Mode Jumping Proposals in MCMC

Markov chain Monte Carlo algorithms generate samples from a target distribution by simulating a Markov chain. Large flexibility exists in specification of transition matrix of the chain. In practice, however, most algorithms used only allow small changes in the state vector in each iteration. This choice typically causes problems for multi-modal distributions as moves between modes become rare and, in turn, results in slow convergence to the target distribution. In this paper we consider continuous distributions on R ⁿ and specify how optimization for local maxima of the target distribution can be incorporated in the specification of the Markov chain. Thereby, we obtain a chain with frequent jumps between modes. We demonstrate the effectiveness of the approach in three examples. The first considers a simple mixture of bivariate normal distributions, whereas the two last examples consider sampling from posterior distributions based on previously analysed data sets.     

Image segmentation using voronoi polygons and MCMC, with application to muscle fibre images

We investigate a Bayesian method for the segmentation of muscle fibre images. The images are reasonably well approximated by a Dirichlet tessellation, and so we use a deformable template model based on Voronoi polygons to represent the segmented image. We consider various prior distributions for the parameters and suggest an appropriate likelihood. Following the Bayesian paradigm, the mathematical form for the posterior distribution is obtained (up to an integrating constant). We introduce a Metropolis-Hastings algorithm and a reversible jump Markov chain Monte Carlo algorithm (RJMCMC) for simulation from the posterior when the number of polygons is fixed or unknown. The particular moves in the RJMCMC algorithm are birth, death and position/colour changes of the point process which determines the location of the polygons. Segmentation of the true image was carried out using the estimated posterior mode and posterior mean. A simulation study is presented which is helpful for tuning the hyperparameters and to assess the accuracy. The algorithms work well on a real image of a muscle fibre cross-section image, and an additional parameter, which models the boundaries of the muscle fibres, is included in the final model.     

Rate estimation in partially observed Markov jump processes with measurement errors

We present a simulation methodology for Bayesian estimation of rate parameters in Markov jump processes arising for example in stochastic kinetic models. To handle the problem of missing components and measurement errors in observed data, we embed the Markov jump process into the framework of a general state space model. We do not use diffusion approximations. Markov chain Monte Carlo and particle filter type algorithms are introduced which allow sampling from the posterior distribution of the rate parameters and the Markov jump process also in data-poor scenarios. The algorithms are illustrated by applying them to rate estimation in a model for prokaryotic auto-regulation and the stochastic Oregonator, respectively.     

Free energy methods for Bayesian inference: efficient exploration of univariate Gaussian mixture posteriors

Because of their multimodality, mixture posterior distributions are difficult to sample with standard Markov chain Monte Carlo (MCMC) methods. We propose a strategy to enhance the sampling of MCMC in this context, using a biasing procedure which originates from computational Statistical Physics. The principle is first to choose a “reaction coordinate”, that is, a “direction” in which the target distribution is multimodal. In a second step, the marginal log-density of the reaction coordinate with respect to the posterior distribution is estimated; minus this quantity is called “free energy” in the computational Statistical Physics literature. To this end, we use adaptive biasing Markov chain algorithms which adapt their targeted invariant distribution on the fly, in order to overcome sampling barriers along the chosen reaction coordinate. Finally, we perform an importance sampling step in order to remove the bias and recover the true posterior. The efficiency factor of the importance sampling step can easily be estimated a priori once the bias is known, and appears to be rather large for the test cases we considered.     

C219. Comment on ”calculating binomial probabilities when the trial probabilities are unequal“

In complex models like hidden Markov chains, the convergence of the MCMC algorithms used to approximate the posterior distribution and the Bayes estimates of the parameters of interest must be controlled in a robust manner. We propose in this paper a series of online controls, which rely on classical non-parametric tests, to evaluate independence from the start-up distribution, stability of the Markov chain, and asymptotic normality. These tests lead to graphical control spreadsheets which arepresentedin the set-up of normalmixture hidden Markov chains to compare the full Gibbs sampler with an aggregated Gibbs sampler based on the forward – backward formulas.     

Parallel algorithms for Markov chain Monte Carlo methods in latent spatial Gaussian models

Markov chain Monte Carlo (MCMC) implementations of Bayesian inference for latent spatial Gaussian models are very computationally intensive, and restrictions on storage and computation time are limiting their application to large problems. Here we propose various parallel MCMC algorithms for such models. The algorithms' performance is discussed with respect to a simulation study, which demonstrates the increase in speed with which the algorithms explore the posterior distribution as a function of the number of processors. We also discuss how feasible problem size is increased by use of these algorithms.     

An online Bayesian mixture labelling method by minimizing deviance of classification probabilities to reference labels

Solving label switching is crucial for interpreting the results of fitting Bayesian mixture models. The label switching originates from the invariance of posterior distribution to permutation of component labels. As a result, the component labels in Markov chain simulation may switch to another equivalent permutation, and the marginal posterior distribution associated with all labels may be similar and useless for inferring quantities relating to each individual component. In this article, we propose a new simple labelling method by minimizing the deviance of the class probabilities to a fixed reference labels. The reference labels can be chosen before running Markov chain Monte Carlo (MCMC) using optimization methods, such as expectation-maximization algorithms, and therefore the new labelling method can be implemented by an online algorithm, which can reduce the storage requirements and save much computation time. Using the Acid data set and Galaxy data set, we demonstrate the success of the proposed labelling method for removing the labelling switching in the raw MCMC samples.     

Stochastic approximation Monte Carlo EM for change-point analysis

In the expectation–maximization (EM) algorithm for maximum likelihood estimation from incomplete data, Markov chain Monte Carlo (MCMC) methods have been used in change-point inference for a long time when the expectation step is intractable. However, the conventional MCMC algorithms tend to get trapped in local mode in simulating from the posterior distribution of change points. To overcome this problem, in this paper we propose a stochastic approximation Monte Carlo version of EM (SAMCEM), which is a combination of adaptive Markov chain Monte Carlo and EM utilizing a maximum likelihood method. SAMCEM is compared with the stochastic approximation version of EM and reversible jump Markov chain Monte Carlo version of EM on simulated and real datasets. The numerical results indicate that SAMCEM can outperform among the three methods by producing much more accurate parameter estimates and the ability to achieve change-point positions and estimates simultaneously.     

On Block Updating in Markov Random Field Models for Disease Mapping

Gaussian Markov random field (GMRF) models are commonly used to model spatial correlation in disease mapping applications. For Bayesian inference by MCMC, so far mainly single-site updating algorithms have been considered. However, convergence and mixing properties of such algorithms can be extremely poor due to strong dependencies of parameters in the posterior distribution. In this paper, we propose various block sampling algorithms in order to improve the MCMC performance. The methodology is rather general, allows for non-standard full conditionals, and can be applied in a modular fashion in a large number of different scenarios. For illustration we consider three different applications: two formulations for spatial modelling of a single disease (with and without additional unstructured parameters respectively), and one formulation for the joint analysis of two diseases. The results indicate that the largest benefits are obtained if parameters and the corresponding hyperparameter are updated jointly in one large block. Implementation of such block algorithms is relatively easy using methods for fast sampling of Gaussian Markov random fields ( Rue, 2001 ). By comparison, Monte Carlo estimates based on single-site updating can be rather misleading, even for very long runs. Our results may have wider relevance for efficient MCMC simulation in hierarchical models with Markov random field components.     

Bayesian analysis of mixture modelling using the multivariate t distribution

A finite mixture model using the multivariate t distribution has been shown as a robust extension of normal mixtures. In this paper, we present a Bayesian approach for inference about parameters of t-mixture models. The specifications of prior distributions are weakly informative to avoid causing nonintegrable posterior distributions. We present two efficient EM-type algorithms for computing the joint posterior mode with the observed data and an incomplete future vector as the sample. Markov chain Monte Carlo sampling schemes are also developed to obtain the target posterior distribution of parameters. The advantages of Bayesian approach over the maximum likelihood method are demonstrated via a set of real data.     

Bayesian Analysis of the Censored Regression Model with an AEPD Error Term

We present the censored regression model with the error term following the asymmetric exponential power distribution. We propose three Markov chain Monte Carlo (MCMC) algorithms: the first one uses the probability integral transformation; the second one uses a combination of the probability integral transformation and random walk draws; while the third one uses random walk draws. Using simulated data we compare the performance of the three MCMC algorithms. Then we compare the posterior means, or Bayes estimates, with maximum likelihood estimates. We estimate the stock option portion of executive compensation as an example of the empirical application.     

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, non-linear 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.     

Bayesian inference for generalized extreme value distributions via Hamiltonian Monte Carlo

In this article, we propose to evaluate and compare Markov chain Monte Carlo (MCMC) methods to estimate the parameters in a generalized extreme value model. We employed the Bayesian approach using traditional Metropolis-Hastings methods, Hamiltonian Monte Carlo (HMC), and Riemann manifold HMC (RMHMC) methods to obtain the approximations to the posterior marginal distributions of interest. Applications to real datasets and simulation studies provide evidence that the extra analytical work involved in Hamiltonian Monte Carlo algorithms is compensated by a more efficient exploration of the parameter space.     

Markov Chain Monte Carlo for Exact Inference for Diffusions

ABSTRACT. We develop exact Markov chain Monte Carlo methods for discretely sampled, directly and indirectly observed diffusions. The qualification ‘exact’ refers to the fact that the invariant and limiting distribution of the Markov chains is the posterior distribution of the parameters free of any discretization error. The class of processes to which our methods directly apply are those which can be simulated using the most general to date exact simulation algorithm. The article introduces various methods to boost the performance of the basic scheme, including reparametrizations and auxiliary Poisson sampling. We contrast both theoretically and empirically how this new approach compares to irreducible high frequency imputation, which is the state‐of‐the‐art alternative for the class of processes we consider, and we uncover intriguing connections. All methods discussed in the article are tested on typical examples.     

Bayesian texture segmentation of weed and crop images using reversible jump Markov chain Monte Carlo methods

Summary. A Bayesian method for segmenting weed and crop textures is described and implemented. The work forms part of a project to identify weeds and crops in images so that selective crop spraying can be carried out. An image is subdivided into blocks and each block is modelled as a single texture. The number of different textures in the image is assumed unknown. A hierarchical Bayesian procedure is used where the texture labels have a Potts model (colour Ising Markov random field) prior and the pixels within a block are distributed according to a Gaussian Markov random field, with the parameters dependent on the type of texture. We simulate from the posterior distribution by using a reversible jump Metropolis–Hastings algorithm, where the number of different texture components is allowed to vary. The methodology is applied to a simulated image and then we carry out texture segmentation on the weed and crop images that motivated the work.     

Bayesian analysis of dynamic magnetic resonance breast images

Summary.  We describe an integrated methodology for analysing dynamic magnetic resonance images of the breast. The problems that motivate this methodology arise from a collaborative study with a tumour institute. The methods are developed within the Bayesian framework and comprise image restoration and classification steps. Two different approaches are proposed for the restoration. Bayesian inference is performed by means of Markov chain Monte Carlo algorithms. We make use of a Metropolis algorithm with a specially chosen proposal distribution that performs better than more commonly used proposals. The classification step is based on a few attribute images yielded by the restoration step that describe the essential features of the contrast agent variation over time. Procedures for hyperparameter estimation are provided, so making our method automatic. The results show the potential of the methodology to extract useful information from acquired dynamic magnetic resonance imaging data about tumour morphology and internal pathophysiological features.     

Non-parametric Bayesian Inference for Integrals with respect to an Unknown Finite Measure

Abstract.  We consider the problem of estimating a collection of integrals with respect to an unknown finite measure μ from noisy observations of some of the integrals. A new method to carry out Bayesian inference for the integrals is proposed. We use a Dirichlet or Gamma process as a prior for μ , and construct an approximation to the posterior distribution of the integrals using the sampling importance resampling algorithm and samples from a new multidimensional version of a Markov chain by Feigin and Tweedie. We prove that the Markov chain is positive Harris recurrent, and that the approximating distribution converges weakly to the posterior as the sample size increases, under a mild integrability condition. Applications to polymer chemistry and mathematical finance are given.     

Approximating cross-validatory predictive evaluation in Bayesian latent variable models with integrated IS and WAIC

Looking at predictive accuracy is a traditional method for comparing models. A natural method for approximating out-of-sample predictive accuracy is leave-one-out cross-validation (LOOCV)—we alternately hold out each case from a full dataset and then train a Bayesian model using Markov chain Monte Carlo without the held-out case; at last we evaluate the posterior predictive distribution of all cases with their actual observations. However, actual LOOCV is time-consuming. This paper introduces two methods, namely iIS and iWAIC, for approximating LOOCV with only Markov chain samples simulated from a posterior based on a full dataset. iIS and iWAIC aim at improving the approximations given by importance sampling (IS) and WAIC in Bayesian models with possibly correlated latent variables. In iIS and iWAIC, we first integrate the predictive density over the distribution of the latent variables associated with the held-out without reference to its observation, then apply IS and WAIC approximations to the integrated predictive density. We compare iIS and iWAIC with other approximation methods in three kinds of models: finite mixture models, models with correlated spatial effects, and a random effect logistic regression model. Our empirical results show that iIS and iWAIC give substantially better approximates than non-integrated IS and WAIC and other methods.     

REVERSIBLE JUMP MARKOV CHAIN MONTE CARLO METHODS AND SEGMENTATION ALGORITHMS IN HIDDEN MARKOV MODELS

We consider hidden Markov models with an unknown number of regimes for the segmentation of the pixel intensities of digital images that consist of a small set of colours. New reversible jump Markov chain Monte Carlo algorithms to estimate both the dimension and the unknown parameters of the model are introduced. Parameters are updated by random walk Metropolis–Hastings moves, without updating the sequence of the hidden Markov chain. The segmentation (i.e. the estimation of the hidden regimes) is a further aim and is performed by means of a number of competing algorithms. We apply our Bayesian inference and segmentation tools to digital images, which are linearized through the Peano–Hilbert scan, and perform experiments and comparisons on both synthetic images and a real brain magnetic resonance image.     

Particle filtering for partially observed Gaussian state space models

Summary. Solving Bayesian estimation problems where the posterior distribution evolves over time through the accumulation of data has many applications for dynamic models. A large number of algorithms based on particle filtering methods, also known as sequential Monte Carlo algorithms, have recently been proposed to solve these problems. We propose a special particle filtering method which uses random mixtures of normal distributions to represent the posterior distributions of partially observed Gaussian state space models. This algorithm is based on a marginalization idea for improving efficiency and can lead to substantial gains over standard algorithms. It differs from previous algorithms which were only applicable to conditionally linear Gaussian state space models. Computer simulations are carried out to evaluate the performance of the proposed algorithm for dynamic tobit and probit models.