Sequential Monte Carlo with transformations

This paper examines methodology for performing Bayesian inference sequentially on a sequence of posteriors on spaces of different dimensions. For this, we use sequential Monte Carlo samplers, introducing the innovation of using deterministic transformations to move particles effectively between target distributions with different dimensions. This approach, combined with adaptive methods, yields an extremely flexible and general algorithm for Bayesian model comparison that is suitable for use in applications where the acceptance rate in reversible jump Markov chain Monte Carlo is low. We use this approach on model comparison for mixture models, and for inferring coalescent trees sequentially, as data arrives.     

Bayesian MARS

A Bayesian approach to multivariate adaptive regression spline (MARS) fitting (Friedman, 1991) is proposed. This takes the form of a probability distribution over the space of possible MARS models which is explored using reversible jump Markov chain Monte Carlo methods (Green, 1995). The generated sample of MARS models produced is shown to have good predictive power when averaged and allows easy interpretation of the relative importance of predictors to the overall fit.     

Multivariate mixtures of normals with unknown number of components

We present full Bayesian analysis of finite mixtures of multivariate normals with unknown number of components. We adopt reversible jump Markov chain Monte Carlo and we construct, in a manner similar to that of Richardson and Green (1997), split and merge moves that produce good mixing of the Markov chains. The split moves are constructed on the space of eigenvectors and eigenvalues of the current covariance matrix so that the proposed covariance matrices are positive definite. Our proposed methodology has applications in classification and discrimination as well as heterogeneity modelling. We test our algorithm with real and simulated data.     

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.     

Generalized multiple-point Metropolis algorithms for approximate Bayesian computation

It is well known that the approximate Bayesian computation algorithm based on Markov chain Monte Carlo methods suffers from the sensitivity to the choice of starting values, inefficiency and a low acceptance rate. To overcome these problems, this study proposes a generalization of the multiple-point Metropolis algorithm, which proceeds by generating multiple-dependent proposals and then by selecting a candidate among the set of proposals on the basis of weights that can be chosen arbitrarily. The performance of the proposed algorithm is illustrated by using both simulated and real data.     

Tuning tempered transitions

The method of tempered transitions was proposed by Neal (Stat. Comput. 6:353–366, 1996) for tackling the difficulties arising when using Markov chain Monte Carlo to sample from multimodal distributions. In common with methods such as simulated tempering and Metropolis-coupled MCMC, the key idea is to utilise a series of successively easier to sample distributions to improve movement around the state space. Tempered transitions does this by incorporating moves through these less modal distributions into the MCMC proposals. Unfortunately the improved movement between modes comes at a high computational cost with a low acceptance rate of expensive proposals. We consider how the algorithm may be tuned to increase the acceptance rates for a given number of temperatures. We find that the commonly assumed geometric spacing of temperatures is reasonable in many but not all applications.     

Markov Chain Monte Carlo model selection for DAG models

We present a methodology for Bayesian model choice and averaging in Gaussian directed acyclic graphs (dags). The dimension-changing move involves adding or dropping a (directed) edge from the graph. The methodology employs the results in Geiger and Heckerman and searches directly in the space of all dags. Model determination is carried out by implementing a reversible jump Markov Chain Monte Carlo sampler. To achieve this aim we rely on the concept of adjacency matrices, which provides a relatively inexpensive check for acyclicity. The performance of our procedure is illustrated by means of two simulated datasets, as well as one real dataset.     

Using the Reversible Jump MCMC Procedure for Identifying and Estimating Univariate TAR Models

One way that has been used for identifying and estimating threshold autoregressive (TAR) models for nonlinear time series follows the Markov chain Monte Carlo (MCMC) approach via the Gibbs sampler. This route has major computational difficulties, specifically, in getting convergence to the parameter distributions. In this article, a new procedure for identifying a TAR model and for estimating its parameters is developed by following the reversible jump MCMC procedure. It is found that the proposed procedure conveys a Markov chain with convergence properties.     

Reversible jump, birth-and-death and more general continuous time Markov chain Monte Carlo samplers

Summary. Reversible jump methods are the most commonly used Markov chain Monte Carlo tool for exploring variable dimension statistical models. Recently, however, an alternative approach based on birth-and-death processes has been proposed by Stephens for mixtures of distributions. We show that the birth-and-death setting can be generalized to include other types of continuous time jumps like split-and-combine moves in the spirit of Richardson and Green. We illustrate these extensions both for mixtures of distributions and for hidden Markov models. We demonstrate the strong similarity of reversible jump and continuous time methodologies by showing that, on appropriate rescaling of time, the reversible jump chain converges to a limiting continuous time birth-and-death process. A numerical comparison in the setting of mixtures of distributions highlights this similarity.     

Learning a multivariate Gaussian mixture model with the reversible jump MCMC algorithm

This paper is a contribution to the methodology of fully Bayesian inference in a multivariate Gaussian mixture model using the reversible jump Markov chain Monte Carlo algorithm. To follow the constraints of preserving the first two moments before and after the split or combine moves, we concentrate on a simplified multivariate Gaussian mixture model, in which the covariance matrices of all components share a common eigenvector matrix. We then propose an approach to the construction of the reversible jump Markov chain Monte Carlo algorithm for this model. Experimental results on several data sets demonstrate the efficacy of our algorithm.     

Bayesian automatic polynomial wavelet regression

This paper considers the problem of Bayesian automatic polynomial wavelet regression (PWR). We propose three different Bayesian methods based on integrated likelihood, conditional empirical Bayes, and reversible jump Markov chain Monte Carlo (MCMC). From the simulation results, we find that the proposed methods are similar to or superior to the existing ones.     

Markov chain Monte Carlo algorithms with sequential proposals

We explore a general framework in Markov chain Monte Carlo (MCMC) sampling where sequential proposals are tried as a candidate for the next state of the Markov chain. This sequential-proposal framework can be applied to various existing MCMC methods, including Metropolis–Hastings algorithms using random proposals and methods that use deterministic proposals such as Hamiltonian Monte Carlo (HMC) or the bouncy particle sampler. Sequential-proposal MCMC methods construct the same Markov chains as those constructed by the delayed rejection method under certain circumstances. In the context of HMC, the sequential-proposal approach has been proposed as extra chance generalized hybrid Monte Carlo (XCGHMC). We develop two novel methods in which the trajectories leading to proposals in HMC are automatically tuned to avoid doubling back, as in the No-U-Turn sampler (NUTS). The numerical efficiency of these new methods compare favorably to the NUTS. We additionally show that the sequential-proposal bouncy particle sampler enables the constructed Markov chain to pass through regions of low target density and thus facilitates better mixing of the chain when the target density is multimodal.

     

Assessing Poisson variation of intestinal tumour multiplicity in mice carrying a Robertsonian translocation

Summary.  Tumour multiplicity is a frequently measured phenotype in animal studies of cancer biology. Poisson variation of this measurement represents a biological and statistical reference point that is usually violated, even in highly controlled experiments, owing to sources of variation in the stochastic process of tumour formation. A recent experiment on murine intestinal tumours presented conditions which seem to generate Poisson-distributed tumour counts. If valid, this would support a claim about mechanisms by which the adenomatous polyposis coli gene is inactivated during tumour initiation. In considering hypothesis testing strategies, model choice and Bayesian approaches, we quantify the positive evidence favouring Poisson variation in this experiment. Statistical techniques used include likelihood ratio testing, the Bayes and Akaike information criteria, negative binomial modelling, reversible jump Markov chain Monte Carlo methods and posterior predictive checking. The posterior approximation that is based on the Bayes information criterion is found to be quite accurate in this small n case-study.     

Estimating models based on Markov jump processes given fragmented observation series

We consider the problem of estimating the rate matrix governing a finite-state Markov jump process given a number of fragmented time series. We propose to concatenate the observed series and to employ the emerging non-Markov process for estimation. We describe the bias arising if standard methods for Markov processes are used for the concatenated process, and provide a post-processing method to correct for this bias. This method applies to discrete-time Markov chains and to more general models based on Markov jump processes where the underlying state process is not observed directly. This is demonstrated in detail for a Markov switching model. We provide applications to simulated time series and to financial market data, where estimators resulting from maximum likelihood methods and Markov chain Monte Carlo sampling are improved using the presented correction.     

Bayesian multivariate Poisson mixtures with an unknown number of components

In this paper we present Bayesian analysis of finite mixtures of multivariate Poisson distributions with an unknown number of components. The multivariate Poisson distribution can be regarded as the discrete counterpart of the multivariate normal distribution, which is suitable for modelling multivariate count data. Mixtures of multivariate Poisson distributions allow for overdispersion and for negative correlations between variables. To perform Bayesian analysis of these models we adopt a reversible jump Markov chain Monte Carlo (MCMC) algorithm with birth and death moves for updating the number of components. We present results obtained from applying our modelling approach to simulated and real data. Furthermore, we apply our approach to a problem in multivariate disease mapping, namely joint modelling of diseases with correlated counts.     

Statistical inference for discretely observed Markov jump processes

Summary.  Likelihood inference for discretely observed Markov jump processes with finite state space is investigated. The existence and uniqueness of the maximum likelihood estimator of the intensity matrix are investigated. This topic is closely related to the imbedding problem for Markov chains. It is demonstrated that the maximum likelihood estimator can be found either by the EM algorithm or by a Markov chain Monte Carlo procedure. When the maximum likelihood estimator does not exist, an estimator can be obtained by using a penalized likelihood function or by the Markov chain Monte Carlo procedure with a suitable prior. The methodology and its implementation are illustrated by examples and simulation studies.     

Bayesian change points analysis on the seismic activity in northeastern Taiwan

Bayesian change points analysis on the seismic activity in northeastern Taiwan is studied via the reversible jump Markov chain Monte Carlo simulation. An epidemic model is considered with Gamma prior distributions for the parameters. The prior distributions are essentially determined based on an earlier period of the seismic data in the same region. It is investigated that there exist two change points during the time period considered. This result is also confirmed by the BIC criteria.     

Bayesian inference for stable Lévy–driven stochastic differential equations with high‐frequency data

In this paper, we consider parametric Bayesian inference for stochastic differential equations driven by a pure‐jump stable Lévy process, which is observed at high frequency. In most cases of practical interest, the likelihood function is not available; hence, we use a quasi‐likelihood and place an associated prior on the unknown parameters. It is shown under regularity conditions that there is a Bernstein–von Mises theorem associated to the posterior. We then develop a Markov chain Monte Carlo algorithm for Bayesian inference, and assisted with theoretical results, we show how to scale Metropolis–Hastings proposals when the frequency of the data grows, in order to prevent the acceptance ratio from going to zero in the large data limit. Our algorithm is presented on numerical examples that help verify our theoretical findings.     

Bayesian estimation in Kibble's bivariate gamma distribution

The authors describe Bayesian estimation for the parameters of the bivariate gamma distribution due to Kibble (1941). The density of this distribution can be written as a mixture, which allows for a simple data augmentation scheme. The authors propose a Markov chain Monte Carlo algorithm to facilitate estimation. They show that the resulting chain is geometrically ergodic, and thus a regenerative sampling procedure is applicable, which allows for estimation of the standard errors of the ergodic means. They develop Bayesian hypothesis testing procedures to test both the dependence hypothesis of the two variables and the hypothesis of equal means. They also propose a reversible jump Markov chain Monte Carlo algorithm to carry out the model selection problem. Finally, they use sets of real and simulated data to illustrate their methodology.     

Interacting multiple try algorithms with different proposal distributions

We introduce a new class of interacting Markov chain Monte Carlo (MCMC) algorithms which is designed to increase the efficiency of a modified multiple-try Metropolis (MTM) sampler. The extension with respect to the existing MCMC literature is twofold. First, the sampler proposed extends the basic MTM algorithm by allowing for different proposal distributions in the multiple-try generation step. Second, we exploit the different proposal distributions to naturally introduce an interacting MTM mechanism (IMTM) that expands the class of population Monte Carlo methods and builds connections with the rapidly expanding world of adaptive MCMC. We show the validity of the algorithm and discuss the choice of the selection weights and of the different proposals. The numerical studies show that the interaction mechanism allows the IMTM to efficiently explore the state space leading to higher efficiency than other competing algorithms.