Reversible Jump Markov Chain Monte Carlo Algorithm Research Articles

Bayesian reconstruction of projection reconstruction NMR (PR-NMR)

Projection reconstruction nuclear magnetic resonance (PR-NMR) is a technique for generating multidimensional NMR spectra. A small number of projections from lower-dimensional NMR spectra are used to reconstruct the multidimensional NMR spectra. In our previous work [1,2], it was shown that multidimensional NMR spectra are efficiently reconstructed using peak-by-peak based reversible jump Markov chain Monte Carlo (RJMCMC) algorithm. We propose an extended and generalized RJMCMC algorithm replacing a simple linear model with a linear mixed model to reconstruct close NMR spectra into true spectra. This statistical method generates samples in a Bayesian scheme. Our proposed algorithm is tested on a set of six projections derived from the three-dimensional 700MHz HNCO spectrum of a protein HasA.

Reversible jump MCMC for inference in a deterministic individual–based model of tree growth for studying forest dynamics

Scientists use deterministic models to study and forecast the behavior of complex environmental processes, with increasing emphasis on incorporating data to inform model input parameters and accounting for parameter uncertainty. We work with a deterministic, individual‐based model (IBM) of tree growth and mortality, which is under development to explore forest dynamics. Some values of IBM input parameters cause premature virtual tree mortality relative to the actual mortality status of an observed tree. This discordance in mortality causes dimension changes in the state of a stochastic implementation of IBM outputs and leads us to address trans‐dimensional moves among states with a novel formulation of reversible jump Markov chain Monte Carlo (RJMCMC). In particular, we present an RJMCMC algorithm that uses a continuously supported, multidimensional index—the IBM input parameter—instead of a discrete index typical of model determination applications. We use both synthetic data and data from the Forest Inventory and Analysis database representing two tree species. We compare results for each dataset and species between our reversible jump (RJ) specification and an alternative, non‐RJ specification. The RJ formulation compares favorably to the non‐RJ formulation with regard to achieving convergence and yielding biologically realistic IBM input parameter estimates. Copyright © 2013 John Wiley & Sons, Ltd.

A Smooth Transition Model for Multiple-Regime Time Series

This study deals with the problem of fitting a time series modeled by a smooth transition regression function. This model extends the standard linear piecewise model. Within the piecewise model, the regression function parameters change abruptly at the changepoints. In the smooth transition model, parametric transition functions are introduced that allow for gradual changes of the regression function around the changepoints. This model can very accurately reproduce both nonregular and smooth random processes. Moreover it allows the extraction of information about the changes in the regression function through the transition function parameters. The estimation of the model is performed through a fully Bayesian framework. Prior distributions are set for each parameter and the full joint posterior distribution is expressed. The computation of standard Bayesian estimates involves intractable multi-dimensional integrals. Therefore, a reversible-jump Markov chain Monte-Carlo algorithm is derived to sample the joint posterior distributions. A comparative simulation study shows that the smooth transition approach achieves competitive performances and provides more sparse representations of standard test functions. Finally, the smooth transition framework is applied to estimate real world electrical transients, which allows the extraction of the relevant features for signal classification.

Random frog: An efficient reversible jump Markov Chain Monte Carlo-like approach for variable selection with applications to gene selection and disease classification

The identification of disease-relevant genes represents a challenge in microarray-based disease diagnosis where the sample size is often limited. Among established methods, reversible jump Markov Chain Monte Carlo (RJMCMC) methods have proven to be quite promising for variable selection. However, the design and application of an RJMCMC algorithm requires, for example, special criteria for prior distributions. Also, the simulation from joint posterior distributions of models is computationally extensive, and may even be mathematically intractable. These disadvantages may limit the applications of RJMCMC algorithms. Therefore, the development of algorithms that possess the advantages of RJMCMC methods and are also efficient and easy to follow for selecting disease-associated genes is required. Here we report a RJMCMC-like method, called random frog that possesses the advantages of RJMCMC methods and is much easier to implement. Using the colon and the estrogen gene expression datasets, we show that random frog is effective in identifying discriminating genes. The top 2 ranked genes for colon and estrogen are Z50753, U00968, and Y10871_at, Z22536_at, respectively. (The source codes with GNU General Public License Version 2.0 are freely available to non-commercial users at: http://code.google.com/p/randomfrog/.).

Free Knot Splines with RJMCMC in Survival Data Analysis

The B-spline representation is a common tool to improve the fitting of smooth nonlinear functions, it offers a fitting as a piecewise polynomial. The regions that define the pieces are separated by a sequence of knots. The main difficulty in this type of modeling is the choice of the number and the locations of these knots. The Reversible Jump Markov Chain Monte Carlo (RJMCMC) algorithm provides a solution to simultaneously select these two parameters by considering the knots as free parameters. This algorithm belongs to the MCMC techniques that allow simulations from target distributions on spaces of varying dimension. The aim of the present investigation is to use this algorithm in the framework of the analysis of survival time, for the Cox model in particular. In fact, the relation between the hazard ratio function and the covariates being assumed to be log-linear, this assumption is too restrictive. Thus, we propose to use the RJMCMC algorithm to model the log hazard ratio function by a B-spline representation with an unknown number of knots at unknown locations. This method is illustrated with two real data sets: the Stanford heart transplant data and lung cancer survival data. Another application of the RJMCMC is selecting the significant covariates, and a simulation study is performed.

Individual Differences Multidimensional Bradley-Terry Model Using Reversible Jump Markov Chain Monte Carlo Algorithm

The Bradley-Terry model has been widely and effectively used to rank stimuli from paired comparison data. Existing approaches for paired comparison data analysis, however, have a number of limitations. First, among applied Bradley-Terry models in which multidimensionality is assumed, the effects of individual differences are not considered. Second, in these multidimensional Bradley-Terry models, the number of dimensions is generally evaluated only after analyzing several models with different dimensions separately, thus causing computational inconvenience. In this study, a multidimensional Bradley-Terry model that considers the effects of individual differences is proposed. The proposed model allows estimation of parameters for both scale values of stimuli and individual differences in multidimensional space. A procedure of parameter estimation is presented that uses a reversible jump Markov chain Monte Carlo (RJMCMC) algorithm to estimate the optimal number of dimensions as well as associated parameters simultaneously. Simulation studies for examining the utility of the RJMCMC algorithm are performed, and real sports data from sumo are used in a representative example, where the time periods of tournaments are used as parameters for individual differences, in order to verify the validity of the proposed model.

Joint Model Selection and Parameter Estimation by Population Monte Carlo Simulation

In this paper, we study the problem of joint model selection and parameter estimation under the Bayesian framework. We propose to use the Population Monte Carlo (PMC) methodology in carrying out Bayesian computations. The PMC methodology has recently been proposed as an efficient sampling technique and an alternative to Markov Chain Monte Carlo (MCMC) sampling. Its flexibility in constructing transition kernels allows for joint sampling of parameter spaces that belong to different models. The proposed method is able to estimate the desired a posteriori distributions accurately. In comparison to the Reversible Jump MCMC (RJMCMC) algorithm, which is popular in solving the same problem, the PMC algorithm does not require burn-in period, it produces approximately uncorrelated samples, and it can be implemented in a parallel fashion. We demonstrate our approach on two examples: sinusoids in white Gaussian noise and direction of arrival (DOA) estimation in colored Gaussian noise, where in both cases the number of signals in the data is a priori unknown. Both simulations show the effectiveness of our proposed algorithm.

Bayesian Changepoint Analysis for Extreme Events (Typhoons, Heavy Rainfall, and Heat Waves): An RJMCMC Approach

Abstract A hierarchical Bayesian framework is developed to identify multiple abrupt regime shifts in an extreme event series. Specifically, extreme events are modeled as a Poisson process with a gamma-distributed rate. Multiple candidate hypotheses are considered, under each of which there presumably exist a certain number of abrupt shifts of the rate. A Bayesian network involving three layers—data, parameter, and hypothesis—is formulated. A reversible jump Markov chain Monte Carlo (RJMCMC) algorithm is developed to calculate posterior probability for each hypothesis as well its associated within-hypothesis parameters. Based on the proposed RJMCMC algorithm, a simulated example is designed to illustrate the effectiveness of the method. Subsequently, the algorithm is applied to three real, rare event time series: the annual typhoon counts over the western North Pacific (WNP), the annual extreme heavy rainfall event counts at the Honolulu airport, and the annual heat wave frequency in the Chicago area. Results indicate that the typhoon activity over the WNP is very likely to have undergone a decadal variation, with two change points occurring around 1972 and 1989 characterized by the active 1960–71 epoch, the inactive 1972–88 epoch, and the moderately active 1989–2006 epoch. For the extreme rainfall case, only one shift around 1970 is found and heavy rainfall frequency has remained stationary since then. There is no evidence that the rate of the annual heat wave counts in the Chicago area has had any abrupt change during the past 50 years.

A new R package for Bayesian estimation of multivariate normal mixtures allowing for selection of the number of components and interval-censored data

An R package mixAK is introduced which implements routines for a semiparametric density estimation through normal mixtures using the Markov chain Monte Carlo (MCMC) methodology. Besides producing the MCMC output, the package computes posterior summary statistics for important characteristics of the fitted distribution or computes and visualizes the posterior predictive density. For the estimated models, penalized expected deviance (PED) and deviance information criterion (DIC) is directly computed which allows for a selection of mixture components. Additionally, multivariate right-, left- and interval-censored observations are allowed. For univariate problems, the reversible jump MCMC algorithm has been implemented and can be used for a joint estimation of the mixture parameters and the number of mixture components. The core MCMC routines have been implemented in C++ and linked to R to ensure a reasonable computational speed. We briefly review the implemented algorithms and illustrate the use of the package on three real examples of different complexity.

Population-Based Reversible Jump Markov Chain Monte Carlo Methods For Bayesian Variable Selection and Evaluation Under Cost Limit Restrictions

Summary The measurement and improvement of the quality of health care are important areas of current research and development. A judgement of appropriateness of medical outcomes in hospital quality-of-care studies must depend on an assessment of patient sickness at admission to hospital. Indicators of patient sickness often must be abstracted from medical records, and some variables are more expensive to measure than others. Quality-of-care studies are frequently undertaken in an environment of cost restriction; thus any scale measuring patient sickness must simultaneously respect two optimality criteria: high predictive accuracy and low cost. Here we examine a variable selection strategy for construction of a scale of sickness in which predictive accuracy is optimized subject to a bound on cost. Conventional model search algorithms (such as those based on standard reversible jump Markov chain Monte Carlo (RJMCMC) sampling) in our setting will often fail, because of the existence of multiple modes of the criterion function with movement paths that are forbidden because of the cost restriction. We develop a population-based trans-dimensional RJMCMC (population RJMCMC) algorithm, in which ideas from the population-based MCMC and simulated tempering algorithms are combined. Comparing our method with standard RJMCMC sampling, we find that the population-based RJMCMC algorithm moves successfully and more efficiently between distant neighbourhoods of ‘good’ models, achieves convergence faster and has smaller Monte Carlo standard errors for a given amount of central processor unit time. In a case-study of n = 2532 pneumonia patients on whom p = 83 sickness indicators were measured, with marginal costs varying from smallest to largest across the predictor variables by a factor of 20, the final model chosen by population RJMCMC sampling, on the basis of both highest posterior probability and specifying the median probability model, was clinically sensible for pneumonia patients and achieved good predictive ability while capping data collection costs.

Reversible jump and the label switching problem in hidden Markov models

Reversible jump Markov chain Monte Carlo (RJMCMC) algorithms can be efficiently applied in Bayesian inference for hidden Markov models (HMMs), when the number of latent regimes is unknown. As for finite mixture models, when priors are invariant to the relabelling of the regimes, HMMs are unidentifiable in data fitting, because multiple ways to label the regimes can alternate during the MCMC iterations; this is the so-called label switching problem. HMMs with an unknown number of regimes are considered here and the goal of this paper is the comparison, both applied and theoretical, of five methods used for tackling label switching within a RJMCMC algorithm; they are: post-processing, partial reordering, permutation sampling, sampling from a Markov prior and rejection sampling. The five strategies we compare have been proposed mostly in the literature of finite mixture models and only two of them, i.e. rejection sampling and partial reordering, have been presented in RJMCMC algorithms for HMMs. We consider RJMCMC algorithms in which the parameters are updated by Gibbs sampling and the dimension of the model changes in split-and-merge and birth-and-death moves. Finally, an example illustrates and compares the five different methodologies.

Bayesian Inference of Genetic Regulatory Networks from Time Series Microarray Data Using Dynamic Bayesian Networks

Reverse engineering of genetic regulatory networks from time series microarray data are investigated. We propose a dynamic Bayesian networks (DBNs) modeling and a full Bayesian learning scheme. The proposed DBN directly models the continuous expression levels and also is associated with parameters that indicate the degree as well as the type of regulations. To learn the network from data, we proposed a reversible jump Markov chain Monte Carlo (RJMCMC) algorithm. The RJMCMC algorithm can provide not only more accurate inference results than the deterministic alternative algorithms but also an estimate of the a posteriori probabilities (APPs) of the network topology. The estimated APPs provide useful information on the confidence of the inferred results and can also be used for efficient Bayesian data integration. The proposed approach is tested on yeast cell cycle microarray data and the results are compared with the KEGG pathway map.

Reversible Jump Markov Chain Monte Carlo for Deconvolution

To solve the problem of estimating an unknown input function to a linear time invariant system we propose an adaptive non-parametric method based on reversible jump Markov chain Monte Carlo (RJMCMC). We use piecewise polynomial functions (splines) to represent the input function. The RJMCMC algorithm allows the exploration of a large space of competing models, in our case the collection of splines corresponding to alternative positions of breakpoints, and it is based on the specification of transition probabilities between the models. RJMCMC determines: the number and the position of the breakpoints, and the coefficients determining the shape of the spline, as well as the corresponding posterior distribution of breakpoints, number of breakpoints, coefficients and arbitrary statistics of interest associated with the estimation problem. Simulation studies show that the RJMCMC method can obtain accurate reconstructions of complex input functions, and obtains better results compared with standard non-parametric deconvolution methods. Applications to real data are also reported.

Bayesian Model Uncertainty In Smooth Transition Autoregressions

Abstract. In this paper, we propose a fully Bayesian approach to the special class of nonlinear time‐series models called the logistic smooth transition autoregressive (LSTAR) model. Initially, a Gibbs sampler is proposed for the LSTAR where the lag length,k, is kept fixed. Then, uncertainty aboutkis taken into account and a novel reversible jump Markov Chain Monte Carlo (RJMCMC) algorithm is proposed. We compared ourRJMCMC algorithmwith well‐known information criteria, such as the Akaikes̀ information criteria, the Bayesian information criteria (BIC) and the deviance information criteria. Our methodology is extensively studied against simulated and real‐time series.

Point processes for unsupervised line network extraction in remote sensing

This paper addresses the problem of unsupervised extraction of line networks (for example, road or hydrographic networks) from remotely sensed images. We model the target line network by an object process, where the objects correspond to interacting line segments. The prior model, called "Quality Candy," is designed to exploit as fully as possible the topological properties of the network under consideration, while the radiometric properties of the network are modeled using a data term based on statistical tests. Two techniques are used to compute this term: one is more accurate, the other more efficient. A calibration technique is used to choose the model parameters. Optimization is done via simulated annealing using a Reversible Jump Markov Chain Monte Carlo (RJMCMC) algorithm. We accelerate convergence of the algorithm by using appropriate proposal kernels. The results obtained on satellite and aerial images are quantitatively evaluated with respect to manual extractions. A comparison with the results obtained using a previous model, called the "Candy" model, shows the interest of adding quality coefficients with respect to interactions in the prior density. The relevance of using an offline computation of the data potential is shown, in particular, when a proposal kernel based on this computation is added in the RJMCMC algorithm.

A unified Markov chain Monte Carlo framework for mapping multiple quantitative trait loci.

In this article, a unified Markov chain Monte Carlo (MCMC) framework is proposed to identify multiple quantitative trait loci (QTL) for complex traits in experimental designs, based on a composite space representation of the problem that has fixed dimension. The proposed unified approach includes the existing Bayesian QTL mapping methods using reversible jump MCMC algorithm as special cases. We also show that a variety of Bayesian variable selection methods using Gibbs sampling can be applied to the composite model space for mapping multiple QTL. The unified framework not only results in some new algorithms, but also gives useful insight into some of the important factors governing the performance of Gibbs sampling and reversible jump for mapping multiple QTL. Finally, we develop strategies to improve the performance of MCMC algorithms.

An application of reversible-jump Markov chain Monte Carlo to spike classification of multi-unit extracellular recordings

Multi-electrode recordings in neural tissue contain the action potential waveforms of many closely spaced neurons. While we can observe the action potential waveforms, we cannot observe which neuron is the source for which waveform nor how many source neurons are being recorded. Current spike-sorting algorithms solve this problem by assuming a fixed number of source neurons and assigning the action potentials given this fixed number. We model the spike waveforms as an anisotropic Gaussian mixture model and present, as an alternative, a reversible-jump Markov chain Monte Carlo (MCMC) algorithm to simultaneously estimate the number of source neurons and to assign each action potential to a source. We derive this MCMC algorithm and illustrate its application using simulated three-dimensional data and real four-dimensional feature vectors extracted from tetrode recordings of rat entorhinal cortex neurons. In the analysis of the simulated data our algorithm finds the correct number of mixture components (sources) and classifies the action potential waveforms with minimal error. In the analysis of real data, our algorithm identifies clusters closely resembling those previously identified by a user-dependent graphical clustering procedure. Our findings suggest that a reversible-jump MCMC algorithm could offer a new strategy for designing automated spike-sorting algorithms.

Classification with Bayesian MARS

We present a new method for classification using a Bayesian version of the Multivariate Adaptive Regression Spline (MARS) model of J.H. Friedman (Annals of Statistics, 19, 1–141, 1991). Special attention is paid to the use of Markov chain Monte Carlo (MCMC) simulation to gain inference under the model. In particular we discuss three important developments in MCMC methodology. First, we describe the reversible jump MCMC algorithm of P.J. Green (Biometrika, 82, 711–732, 1995) which allows inference on a varying dimensional, possibly uncountable, model space. This allows us to consider MARS models of differing numbers and positions of splines. Secondly, we discuss marginalisation which is used to reduce the effective dimension of the parameter space under consideration. Thirdly, we describe the use of latent variables to improve the MCMC computation. Our methods are generic and can be applied to any basis function model including, wavelets, artificial neural nets and radial basis functions. We present examples to show that the Bayesian MARS classifier is competitive with other approaches on a number of benchmark data sets.

Linkage analysis of quantitative trait loci in multiple line crosses.

Simple line crosses, for example, backcross and F2, are commonly used in mapping quantitative trait loci (QTL). However, these simple crosses are rarely used alone in commercial plant breeding; rather, crosses involving multiple inbred lines or several simple crosses but connected by shared inbred lines may be common in plant breeding. Mapping QTL using crosses of multiple lines is more relevant to plant breeding. Unfortunately, current statistical methods and computer programs of QTL mapping are all designed for simple line crosses or multiple line crosses but under a regular mating system. It is not straightforward to extend the existing methods to handle multiple line crosses under irregular and complicated mating designs. The major hurdle comes from irregular inbreeding, multiple generations, and multiple alleles. In this study, we develop a Bayesian method implemented via the Markov chain Monte Carlo (MCMC) algorithm for mapping QTL using complicated multiple line crosses. With the MCMC algorithm, we are able to draw a complete path of the gene flow from founder alleles to their descendents via a recursive process. This has greatly simplified the problem caused by irregular mating and inbreeding in the mapping population. Adopting the reversible jump MCMC algorithm, we are able to simultaneously search for multiple QTL along the genome. We can even infer the posterior distribution of the number of QTL, one of the most important parameters in QTL study. Application of the new MCMC based QTL mapping procedure is demonstrated using two different mating designs. Design I involves two inbred lines and their derived F1, F2, and BC populations. Design II is a half-diallel cross involving three inbred lines. The two designs appear different, but can be handled with the same robust computer program.

Markov Chain Monte Carlo Convergence Assessment via Two-Way Analysis of Variance

In this article we discuss the problem of assessing the performance of Markov chain Monte Carlo (MCMC) algorithms on the basis of simulation output. In essence, we extend the original ideas of Gelman and Rubin and, more recently, Brooks and Gelman, to problems where we are able to split the variation inherent within the MCMC simulation output into two distinct groups. We show how such a diagnostic may be useful in assessing the performance of MCMC samplers addressing model choice problems, such as the reversible jump MCMC algorithm. In the model choice context, we show how the reversible jump MCMC simulation output for parameters that retain a coherent interpretation throughout the simulation, can be used to assess convergence. By considering various decompositions of the sampling variance of this parameter, we can assess the performance of our MCMC sampler in terms of its mixing properties both within and between models and we illustrate our approach in both the graphical Gaussian models and normal mixtures context. Finally, we provide an example of the application of our diagnostic to the assessment of the influence of different starting values on MCMC simulation output, thereby illustrating the wider utility of our method beyond the Bayesian model choice and reversible jump MCMC context.