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.     

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.     

Bayesian clustering for continuous-time hidden Markov models

We develop clustering procedures for longitudinal trajectories based on a continuous-time hidden Markov model (CTHMM) and a generalized linear observation model. Specifically, in this article we carry out finite and infinite mixture model-based clustering for a CTHMM and achieve inference using Markov chain Monte Carlo (MCMC). For a finite mixture model with a prior on the number of components, we implement reversible-jump MCMC to facilitate the trans-dimensional move between models with different numbers of clusters. For a Dirichlet process mixture model, we utilize restricted Gibbs sampling split–merge proposals to improve the performance of the MCMC algorithm. We apply our proposed algorithms to simulated data as well as a real-data example, and the results demonstrate the desired performance of the new sampler.     

Layered adaptive importance sampling

Monte Carlo methods represent the de facto standard for approximating complicated integrals involving multidimensional target distributions. In order to generate random realizations from the target distribution, Monte Carlo techniques use simpler proposal probability densities to draw candidate samples. The performance of any such method is strictly related to the specification of the proposal distribution, such that unfortunate choices easily wreak havoc on the resulting estimators. In this work, we introduce a layered (i.e., hierarchical) procedure to generate samples employed within a Monte Carlo scheme. This approach ensures that an appropriate equivalent proposal density is always obtained automatically (thus eliminating the risk of a catastrophic performance), although at the expense of a moderate increase in the complexity. Furthermore, we provide a general unified importance sampling (IS) framework, where multiple proposal densities are employed and several IS schemes are introduced by applying the so-called deterministic mixture approach. Finally, given these schemes, we also propose a novel class of adaptive importance samplers using a population of proposals, where the adaptation is driven by independent parallel or interacting Markov chain Monte Carlo (MCMC) chains. The resulting algorithms efficiently combine the benefits of both IS and MCMC methods.     

Monte Carlo integration with Markov chain

There are two conceptually distinct tasks in Markov chain Monte Carlo (MCMC): a sampler is designed for simulating a Markov chain and then an estimator is constructed on the Markov chain for computing integrals and expectations. In this article, we aim to address the second task by extending the likelihood approach of Kong et al. for Monte Carlo integration. We consider a general Markov chain scheme and use partial likelihood for estimation. Basically, the Markov chain scheme is treated as a random design and a stratified estimator is defined for the baseline measure. Further, we propose useful techniques including subsampling, regulation, and amplification for achieving overall computational efficiency. Finally, we introduce approximate variance estimators for the point estimators. The method can yield substantially improved accuracy compared with Chib's estimator and the crude Monte Carlo estimator, as illustrated with three examples.     

Bayesian analysis of a linear mixed model with AR(p) errors via MCMC

We develop Bayesian procedures to make inference about parameters of a statistical design with autocorrelated error terms. Modelling treatment effects can be complex in the presence of other factors such as time; for example in longitudinal data. In this paper, Markov chain Monte Carlo methods (MCMC), the Metropolis-Hastings algorithm and Gibbs sampler are used to facilitate the Bayesian analysis of real life data when the error structure can be expressed as an autoregressive model of order p. We illustrate our analysis with real data.     

Stochastic adaptation of importance sampler

Improving efficiency of the importance sampler is at the centre of research on Monte Carlo methods. While the adaptive approach is usually not so straightforward within the Markov chain Monte Carlo framework, the counterpart in importance sampling can be justified and validated easily. We propose an iterative adaptation method for learning the proposal distribution of an importance sampler based on stochastic approximation. The stochastic approximation method can recruit general iterative optimization techniques like the minorization–maximization algorithm. The effectiveness of the approach in optimizing the Kullback divergence between the proposal distribution and the target is demonstrated using several examples.     

A flexible particle Markov chain Monte Carlo method

Particle Markov Chain Monte Carlo methods are used to carry out inference in nonlinear and non-Gaussian state space models, where the posterior density of the states is approximated using particles. Current approaches usually perform Bayesian inference using either a particle marginal Metropolis–Hastings (PMMH) algorithm or a particle Gibbs (PG) sampler. This paper shows how the two ways of generating variables mentioned above can be combined in a flexible manner to give sampling schemes that converge to a desired target distribution. The advantage of our approach is that the sampling scheme can be tailored to obtain good results for different applications. For example, when some parameters and the states are highly correlated, such parameters can be generated using PMMH, while all other parameters are generated using PG because it is easier to obtain good proposals for the parameters within the PG framework. We derive some convergence properties of our sampling scheme and also investigate its performance empirically by applying it to univariate and multivariate stochastic volatility models and comparing it to other PMCMC methods proposed in the literature.     

Automating and evaluating reversible jump MCMC proposal distributions

The reversible jump Markov chain Monte Carlo (MCMC) sampler (Green in Biometrika 82:711–732, 1995) has become an invaluable device for Bayesian practitioners. However, the primary difficulty with the sampler lies with the efficient construction of transitions between competing models of possibly differing dimensionality and interpretation. We propose the use of a marginal density estimator to construct between-model proposal distributions. This provides both a step towards black-box simulation for reversible jump samplers, and a tool to examine the utility of common between-model mapping strategies. We compare the performance of our approach to well established alternatives in both time series and mixture model examples.     

Following a moving target—Monte Carlo inference for dynamic Bayesian models

Markov chain Monte Carlo (MCMC) sampling is a numerically intensive simulation technique which has greatly improved the practicality of Bayesian inference and prediction. However, MCMC sampling is too slow to be of practical use in problems involving a large number of posterior (target) distributions, as in dynamic modelling and predictive model selection. Alternative simulation techniques for tracking moving target distributions, known as particle filters, which combine importance sampling, importance resampling and MCMC sampling, tend to suffer from a progressive degeneration as the target sequence evolves. We propose a new technique, based on these same simulation methodologies, which does not suffer from this progressive degeneration.     

An automated (Markov chain) Monte Carlo EM algorithm

We present an automated Monte Carlo EM (MCEM) algorithm which efficiently assesses Monte Carlo error in the presence of dependent Monte Carlo, particularly Markov chain Monte Carlo, E-step samples and chooses an appropriate Monte Carlo sample size to minimize this Monte Carlo error with respect to progressive EM step estimates. Monte Carlo error is gauged though an application of the central limit theorem during renewal periods of the MCMC sampler used in the E-step. The resulting normal approximation allows us to construct a rigorous and adaptive rule for updating the Monte Carlo sample size each iteration of the MCEM algorithm. We illustrate our automated routine and compare the performance with competing MCEM algorithms in an analysis of a data set fit by a generalized linear mixed model.     

Bayesian Analysis of Two-Regime Threshold Autoregressive Moving Average Model with Exogenous Inputs

We consider Bayesian analysis of threshold autoregressive moving average model with exogenous inputs (TARMAX). In order to obtain the desired marginal posterior distributions of all parameters including the threshold value of the two-regime TARMAX model, we use two different Markov chain Monte Carlo (MCMC) methods to apply Gibbs sampler with Metropolis-Hastings algorithm. The first one is used to obtain iterative least squares estimates of the parameters. The second one includes two MCMC stages for estimate the desired marginal posterior distributions and the parameters. Simulation experiments and a real data example show support to our approaches.     

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.     

Markov chain importance sampling with applications to rare event probability estimation

We present a versatile Monte Carlo method for estimating multidimensional integrals, with applications to rare-event probability estimation. The method fuses two distinct and popular Monte Carlo simulation methods—Markov chain Monte Carlo and importance sampling—into a single algorithm. We show that for some applied numerical examples the proposed Markov Chain importance sampling algorithm performs better than methods based solely on importance sampling or MCMC.     

On parallelizable Markov chain Monte Carlo algorithms with waste-recycling

Parallelizable Markov chain Monte Carlo (MCMC) generates multiple proposals and parallelizes the evaluations of the likelihood function on different cores at each MCMC iteration. Inspired by Calderhead (Proc Natl Acad Sci 111(49):17408–17413, 2014), we introduce a general ‘waste-recycling’ framework for parallelizable MCMC, under which we show that using weighted samples from waste-recycling is preferable to resampling in terms of both statistical and computational efficiencies. We also provide a simple-to-use criteria, the generalized effective sample size, for evaluating efficiencies of parallelizable MCMC algorithms, which applies to both the waste-recycling and the vanilla versions. A moment estimator of the generalized effective sample size is provided and shown to be reasonably accurate by simulations.     

Parallel tempering for dynamic generalized linear models

ABSTRACT

Markov chain Monte Carlo (MCMC) methods can be used for statistical inference. The methods are time-consuming due to time-vary. To resolve these problems, parallel tempering (PT), as a parallel MCMC method, is tried, for dynamic generalized linear models (DGLMs), as well as the several optimal properties of our proposed method. In PT, two or more samples are drawn at the same time, and samples can exchange information with each other. We also present some simulations of the DGLMs in the case and provide two applications of Poisson-type DGLMs in financial research.     

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.     

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.     

Importance sampling type estimators based on approximate marginal Markov chain Monte Carlo

We consider importance sampling (IS) type weighted estimators based on Markov chain Monte Carlo (MCMC) targeting an approximate marginal of the target distribution. In the context of Bayesian latent variable models, the MCMC typically operates on the hyperparameters, and the subsequent weighting may be based on IS or sequential Monte Carlo (SMC), but allows for multilevel techniques as well. The IS approach provides a natural alternative to delayed acceptance (DA) pseudo-marginal/particle MCMC, and has many advantages over DA, including a straightforward parallelization and additional flexibility in MCMC implementation. We detail minimal conditions which ensure strong consistency of the suggested estimators, and provide central limit theorems with expressions for asymptotic variances. We demonstrate how our method can make use of SMC in the state space models context, using Laplace approximations and time-discretized diffusions. Our experimental results are promising and show that the IS-type approach can provide substantial gains relative to an analogous DA scheme, and is often competitive even without parallelization.     

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.