On the Flexibility of Metropolis–Hastings Acceptance Probabilities in Auxiliary Variable Proposal Generation

Abstract. Use of auxiliary variables for generating proposal variables within a Metropolis–Hastings setting has been suggested in many different settings. This has in particular been of interest for simulation from complex distributions such as multimodal distributions or in transdimensional approaches. For many of these approaches, the acceptance probabilities that are used turn up somewhat magic and different proofs for their validity have been given in each case. In this article, we will present a general framework for construction of acceptance probabilities in auxiliary variable proposal generation. In addition to showing the similarities between many of the proposed algorithms in the literature, the framework also demonstrates that there is a great flexibility in how to construct acceptance probabilities. With this flexibility, alternative acceptance probabilities are suggested. Some numerical experiments are also reported.     

A simulation approach to convergence rates for Markov chain Monte Carlo algorithms

Markov chain Monte Carlo (MCMC) methods, including the Gibbs sampler and the Metropolis–Hastings algorithm, are very commonly used in Bayesian statistics for sampling from complicated, high-dimensional posterior distributions. A continuing source of uncertainty is how long such a sampler must be run in order to converge approximately to its target stationary distribution. A method has previously been developed to compute rigorous theoretical upper bounds on the number of iterations required to achieve a specified degree of convergence in total variation distance by verifying drift and minorization conditions. We propose the use of auxiliary simulations to estimate the numerical values needed in this theorem. Our simulation method makes it possible to compute quantitative convergence bounds for models for which the requisite analytical computations would be prohibitively difficult or impossible. On the other hand, although our method appears to perform well in our example problems, it cannot provide the guarantees offered by analytical proof.     

Understanding the Hastings Algorithm

The Hastings algorithm is a key tool in computational science. While mathematically justified by detailed balance, it can be conceptually difficult to grasp. Here, we present two complementary and intuitive ways to derive and understand the algorithm. In our framework, it is straightforward to see that the celebrated Metropolis–Hastings algorithm has the highest acceptance probability of all Hastings algorithms.     

A joint modeling approach for spatial earthquake risk variations

Modeling spatial patterns and processes to assess the spatial variations of data over a study region is an important issue in many fields. In this paper, we focus on investigating the spatial variations of earthquake risks after a main shock. Although earthquake risks have been extensively studied in the literatures, to our knowledge, there does not exist a suitable spatial model for assessing the problem. Therefore, we propose a joint modeling approach based on spatial hierarchical Bayesian models and spatial conditional autoregressive models to describe the spatial variations in earthquake risks over the study region during two periods. A family of stochastic algorithms based on a Markov chain Monte Carlo technique is then performed for posterior computations. The probabilistic issue for the changes of earthquake risks after a main shock is also discussed. Finally, the proposed method is applied to the earthquake records for Taiwan before and after the Chi-Chi earthquake.     

Acceleration of the Multiple-Try Metropolis algorithm using antithetic and stratified sampling

The Multiple-Try Metropolis is a recent extension of the Metropolis algorithm in which the next state of the chain is selected among a set of proposals. We propose a modification of the Multiple-Try Metropolis algorithm which allows for the use of correlated proposals, particularly antithetic and stratified proposals. The method is particularly useful for random walk Metropolis in high dimensional spaces and can be used easily when the proposal distribution is Gaussian. We explore the use of quasi Monte Carlo (QMC) methods to generate highly stratified samples. A series of examples is presented to evaluate the potential of the method.     

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.     

Equation-solving estimator based on the general n-step MHDR algorithm

Markov chain Monte Carlo (MCMC) methods provide an important means to simulate from almost any probability density. To approximate non-standard discrete distributions, the equation-solving MCMC estimator was developed as an alternative to the classical frequency estimator. The used simulation scheme is the Metropolis–Hastings (M–H) algorithm. Recently, this estimator has been extended to the specific context of 2-step Metropolis-Hastings with delayed rejection (MHDR) algorithm, which allowed a considerable reduction in asymptotic variance. In this paper, we propose an adaptation of equation-solving estimator to the case of general n-step MHDR sampler. The aim is to further improve the precision. An application to a Bayesian hypothesis test problem shows the high performance, in terms of accuracy, of the equation-solving estimator, based on a MHDR algorithm with more than two stages.     

A Simulator for Twenty20 Cricket

This paper develops a Twenty20 cricket simulator for matches between sides belonging to the International Cricket Council. As input, the simulator requires the probabilities of batting outcomes which are dependent on the batsman, the bowler, the number of overs consumed and the number of wickets lost. The determination of batting probabilities is based on an amalgam of standard classical estimation techniques and a hierarchical empirical Bayes approach where the probabilities of batting outcomes borrow information from related scenarios. Initially, the probabilities of batting outcomes are obtained for the first innings. In the second innings, the target score obtained from the first innings affects the aggressiveness of batting during the second innings. We use the target score to modify batting probabilities in the second innings simulation. This gives rise to the suggestion that teams may not be adjusting their second innings batting aggressiveness in an optimal way. The adequacy of the simulator is addressed through various goodness‐of‐fit diagnostics.     

On the Value of derivative evaluations and random walk suppression in Markov Chain Monte Carlo algorithms

Two strategies that can potentially improve Markov Chain Monte Carlo algorithms are to use derivative evaluations of the target density, and to suppress random walk behaviour in the chain. The use of one or both of these strategies has been investigated in a few specific applications, but neither is used routinely. We undertake a broader evaluation of these techniques, with a view to assessing their utility for routine use. In addition to comparing different algorithms, we also compare two different ways in which the algorithms can be applied to a multivariate target distribution. Specifically, the univariate version of an algorithm can be applied repeatedly to one-dimensional conditional distributions, or the multivariate version can be applied directly to the target distribution.     

Bayesian Inference for a Stochastic Epidemic Model with Uncertain Numbers of Susceptibles of Several Types

A stochastic epidemic model with several kinds of susceptible is used to analyse temporal disease outbreak data from a Bayesian perspective. Prior distributions are used to model uncertainty in the actual numbers of susceptibles initially present. The posterior distribution of the parameters of the model is explored via Markov chain Monte Carlo methods. The methods are illustrated using two datasets, and the results are compared where possible to results obtained by previous analyses.     

Bayesian mixture models for complex high dimensional count data in phage display experiments

Summary.  Phage display is a biological process that is used to screen random peptide libraries for ligands that bind to a target of interest with high affinity. On the basis of a count data set from an innovative multistage phage display experiment, we propose a class of Bayesian mixture models to cluster peptide counts into three groups that exhibit different display patterns across stages. Among the three groups, the investigators are particularly interested in that with an ascending display pattern in the counts, which implies that the peptides are likely to bind to the target with strong affinity. We apply a Bayesian false discovery rate approach to identify the peptides with the strongest affinity within the group. A list of peptides is obtained, among which important ones with meaningful functions are further validated by biologists. To examine the performance of the Bayesian model, we conduct a simulation study and obtain desirable results.     

Multivariate Stochastic Volatility via Wishart Processes: A Comment

This comment refers to an error in the methodology for estimating the parameters of the model developed by Philipov and Glickman for modeling multivariate stochastic volatility via Wishart processes. For estimation they used Bayesian techniques. The derived expressions for the full conditionals of the model parameters as well as the expression for the acceptance ratio of the covariance matrix are erroneous. In this erratum all necessary formulae are given to guarantee an appropriate implementation and application of the model.     

A robust multivariate measurement error model with skew-normal/independent distributions and Bayesian MCMC implementation

Skew-normal/independent distributions are a class of asymmetric thick-tailed distributions that include the skew-normal distribution as a special case. In this paper, we explore the use of Markov Chain Monte Carlo (MCMC) methods to develop a Bayesian analysis in multivariate measurement errors models. We propose the use of skew-normal/independent distributions to model the unobserved value of the covariates (latent variable) and symmetric normal/independent distributions for the random errors term, providing an appealing robust alternative to the usual symmetric process in multivariate measurement errors models. Among the distributions that belong to this class of distributions, we examine univariate and multivariate versions of the skew-normal, skew-t, skew-slash and skew-contaminated normal distributions. The results and methods are applied to a real data set.     

A Bayesian random effects model for analyzing mixed negative binomial and continuous longitudinal responses

ABSTRACT

A general Bayesian random effects model for analyzing longitudinal mixed correlated continuous and negative binomial responses with and without missing data is presented. This Bayesian model, given some random effects, uses a normal distribution for the continuous response and a negative binomial distribution for the count response. A Markov Chain Monte Carlo sampling algorithm is described for estimating the posterior distribution of the parameters. This Bayesian model is illustrated by a simulation study. For sensitivity analysis to investigate the change of parameter estimates with respect to the perturbation from missing at random to not missing at random assumption, the use of posterior curvature is proposed. The model is applied to a medical data, obtained from an observational study on women, where the correlated responses are the negative binomial response of joint damage and continuous response of body mass index. The simultaneous effects of some covariates on both responses are also investigated.     

Bayesian Truncated Poisson Regression with Application to Dutch Illegal Immigrant Data

This article presents a Bayesian approach to the regression analysis of truncated data, with a focus on zero-truncated counts from the Poisson distribution. The approach provides inference not only on the regression coefficients but also on the total sample size and the parameters of the covariate distribution. The theory is applied to some illegal immigrant data from The Netherlands. Several models are fitted with the aid of Markov chain Monte Carlo methods and assessed via posterior predictive p-values. Inferences are compared with those obtained elsewhere using other approaches.     

Generalized bent-cable methodology for changepoint data: a Bayesian approach

The choice of the model framework in a regression setting depends on the nature of the data. The focus of this study is on changepoint data, exhibiting three phases: incoming and outgoing, both of which are linear, joined by a curved transition. Bent-cable regression is an appealing statistical tool to characterize such trajectories, quantifying the nature of the transition between the two linear phases by modeling the transition as a quadratic phase with unknown width. We demonstrate that a quadratic function may not be appropriate to adequately describe many changepoint data. We then propose a generalization of the bent-cable model by relaxing the assumption of the quadratic bend. The properties of the generalized model are discussed and a Bayesian approach for inference is proposed. The generalized model is demonstrated with applications to three data sets taken from environmental science and economics. We also consider a comparison among the quadratic bent-cable, generalized bent-cable and piecewise linear models in terms of goodness of fit in analyzing both real-world and simulated data. This study suggests that the proposed generalization of the bent-cable model can be valuable in adequately describing changepoint data that exhibit either an abrupt or gradual transition over time.     

Comment

The article provides detailed and accurate illustrations of Bayesian analysis of DSGE models that are likely to be used increasingly in support of central bank policy making. These comments identify a dozen aspects of these methods, discussing how their application and improvement can contribute to effective support of policy.     

Assessing convergence diagnostic tests for Bayesian Cox regression

The Markov chain Monte Carlo (MCMC) method generates samples from the posterior distribution and uses these samples to approximate expectations of quantities of interest. For the process, researchers have to decide whether the Markov chain has reached the desired posterior distribution. Using convergence diagnostic tests are very important to decide whether the Markov chain has reached the target distribution. Our interest in this study was to compare the performances of convergence diagnostic tests for all parameters of Bayesian Cox regression model with different number of iterations by using a simulation and a real lung cancer dataset.     

A Markov Chain Monte Carlo version of the genetic algorithm Differential Evolution: easy Bayesian computing for real parameter spaces

Differential Evolution (DE) is a simple genetic algorithm for numerical optimization in real parameter spaces. In a statistical context one would not just want the optimum but also its uncertainty. The uncertainty distribution can be obtained by a Bayesian analysis (after specifying prior and likelihood) using Markov Chain Monte Carlo (MCMC) simulation. This paper integrates the essential ideas of DE and MCMC, resulting in Differential Evolution Markov Chain (DE-MC). DE-MC is a population MCMC algorithm, in which multiple chains are run in parallel. DE-MC solves an important problem in MCMC, namely that of choosing an appropriate scale and orientation for the jumping distribution. In DE-MC the jumps are simply a fixed multiple of the differences of two random parameter vectors that are currently in the population. The selection process of DE-MC works via the usual Metropolis ratio which defines the probability with which a proposal is accepted. In tests with known uncertainty distributions, the efficiency of DE-MC with respect to random walk Metropolis with optimal multivariate Normal jumps ranged from 68% for small population sizes to 100% for large population sizes and even to 500% for the 97.5% point of a variable from a 50-dimensional Student distribution. Two Bayesian examples illustrate the potential of DE-MC in practice. DE-MC is shown to facilitate multidimensional updates in a multi-chain “Metropolis-within-Gibbs” sampling approach. The advantage of DE-MC over conventional MCMC are simplicity, speed of calculation and convergence, even for nearly collinear parameters and multimodal densities.     

Bayesian neural networks for nonlinear time series forecasting

In this article, we apply Bayesian neural networks (BNNs) to time series analysis, and propose a Monte Carlo algorithm for BNN training. In addition, we go a step further in BNN model selection by putting a prior on network connections instead of hidden units as done by other authors. This allows us to treat the selection of hidden units and the selection of input variables uniformly. The BNN model is compared to a number of competitors, such as the Box-Jenkins model, bilinear model, threshold autoregressive model, and traditional neural network model, on a number of popular and challenging data sets. Numerical results show that the BNN model has achieved a consistent improvement over the competitors in forecasting future values. Insights on how to improve the generalization ability of BNNs are revealed in many respects of our implementation, such as the selection of input variables, the specification of prior distributions, and the treatment of outliers.