Computing Normalizing Constants for Finite Mixture Models via Incremental Mixture Importance Sampling (IMIS)

This article proposes a method for approximating integrated likelihoods in finite mixture models. We formulate the model in terms of the unobserved group memberships, z, and make them the variables of integration. The integral is then evaluated using importance sampling over the z. We propose an adaptive importance sampling function which is itself a mixture, with two types of component distributions, one concentrated and one diffuse. The more concentrated type of component serves the usual purpose of an importance sampling function, sampling mostly group assignments of high posterior probability. The less concentrated type of component allows for the importance sampling function to explore the space in a controlled way to find other, unvisited assignments with high posterior probability. Components are added adaptively, one at a time, to cover areas of high posterior probability not well covered by the current importance sampling function. The method is called incremental mixture importance sampling (IMIS).

IMIS is easy to implement and to monitor for convergence. It scales easily for higher dimensional mixture distributions when a conjugate prior is specified for the mixture parameters. The simulated values on which the estimate is based are independent, which allows for straightforward estimation of standard errors. The self-monitoring aspects of the method make it easier to adjust tuning parameters in the course of estimation than standard Markov chain Monte Carlo algorithms. With only small modifications to the code, one can use the method for a wide variety of mixture distributions of different dimensions. The method performed well in simulations and in mixture problems in astronomy and medical research.     

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.     

Hybrid schemes for exact conditional inference in discrete exponential families

Exact conditional goodness-of-fit tests for discrete exponential family models can be conducted via Monte Carlo estimation of p values by sampling from the conditional distribution of multiway contingency tables. The two most popular methods for such sampling are Markov chain Monte Carlo (MCMC) and sequential importance sampling (SIS). In this work we consider various ways to hybridize the two schemes and propose one standout strategy as a good general purpose method for conducting inference. The proposed method runs many parallel chains initialized at SIS samples across the fiber. When a Markov basis is unavailable, the proposed scheme uses a lattice basis with intermittent SIS proposals to guarantee irreducibility and asymptotic unbiasedness. The scheme alleviates many of the challenges faced by the MCMC and SIS schemes individually while largely retaining their strengths. It also provides diagnostics that guide and lend credibility to the procedure. Simulations demonstrate the viability of the approach.     

Probabilities for Queueing Systems with Embedded Markov Chains

Abstract

Transition probabilities of embedded Markov chain for single-server queues are considered when the distribution of the inter-arrival time or that of the service time is specified. A comprehensive collection of formulas is derived for the transition probabilities, covering some seventeen flexible families. The corresponding estimation procedures are also derived by the method of moments. It is expected that this work could serve as a useful reference for the modeling of queuing systems with embedded Markov chains.     

Regenerative simulation of TES processes

Regenerative simulation has become a familiar and established tool for simulation-based estimation. However, many applications (e.g., traffic in high-speed communications networks) call for autocorrelated stochastic models to which traditional regenerative theory is not directly applicable. Consequently, extensions of regenerative simulation to dependent time series is increasingly gaining in theoretical and practical interest, with Markov chains constituting an important case. Fortunately, a regenerative structure can be identified in Harris-recurrent Markov chains with minor modification, and this structure can be exploited for standard regenerative estimation. In this paper we focus on a versatile class of Harris-recurrent Markov chains, called TES (Transform-Expand-Sample). TES processes can generate a variety of sample paths with arbitrary marginal distributions, and autocorrelation functions with a variety of functional forms (monotone, oscillating and alternating). A practical advantage of TES processes is that they can simultaneously capture the first and second order statistics of empirical sample paths (raw field measurements). Specifically, the TES modeling methodology can simultaneously match the empirical marginal distribution (histogram), as well as approximate the empirical autocorrelation function. We explicitly identify regenerative structures in TES processes and proceed to address efficiency and accuracy issues of prospective simulations. To show the efficacy of our approach, we report on a TES/M/1 case study. In this study, we used the likelihood ratio method to calculate the mean waiting time performance as a function of the regenerative structure and the intrinsic TES parameter controlling burstiness (degree of autocorrelation) in the arrival process. The score function method was used to estimate the corresponding sensitivity (gradient) with respect to the service rate. Finally, we demonstrated the importance of the particular regenerative structure selected in regard to the estimation efficiency and accuracy induced by the regeneration cycle length.     

Shuffling algorithm for boxed plane partitions

We introduce discrete time Markov chains that preserve uniform measures on boxed plane partitions. Elementary Markov steps change the size of the box from a×b×c to (a−1)×(b+1)×c or (a+1)×(b−1)×c. Algorithmic realization of each step involves O((a+b)c) operations. One application is an efficient perfect random sampling algorithm for uniformly distributed boxed plane partitions.Trajectories of our Markov chains can be viewed as random point configurations in the three-dimensional lattice. We compute the bulk limits of the correlation functions of the resulting random point process on suitable two-dimensional sections. The limiting correlation functions define a two-dimensional determinantal point processes with certain Gibbs properties.     

Schur dynamics of the Schur processes

We construct discrete time Markov chains that preserve the class of Schur processes on partitions and signatures.One application is a simple exact sampling algorithm for qvolume-distributed skew plane partitions with an arbitrary back wall. Another application is a construction of Markov chains on infinite Gelfand–Tsetlin schemes that represent deterministic flows on the space of extreme characters of the infinite-dimensional unitary group.     

An Efficient Algorithm for Rare-event Probability Estimation,Combinatorial Optimization,and Counting

Although importance sampling is an established and effective sampling and estimation technique, it becomes unstable and unreliable for high-dimensional problems. The main reason is that the likelihood ratio in the importance sampling estimator degenerates when the dimension of the problem becomes large. Various remedies to this problem have been suggested, including heuristics such as resampling. Even so, the consensus is that for large-dimensional problems, likelihood ratios (and hence importance sampling) should be avoided. In this paper we introduce a new adaptive simulation approach that does away with likelihood ratios, while retaining the multi-level approach of the cross-entropy method. Like the latter, the method can be used for rare-event probability estimation, optimization, and counting. Moreover, the method allows one to sample exactly from the target distribution rather than asymptotically as in Markov chain Monte Carlo. Numerical examples demonstrate the effectiveness of the method for a variety of applications.      

Computational Bayesian Analysis of Hidden Markov Models

Abstract

Versions of the Gibbs Sampler are derived for the analysis of data from hidden Markov chains and hidden Markov random fields. The principal new development is to use the pseudolikelihood function associated with the underlying Markov process in place of the likelihood, which is intractable in the case of a Markov random field, in the simulation step for the parameters in the Markov process. Theoretical aspects are discussed and a numerical study is reported.     

Markov Integrated Semigroups and their Applications to Continuous-Time Markov Chains

A Markov integrated semigroup G(t) is by definition a weaklystar differentiable and increasing contraction integrated semigroup on l _∞. We obtain a generation theorem for such semigroups and find that they are not integrated C ₀-semigroups unless the generators are bounded. To link up with the continuous-time Markov chains (CTMCs), we show that there exists a one-to-one relationship between Markov integrated semigroups and transition functions. This gives a clear probability explanation of G(t): it is just the mean transition time, and allows us to define and to investigate its q-matrix. For a given q-matrix Q, we give a criterion for the minimal Q-function to be a Feller-Reuter-Riley (FRR) transition function, this criterion gives an answer to a long-time question raised by Reuter and Riley (1972). This research was supported by the China Postdoctoral Science Foundation (No.2005038326).     

Coupling vs. conductance for the Jerrum–Sinclair chain*

We address the following question: is the causal coupling method as strong as the conductance method in showing rapid mixing of Markov chains? A causal coupling is a coupling which uses only past and present information, but not information about the future. We answer the above question in the negative by showing that there exists a bipartite graph G such that any causal coupling argument on the Jerrum–Sinclair Markov chain for sampling almost uniformly from the set of perfect and near perfect matchings of G must necessarily take time exponential in the number of vertices in G. In contrast, the above Markov chain on G has been shown to mix in polynomial time using conductance arguments. © 2001 John Wiley & Sons, Inc. Random Struct. Alg., 18: 1–17, 2001     

Computational Aspects of Nonparametric Bayesian Analysis with Applications to the Modeling of Multiple Binary Sequences

Abstract

We consider Markov mixture models for multiple longitudinal binary sequences. Prior uncertainty in the mixing distribution is characterized by a Dirichlet process centered on a matrix beta measure. We use this setting to evaluate and compare the performance of three competing algorithms that arise more generally in Dirichlet process mixture calculations: sequential imputations, Gibbs sampling, and a predictive recursion, for which an extension of the sequential calculations is introduced. This facilitates the estimation of quantities related to clustering structure which is not available in the original formulation. A numerical comparison is carried out in three examples. Our findings suggest that the sequential imputations method is most useful for relatively small problems, and that the predictive recursion can be an efficient preliminary tool for more reliable, but computationally intensive, Gibbs sampling implementations.     

The Cross-Entropy Method for Combinatorial and Continuous Optimization

We present a new and fast method, called the cross-entropy method, for finding the optimal solution of combinatorial and continuous nonconvex optimization problems with convex bounded domains. To find the optimal solution we solve a sequence of simple auxiliary smooth optimization problems based on Kullback-Leibler cross-entropy, importance sampling, Markov chain and Boltzmann distribution. We use importance sampling as an important ingredient for adaptive adjustment of the temperature in the Boltzmann distribution and use Kullback-Leibler cross-entropy to find the optimal solution. In fact, we use the mode of a unimodal importance sampling distribution, like the mode of beta distribution, as an estimate of the optimal solution for continuous optimization and Markov chains approach for combinatorial optimization. In the later case we show almost surely convergence of our algorithm to the optimal solution. Supporting numerical results for both continuous and combinatorial optimization problems are given as well. Our empirical studies suggest that the cross-entropy method has polynomial in the size of the problem running time complexity.     

Functional Compatibility,Markov Chains,and Gibbs Sampling with Improper Posteriors

Abstract

The members of a set of conditional probability density functions are called compatible if there exists a joint probability density function that generates them. We generalize this concept by calling the conditionals functionally compatible if there exists a non-negative function that behaves like a joint density as far as generating the conditionals according to the probability calculus, but whose integral over the whole space is not necessarily finite. A necessary and sufficient condition for functional compatibility is given that provides a method of calculating this function, if it exists. A Markov transition function is then constructed using a set of functionally compatible conditional densities and it is shown, using the compatibility results, that the associated Markov chain is positive recurrent if and only if the conditionals are compatible. A Gibbs Markov chain, constructed via “Gibbs conditionals” from a hierarchical model with an improper posterior, is a special case. Therefore, the results of this article can be used to evaluate the consequences of applying the Gibbs sampler when the posterior's impropriety is unknown to the user. Our results cannot, however, be used to detect improper posteriors. Monte Carlo approximations based on Gibbs chains are shown to have undesirable limiting behavior when the posterior is improper. The results are applied to a Bayesian hierarchical one-way random effects model with an improper posterior distribution. The model is simple, but also quite similar to some models with improper posteriors that have been used in conjunction with the Gibbs sampler in the literature.     

Evolving sets, mixing and heat kernel bounds

We show that a new probabilistic technique, recently introduced by the first author, yields the sharpest bounds obtained to date on mixing times of Markov chains in terms of isoperimetric properties of the state space (also known as conductance bounds or Cheeger inequalities). We prove that the bounds for mixing time in total variation obtained by Lovász and Kannan, can be refined to apply to the maximum relative deviation |pⁿ(x,y)/π(y)−1| of the distribution at time n from the stationary distribution π. We then extend our results to Markov chains on infinite state spaces and to continuous-time chains. Our approach yields a direct link between isoperimetric inequalities and heat kernel bounds; previously, this link rested on analytic estimates known as Nash inequalities.Research supported in part by NSF Grants #DMS-0104073 and #DMS-0244479.     

Generalizing Swendsen–Wang for Image Analysis

Markov chain Monte Carlo (MCMC) methods have been used in many fields (physics, chemistry, biology, and computer science) for simulation, inference, and optimization. In many applications, Markov chains are simulated for sampling from target probabilities π(X) defined on graphs G. The graph vertices represent elements of the system, the edges represent spatial relationships, while X is a vector of variables on the vertices which often take discrete values called labels or colors. Designing efficient Markov chains is a challenging task when the variables are strongly coupled. Because of this, methods such as the single-site Gibbs sampler often experience suboptimal performance. A well-celebrated algorithm, the Swendsen–Wang (SW) method, can address the coupling problem. It clusters the vertices as connected components after turning off some edges probabilistically, and changes the color of one cluster as a whole. It is known to mix rapidly under certain conditions. Unfortunately, the SW method has limited applicability and slows down in the presence of “external fields;” for example, likelihoods in Bayesian inference. In this article, we present a general cluster algorithm that extends the SW algorithm to general Bayesian inference on graphs. We focus on image analysis problems where the graph sizes are in the order of 10³–10⁶ with small connectivity. The edge probabilities for clustering are computed using discriminative probabilities from data. We design versions of the algorithm to work on multi grid and multilevel graphs, and present applications to two typical problems in image analysis, namely image segmentation and motion analysis. In our experiments, the algorithm is at least two orders of magnitude faster (in CPU time) than the single-site Gibbs sampler.     

The Kumar-Becker-Lin scheme revisited

The Kumar-Becker-Lin scheme introduces a slowly vanishing cost bias in the parameter estimation part of self-tuning control in order to improve its performance. This paper establishes the a.s. optimality of a variant of this scheme for Markov chains on a countable state space when the action space is compact metric and the parameter space is a compact subset ofR ^m .     

Combinatorial criteria for uniqueness of Gibbs measures

We generalize previously known conditions for uniqueness of the Gibbs measure in statistical physics models by presenting conditions of any finite size for models on any underlying graph. We give two dual conditions, one requiring that the total influence on a site is small, and the other that the total influence of a site is small. Our proofs are combinatorial in nature and use tools from the analysis of discrete Markov chains, in particular the path coupling method. The implications of our conditions for the mixing time of natural Markov chains associated with the models are discussed as well. We also present some examples of models for which the conditions hold. © 2005 Wiley Periodicals, Inc. Random Struct. Alg., 2005     

Nash inequalities for finite Markov chains

This paper develops bounds on the rate of decay of powers of Markov kernels on finite state spaces. These are combined with eigenvalue estimates to give good bounds on the rate of convergence to stationarity for finite Markov chains whose underlying graph has moderate volume growth. Roughly, for such chains, order (diameter) steps are necessary and suffice to reach stationarity. We consider local Poincaré inequalities and use them to prove Nash inequalities. These are bounds onl ₂-norms in terms of Dirichlet forms andl ₁-norms which yield decay rates for iterates of the kernel. This method is adapted from arguments developed by a number of authors in the context of partial differential equations and, later, in the study of random walks on infinite graphs. The main results do not require reversibility.     

Infinitely dimensional control Markov branching chains in random environments

First of all we introduce the concepts of infinitely dimensional control Markov branching chains in random environments (β-MBCRE) and prove the existence of such chains, then we introduce the concepts of conditional generating functionals and random Markov transition functions of such chains and investigate their branching property. Base on these concepts we calculate the moments of the β-MBCRE and obtain the main results of this paper such as extinction probabilities, polarization and proliferation rate. Finally we discuss the classification of β-MBCRE according to the different standards.