Geometric ergodicity of the Gibbs sampler for the Poisson change-point model

Poisson change-point models have been widely used for modelling inhomogeneous time-series of count data. There are a number of methods available for estimating the parameters in these models using iterative techniques such as MCMC. Many of these techniques share the common problem that there does not seem to be a definitive way of knowing the number of iterations required to obtain sufficient convergence. In this paper, we show that the Gibbs sampler of the Poisson change-point model is geometrically ergodic. Establishing geometric ergodicity is crucial from a practical point of view as it implies the existence of a Markov chain central limit theorem, which can be used to obtain standard error estimates. We prove that the transition kernel is a trace-class operator, which implies geometric ergodicity of the sampler. We then provide a useful application of the sampler to a model for the quarterly driver fatality counts for the state of Victoria, Australia.     

Approximate Predetermined Convergence Properties of the Gibbs Sampler

This article aims to provide a method for approximately predetermining convergence properties of the Gibbs sampler. This is to be done by first finding an approximate rate of convergence for a normal approximation of the target distribution. The rates of convergence for different implementation strategies of the Gibbs sampler are compared to find the best one. In general, the limiting convergence properties of the Gibbs sampler on a sequence of target distributions (approaching a limit) are not the same as the convergence properties of the Gibbs sampler on the limiting target distribution. Theoretical results are given in this article to justify that under conditions, the convergence properties of the Gibbs sampler can be approximated as well. A number of practical examples are given for illustration.     

A Generalized Markov Sampler

A recent development of the Markov chain Monte Carlo (MCMC) technique is the emergence of MCMC samplers that allow transitions between different models. Such samplers make possible a range of computational tasks involving models, including model selection, model evaluation, model averaging and hypothesis testing. An example of this type of sampler is the reversible jump MCMC sampler, which is a generalization of the Metropolis–Hastings algorithm. Here, we present a new MCMC sampler of this type. The new sampler is a generalization of the Gibbs sampler, but somewhat surprisingly, it also turns out to encompass as particular cases all of the well-known MCMC samplers, including those of Metropolis, Barker, and Hastings. Moreover, the new sampler generalizes the reversible jump MCMC. It therefore appears to be a very general framework for MCMC sampling. This paper describes the new sampler and illustrates its use in three applications in Computational Biology, specifically determination of consensus sequences, phylogenetic inference and delineation of isochores via multiple change-point analysis.     

Statistically Efficient Thinning of a Markov Chain Sampler

It is common to subsample Markov chain output to reduce the storage burden. Geyer shows that discarding k ? 1 out of every k observations will not improve statistical efficiency, as quantified through variance in a given computational budget. That observation is often taken to mean that thinning Markov chain Monte Carlo (MCMC) output cannot improve statistical efficiency. Here, we suppose that it costs one unit of time to advance a Markov chain and then θ > 0 units of time to compute a sampled quantity of interest. For a thinned process, that cost θ is incurred less often, so it can be advanced through more stages. Here, we provide examples to show that thinning will improve statistical efficiency if θ is large and the sample autocorrelations decay slowly enough. If the lag ? ? 1 autocorrelations of a scalar measurement satisfy ρ_? > ρ_{? + 1} > 0, then there is always a θ < ∞ at which thinning becomes more efficient for averages of that scalar. Many sample autocorrelation functions resemble first order AR(1) processes with ρ_? = ρ^|?| for some ? 1 < ρ < 1. For an AR(1) process, it is possible to compute the most efficient subsampling frequency k. The optimal k grows rapidly as ρ increases toward 1. The resulting efficiency gain depends primarily on θ, not ρ. Taking k = 1 (no thinning) is optimal when ρ ? 0. For ρ > 0, it is optimal if and only if θ ? (1 ? ρ)²/(2ρ). This efficiency gain never exceeds 1 + θ. This article also gives efficiency bounds for autocorrelations bounded between those of two AR(1) processes. Supplementary materials for this article are available online.     

Convergence rates of symmetric scan Gibbs sampler

We consider the symmetric scan Gibbs sampler, and give some explicit estimates of convergence rates on the Wasserstein distance for this Markov chain Monte Carlo under the Dobrushin uniqueness condition.     

Inference for the Number of Topics in the Latent Dirichlet Allocation Model via Bayesian Mixture Modeling

In latent Dirichlet allocation, the number of topics, T, is a hyperparameter of the model that must be specified before one can fit the model. The need to specify T in advance is restrictive. One way of dealing with this problem is to put a prior on T, but unfortunately the distribution on the latent variables of the model is then a mixture of distributions on spaces of different dimensions, and estimating this mixture distribution by Markov chain Monte Carlo is very difficult. We present a variant of the Metropolis–Hastings algorithm that can be used to estimate this mixture distribution, and in particular the posterior distribution of the number of topics. We evaluate our methodology on synthetic data and compare it with procedures that are currently used in the machine learning literature. We also give an illustration on two collections of articles from Wikipedia. Supplemental materials for this article are available online.     

A Repelling–Attracting Metropolis Algorithm for Multimodality

Although the Metropolis algorithm is simple to implement, it often has difficulties exploring multimodal distributions. We propose the repelling–attracting Metropolis (RAM) algorithm that maintains the simple-to-implement nature of the Metropolis algorithm, but is more likely to jump between modes. The RAM algorithm is a Metropolis-Hastings algorithm with a proposal that consists of a downhill move in density that aims to make local modes repelling, followed by an uphill move in density that aims to make local modes attracting. The downhill move is achieved via a reciprocal Metropolis ratio so that the algorithm prefers downward movement. The uphill move does the opposite using the standard Metropolis ratio which prefers upward movement. This down-up movement in density increases the probability of a proposed move to a different mode. Because the acceptance probability of the proposal involves a ratio of intractable integrals, we introduce an auxiliary variable which creates a term in the acceptance probability that cancels with the intractable ratio. Using several examples, we demonstrate the potential for the RAM algorithm to explore a multimodal distribution more efficiently than a Metropolis algorithm and with less tuning than is commonly required by tempering-based methods. Supplementary materials are available online.     

Conditioning in Markov Chain Monte Carlo

Abstract

The so-called “Rao-Blackwellized” estimators proposed by Gelfand and Smith do not always reduce variance in Markov chain Monte Carlo when the dependence in the Markov chain is taken into account. An illustrative example is given, and a theorem characterizing the necessary and sufficient condition for such an estimator to always reduce variance is proved.     

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Most regression modeling is based on traditional mean regression which results in non-robust estimation results for non-normal errors. Compared to conventional mean regression, composite quantile regression (CQR) may produce more robust parameters estimation. Based on a composite asymmetric Laplace distribution (CALD), we build a Bayesian hierarchical model for the weighted CQR (WCQR). The Gibbs sampler algorithm of Bayesian WCQR is developed to implement posterior inference. Finally, the proposed method are illustrated by some simulation studies and a real data analysis.     

General Methods for Monitoring Convergence of Iterative Simulations

Abstract

We generalize the method proposed by Gelman and Rubin (1992a) for monitoring the convergence of iterative simulations by comparing between and within variances of multiple chains, in order to obtain a family of tests for convergence. We review methods of inference from simulations in order to develop convergence-monitoring summaries that are relevant for the purposes for which the simulations are used. We recommend applying a battery of tests for mixing based on the comparison of inferences from individual sequences and from the mixture of sequences. Finally, we discuss multivariate analogues, for assessing convergence of several parameters simultaneously.     

Reversible algorithm of simulating multivariate densities with multi-hump

To simulate a multivariate density with multi-hump, Markov chain Monte Carlo method encounters the obstacle of escaping from one hump to another, since it usually takes extraordinately long time and then becomes practically impossible to perform. To overcome these difficulties, a reversible scheme to generate a Markov chain, in terms of which the simulated density may be successful in rather general cases of practically avoiding being trapped in local humps, was suggested.     

Bayesian Analysis of a Two-State Markov Modulated Poisson Process

Abstract

We postulate observations from a Poisson process whose rate parameter modulates between two values determined by an unobserved Markov chain. The theory switches from continuous to discrete time by considering the intervals between observations as a sequence of dependent random variables. A result from hidden Markov models allows us to sample from the posterior distribution of the model parameters given the observed event times using a Gibbs sampler with only two steps per iteration.     

On reparametrization and the Gibbs sampler

Gibbs samplers derived under different parametrizations of the target density can have radically different rates of convergence. In this article, we specify conditions under which reparametrization leaves the convergence rate of a Gibbs chain unchanged. An example illustrates how these results can be exploited in convergence rate analyses.     

A theory for the multiset sampler

The multiset sampler (MSS) can be viewed as a new data augmentation scheme and it has been applied successfully to a wide range of statistical inference problems. The key idea of the MSS is to augment the system with a multiset of the missing components, and construct an appropriate joint distribution of the parameters of interest and the missing components to facilitate the inference based on Markov chain Monte Carlo. The standard data augmentation strategy corresponds to the MSS with multiset size one. This paper provides a theoretical comparison of the MSS with different multiset sizes. We show that the MSS converges to the target distribution faster as the multiset size increases. This explains the improvement in convergence rate for the MSS with large multiset sizes over the standard data augmentation scheme.     

Antithetic Methods for Gibbs Samplers

In this article we propose a modification to the output from Metropolis-within-Gibbs samplers that can lead to substantial reductions in the variance over standard estimates. The idea is simple: at each time step of the algorithm, introduce an extra sample into the estimate that is negatively correlated with the current sample, the rationale being that this provides a two-sample numerical approximation to a Rao–Blackwellized estimate. As the conditional sampling distribution at each step has already been constructed, the generation of the antithetic sample often requires negligible computational effort. Our method is implementable whenever one subvector of the state can be sampled from its full conditional and the corresponding distribution function may be inverted, or the full conditional has a symmetric density. We demonstrate our approach in the context of logistic regression and hierarchical Poisson models. The data and computer code used in this article are available online.     

On solving integral equations using Markov chain Monte Carlo methods

In this paper, we propose an original approach to the solution of Fredholm equations of the second kind. We interpret the standard Von Neumann expansion of the solution as an expectation with respect to a probability distribution defined on a union of subspaces of variable dimension. Based on this representation, it is possible to use trans-dimensional Markov chain Monte Carlo (MCMC) methods such as Reversible Jump MCMC to approximate the solution numerically. This can be an attractive alternative to standard Sequential Importance Sampling (SIS) methods routinely used in this context. To motivate our approach, we sketch an application to value function estimation for a Markov decision process. Two computational examples are also provided.     

A Pivotal Allocation-Based Algorithm for Solving the Label-Switching Problem in Bayesian Mixture Models

In Bayesian analysis of mixture models, the label-switching problem occurs as a result of the posterior distribution being invariant to any permutation of cluster indices under symmetric priors. To solve this problem, we propose a novel relabeling algorithm and its variants by investigating an approximate posterior distribution of the latent allocation variables instead of dealing with the component parameters directly. We demonstrate that our relabeling algorithm can be formulated in a rigorous framework based on information theory. Under some circumstances, it is shown to resemble the classical Kullback-Leibler relabeling algorithm and include the recently proposed equivalence classes representatives relabeling algorithm as a special case. Using simulation studies and real data examples, we illustrate the efficiency of our algorithm in dealing with various label-switching phenomena. Supplemental materials for this article are available online.     

Honest Importance Sampling With Multiple Markov Chains

Importance sampling is a classical Monte Carlo technique in which a random sample from one probability density, π₁, is used to estimate an expectation with respect to another, π. The importance sampling estimator is strongly consistent and, as long as two simple moment conditions are satisfied, it obeys a central limit theorem (CLT). Moreover, there is a simple consistent estimator for the asymptotic variance in the CLT, which makes for routine computation of standard errors. Importance sampling can also be used in the Markov chain Monte Carlo (MCMC) context. Indeed, if the random sample from π₁ is replaced by a Harris ergodic Markov chain with invariant density π₁, then the resulting estimator remains strongly consistent. There is a price to be paid, however, as the computation of standard errors becomes more complicated. First, the two simple moment conditions that guarantee a CLT in the iid case are not enough in the MCMC context. Second, even when a CLT does hold, the asymptotic variance has a complex form and is difficult to estimate consistently. In this article, we explain how to use regenerative simulation to overcome these problems. Actually, we consider a more general setup, where we assume that Markov chain samples from several probability densities, π₁, …, π_k, are available. We construct multiple-chain importance sampling estimators for which we obtain a CLT based on regeneration. We show that if the Markov chains converge to their respective target distributions at a geometric rate, then under moment conditions similar to those required in the iid case, the MCMC-based importance sampling estimator obeys a CLT. Furthermore, because the CLT is based on a regenerative process, there is a simple consistent estimator of the asymptotic variance. We illustrate the method with two applications in Bayesian sensitivity analysis. The first concerns one-way random effect models under different priors. The second involves Bayesian variable selection in linear regression, and for this application, importance sampling based on multiple chains enables an empirical Bayes approach to variable selection.     

Volterra积分方程的蒙特卡罗数值求解方法

本文讨论了第一类、第二类以及具有奇异核的Volterra积分方程的数值解问题.利用重要抽样蒙特卡罗随机模拟方法获得积分方程解的近似计算结果.通过对文献中算例的实现表明文中所提方法扩展了Volterra型积分方程的数值求解方法,     

Langevin-Type Models I: Diffusions with Given Stationary Distributions and their Discretizations*

We describe algorithms for estimating a given measure known up to a constant of proportionality, based on a large class of diffusions (extending the Langevin model) for which is invariant. We show that under weak conditions one can choose from this class in such a way that the diffusions converge at exponential rate to , and one can even ensure that convergence is independent of the starting point of the algorithm. When convergence is less than exponential we show that it is often polynomial at verifiable rates. We then consider methods of discretizing the diffusion in time, and find methods which inherit the convergence rates of the continuous time process. These contrast with the behavior of the naive or Euler discretization, which can behave badly even in simple cases. Our results are described in detail in one dimension only, although extensions to higher dimensions are also briefly described.