Exact Sampling from a Continuous State Space

Propp & Wilson (1996) described a protocol, called coupling from the past, for exact sampling from a target distribution using a coupled Markov chain Monte Carlo algorithm. In this paper we extend coupling from the past to various MCMC samplers on a continuous state space; rather than following the monotone sampling device of Propp & Wilson, our approach uses methods related to gamma-coupling and rejection sampling to simulate the chain, and direct accounting of sample paths.     

On convergence of properly weighted samples to the target distribution

We consider importance sampling as well as other properly weighted samples with respect to a target distribution π$\pi$ from a different point of view. By considering the associated weights as sojourn times until the next jump, we define appropriate jump processes. When the original sample sequence forms an ergodic Markov chain, the associated jump process is an ergodic semi-Markov process with stationary distribution π$\pi$. In this respect, properly weighted samples behave very similarly to standard Markov chain Monte Carlo (MCMC) schemes in that they exhibit convergence to the target distribution as well. Indeed, some standard MCMC procedures like the Metropolis–Hastings algorithm are included in this context. Moreover, when the samples are independent and the mean weight is bounded above, we describe a slight modification in order to achieve exact (weighted) samples from the target distribution.     

Markov chain Monte Carlo methods for parameter estimation of the modified Weibull distribution

In this paper, the Markov chain Monte Carlo (MCMC) method is used to estimate the parameters of a modified Weibull distribution based on a complete sample. While maximum-likelihood estimation (MLE) is the most used method for parameter estimation, MCMC has recently emerged as a good alternative. When applied to parameter estimation, MCMC methods have been shown to be easy to implement computationally, the estimates always exist and are statistically consistent, and their probability intervals are convenient to construct. Details of applying MCMC to parameter estimation for the modified Weibull model are elaborated and a numerical example is presented to illustrate the methods of inference discussed in this paper. To compare MCMC with MLE, a simulation study is provided, and the differences between the estimates obtained by the two algorithms are examined.     

Bayesian inference for circular distributions with unknown normalising constants

Very often, the likelihoods for circular data sets are of quite complicated forms, and the functional forms of the normalising constants, which depend upon the unknown parameters, are unknown. This latter problem generally precludes rigorous, exact inference (both classical and Bayesian) for circular data.Noting the paucity of literature on Bayesian circular data analysis, and also because realistic data analysis is naturally permitted by the Bayesian paradigm, we address the above problem taking a Bayesian perspective. In particular, we propose a methodology that combines importance sampling and Markov chain Monte Carlo (MCMC) in a very effective manner to sample from the posterior distribution of the parameters, given the circular data. With simulation study and real data analysis, we demonstrate the considerable reliability and flexibility of our proposed methodology in analysing circular data.     

Adaptive multiple importance sampling for Gaussian processes

In applications of Gaussian processes (GPs) where quantification of uncertainty is a strict requirement, it is necessary to accurately characterize the posterior distribution over Gaussian process covariance parameters. This is normally done by means of standard Markov chain Monte Carlo (MCMC) algorithms, which require repeated expensive calculations involving the marginal likelihood. Motivated by the desire to avoid the inefficiencies of MCMC algorithms rejecting a considerable amount of expensive proposals, this paper develops an alternative inference framework based on adaptive multiple importance sampling (AMIS). In particular, this paper studies the application of AMIS for GPs in the case of a Gaussian likelihood, and proposes a novel pseudo-marginal-based AMIS algorithm for non-Gaussian likelihoods, where the marginal likelihood is unbiasedly estimated. The results suggest that the proposed framework outperforms MCMC-based inference of covariance parameters in a wide range of scenarios.     

A new strategy for speeding Markov chain Monte Carlo algorithms

Markov chain Monte Carlo (MCMC) methods have become popular as a basis for drawing inference from complex statistical models. Two common difficulties with MCMC algorithms are slow mixing and long run-times, which are frequently closely related. Mixing over the entire state space can often be aided by careful tuning of the chain's transition kernel. In order to preserve the algorithm's stationary distribution, however, care must be taken when updating a chain's transition kernel based on that same chain's history. In this paper we introduce a technique that allows the transition kernel of the Gibbs sampler to be updated at user specified intervals, while preserving the chain's stationary distribution. This technique seems to be beneficial both in increasing efficiency of the resulting estimates (via Rao-Blackwellization) and in reducing the run-time. A reinterpretation of the modified Gibbs sampling scheme introduced in terms of auxiliary samples allows its extension to the more general Metropolis-Hastings framework. The strategies we develop are particularly helpful when calculation of the full conditional (for a Gibbs algorithm) or of the proposal distribution (for a Metropolis-Hastings algorithm) is computationally expensive. Partial financial support from FAR 2002-3, University of Insubria is gratefully acknowledged.     

A geometric approach to transdimensional markov chain monte carlo

The authors present theoretical results that show how one can simulate a mixture distribution whose components live in subspaces of different dimension by reformulating the problem in such a way that observations may be drawn from an auxiliary continuous distribution on the largest subspace and then transformed in an appropriate fashion. Motivated by the importance of enlarging the set of available Markov chain Monte Carlo (MCMC) techniques, the authors show how their results can be fruitfully employed in problems such as model selection (or averaging) of nested models, or regeneration of Markov chains for evaluating standard deviations of estimated expectations derived from MCMC simulations.     

Approximate Bayesian Computation for Exponential Random Graph Models for Large Social Networks

We consider the issue of sampling from the posterior distribution of exponential random graph (ERG) models and other statistical models with intractable normalizing constants. Existing methods based on exact sampling are either infeasible or require very long computing time. We study a class of approximate Markov chain Monte Carlo (MCMC) sampling schemes that deal with this issue. We also develop a new Metropolis–Hastings kernel to sample sparse large networks from ERG models. We illustrate the proposed methods on several examples.     

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.     

Inference in molecular population genetics

Full likelihood-based inference for modern population genetics data presents methodological and computational challenges. The problem is of considerable practical importance and has attracted recent attention, with the development of algorithms based on importance sampling (IS) and Markov chain Monte Carlo (MCMC) sampling. Here we introduce a new IS algorithm. The optimal proposal distribution for these problems can be characterized, and we exploit a detailed analysis of genealogical processes to develop a practicable approximation to it. We compare the new method with existing algorithms on a variety of genetic examples. Our approach substantially outperforms existing IS algorithms, with efficiency typically improved by several orders of magnitude. The new method also compares favourably with existing MCMC methods in some problems, and less favourably in others, suggesting that both IS and MCMC methods have a continuing role to play in this area. We offer insights into the relative advantages of each approach, and we discuss diagnostics in the IS framework.     

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.     

Markov Chain Monte Carlo in small worlds

As the number of applications for Markov Chain Monte Carlo (MCMC) grows, the power of these methods as well as their shortcomings become more apparent. While MCMC yields an almost automatic way to sample a space according to some distribution, its implementations often fall short of this task as they may lead to chains which converge too slowly or get trapped within one mode of a multi-modal space. Moreover, it may be difficult to determine if a chain is only sampling a certain area of the space or if it has indeed reached stationarity. In this paper, we show how a simple modification of the proposal mechanism results in faster convergence of the chain and helps to circumvent the problems described above. This mechanism, which is based on an idea from the field of “small-world” networks, amounts to adding occasional “wild” proposals to any local proposal scheme. We demonstrate through both theory and extensive simulations, that these new proposal distributions can greatly outperform the traditional local proposals when it comes to exploring complex heterogenous spaces and multi-modal distributions. Our method can easily be applied to most, if not all, problems involving MCMC and unlike many other remedies which improve the performance of MCMC it preserves the simplicity of the underlying algorithm.     

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.     

Collapsing of Non‐centred Parameterized MCMC Algorithms with Applications to Epidemic Models

Data augmentation is required for the implementation of many Markov chain Monte Carlo (MCMC) algorithms. The inclusion of augmented data can often lead to conditional distributions from well‐known probability distributions for some of the parameters in the model. In such cases, collapsing (integrating out parameters) has been shown to improve the performance of MCMC algorithms. We show how integrating out the infection rate parameter in epidemic models leads to efficient MCMC algorithms for two very different epidemic scenarios, final outcome data from a multitype SIR epidemic and longitudinal data from a spatial SI epidemic. The resulting MCMC algorithms give fresh insight into real‐life epidemic data sets.     

Maximizing generalized linear mixed model likelihoods with an automated Monte Carlo EM algorithm

Two new implementations of the EM algorithm are proposed for maximum likelihood fitting of generalized linear mixed models. Both methods use random (independent and identically distributed) sampling to construct Monte Carlo approximations at the E-step. One approach involves generating random samples from the exact conditional distribution of the random effects (given the data) by rejection sampling, using the marginal distribution as a candidate. The second method uses a multivariate t importance sampling approximation. In many applications the two methods are complementary. Rejection sampling is more efficient when sample sizes are small, whereas importance sampling is better with larger sample sizes. Monte Carlo approximation using random samples allows the Monte Carlo error at each iteration to be assessed by using standard central limit theory combined with Taylor series methods. Specifically, we construct a sandwich variance estimate for the maximizer at each approximate E-step. This suggests a rule for automatically increasing the Monte Carlo sample size after iterations in which the true EM step is swamped by Monte Carlo error. In contrast, techniques for assessing Monte Carlo error have not been developed for use with alternative implementations of Monte Carlo EM algorithms utilizing Markov chain Monte Carlo E-step approximations. Three different data sets, including the infamous salamander data of McCullagh and Nelder, are used to illustrate the techniques and to compare them with the alternatives. The results show that the methods proposed can be considerably more efficient than those based on Markov chain Monte Carlo algorithms. However, the methods proposed may break down when the intractable integrals in the likelihood function are of high dimension.     

Quantifying uncertainty in transdimensional Markov chain Monte Carlo using discrete Markov models

Bayesian analysis often concerns an evaluation of models with different dimensionality as is necessary in, for example, model selection or mixture models. To facilitate this evaluation, transdimensional Markov chain Monte Carlo (MCMC) relies on sampling a discrete indexing variable to estimate the posterior model probabilities. However, little attention has been paid to the precision of these estimates. If only few switches occur between the models in the transdimensional MCMC output, precision may be low and assessment based on the assumption of independent samples misleading. Here, we propose a new method to estimate the precision based on the observed transition matrix of the model-indexing variable. Assuming a first-order Markov model, the method samples from the posterior of the stationary distribution. This allows assessment of the uncertainty in the estimated posterior model probabilities, model ranks, and Bayes factors. Moreover, the method provides an estimate for the effective sample size of the MCMC output. In two model selection examples, we show that the proposed approach provides a good assessment of the uncertainty associated with the estimated posterior model probabilities.

     

Bayesian weighted inference from surveys

Data from large surveys are often supplemented with sampling weights that are designed to reflect unequal probabilities of response and selection inherent in complex survey sampling methods. We propose two methods for Bayesian estimation of parametric models in a setting where the survey data and the weights are available, but where information on how the weights were constructed is unavailable. The first approach is to simply replace the likelihood with the pseudo likelihood in the formulation of Bayes theorem. This is proven to lead to a consistent estimator but also leads to credible intervals that suffer from systematic undercoverage. Our second approach involves using the weights to generate a representative sample which is integrated into a Markov chain Monte Carlo (MCMC) or other simulation algorithms designed to estimate the parameters of the model. In the extensive simulation studies, the latter methodology is shown to achieve performance comparable to the standard frequentist solution of pseudo maximum likelihood, with the added advantage of being applicable to models that require inference via MCMC. The methodology is demonstrated further by fitting a mixture of gamma densities to a sample of Australian household income.     

Sampling-based Inference of Time Deformation Models with Heavy Tail Distributions

This article focuses on simulation-based inference for the time-deformation models directed by a duration process. In order to better capture the heavy tail property of the time series of financial asset returns, the innovation of the observation equation is subsequently assumed to have a Student-t distribution. Suitable Markov chain Monte Carlo (MCMC) algorithms, which are hybrids of Gibbs and slice samplers, are proposed for estimation of the parameters of these models. In the algorithms, the parameters of the models can be sampled either directly from known distributions or through an efficient slice sampler. The states are simulated one at a time by using a Metropolis-Hastings method, where the proposal distributions are sampled through a slice sampler. Simulation studies conducted in this article suggest that our extended models and accompanying MCMC algorithms work well in terms of parameter estimation and volatility forecast.     

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.     

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.