter values from observed image data many of these procedures attempt to maximize the joint posterior distribution on the image scene to implement these methods approximations to the joint posterior densities are required because the dependence of the gibbs partition function on the hyperparameter values is unknown here the authors use recent results in markov chain monte carlo mcmc sampling to estimate the relative values of gibbs partition functions and using these values sample from joint posterior distributions on image scenes this allows for a fully bayesian procedure which does not fix the hyperparameters at some estimated or specified value but enables uncertainty about these values to be propagated through to the estimated intensities the authors utilize realizations from the posterior distribution for determining credible regions for the intensity of the emission source the authors consider two different markov random field mrf models the power model and a line site model as applications they estimate the posterior distribution of source intensities from computer simulated data as well as data collected from a physical single photon emission computed tomography spect phantom

the bayesian bridge summary we propose the bayesian bridge estimator for regularized regression and classification two key mixture representations for the bayesian bridge model are developed a scale mixture of normal distributions with respect to an stable random variable a mixture of bartlett fejer kernels or triangle densities with respect to a two component mixture of gamma random variables both lead to markov chain monte carlo methods for posterior simulation and these methods turn out to have complementary domains of maximum efficiency the first representation is a well known result due to west and is the better choice for collinear design matrices the second representation is new and is more efficient for orthogonal problems largely because it avoids the need to deal with exponentially tilted stable random variables it also provides insight into the multimodality of the joint posterior distribution which is a feature of the bridge model that is notably absent under ridge or lasso type priors we prove a theorem that extends this representation to a wider class of densities representable as scale mixtures of beta distributions and we provide an explicit inversion formula for the mixing distribution the connections with slice sampling and scale mixtures of normal distributions are explored on the practical side we find that the bayesian bridge model outperforms its classical cousin in estimation and prediction across a variety of data sets both simulated and real we also show that the markov chain monte carlo algorithm for fitting the bridge model exhibits excellent mixing properties particularly for the global scale parameter this makes for a favourable contrast with analogous markov chain monte carlo algorithms for other sparse bayesian models all methods described in this paper are implemented in the r package bayesbridge an extensive set of simulation results is provided in two on line supplemental files

spatial statistics and bayesian computation on wednesday may 6th 1992 professor b w silverman in the chair summary markov chain monte carlo mcmc algorithms such as the gibbs sampler have provided a bayesian inference machine in image analysis and in other areas of spatial statistics for several years founded on the pioneering ideas of ulf grenander more recently the observation that hyperparameters can be included as part of the updating schedule and the fact that almost any multivariate distribution is equivalently a markov random field has opened the way to the use of mcmc in general bayesian computation in this paper we trace the early development of mcmc in bayesian inference review some recent computational progress in statistical physics based on the introduction of auxiliary variables and discuss its current and future relevance in bayesian applications we briefly describe a simple mcmc implementation for the bayesian analysis of agricultural field experiments with which we have some practical experience

perfect slice samplers perfect sampling allows the exact simulation of random variables from the stationary measure of a markov chain by exploiting monotonicity properties of the slice sampler we show that a perfect version of the algorithm can be easily implemented at least when the target distribution is bounded various extensions including perfect product slice samplers and examples of applications are discussed

likelihood analysis of spatial point patterns summary the likelihood procedure for estimating the pairwise interaction potential function is developed for statistical analysis of homogeneous spatial point patterns approximation methods of the normalizing factor of gibbs canonical distribution are discussed both to estimate a scale parameter and to measure the softness or hardness of repulsive interactions the approximations are useful up to a considerably high density the validity of our procedure is demonstrated by some computer experi ments some real data are analysed