Model based labeling for mixture models

Label switching is one of the fundamental problems for Bayesian mixture model analysis. Due to the permutation invariance of the mixture posterior, we can consider that the posterior of a m-component mixture model is a mixture distribution with m! symmetric components and therefore the object of labeling is to recover one of the components. In order to do labeling, we propose to first fit a symmetric m!-component mixture model to the Markov chain Monte Carlo (MCMC) samples and then choose the label for each sample by maximizing the corresponding classification probabilities, which are the probabilities of all possible labels for each sample. Both parametric and semi-parametric ways are proposed to fit the symmetric mixture model for the posterior. Compared to the existing labeling methods, our proposed method aims to approximate the posterior directly and provides the labeling probabilities for all possible labels and thus has a model explanation and theoretical support. In addition, we introduce a situation in which the “ideally” labeled samples are available and thus can be used to compare different labeling methods. We demonstrate the success of our new method in dealing with the label switching problem using two examples.     

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.     

An online Bayesian mixture labelling method by minimizing deviance of classification probabilities to reference labels

Solving label switching is crucial for interpreting the results of fitting Bayesian mixture models. The label switching originates from the invariance of posterior distribution to permutation of component labels. As a result, the component labels in Markov chain simulation may switch to another equivalent permutation, and the marginal posterior distribution associated with all labels may be similar and useless for inferring quantities relating to each individual component. In this article, we propose a new simple labelling method by minimizing the deviance of the class probabilities to a fixed reference labels. The reference labels can be chosen before running Markov chain Monte Carlo (MCMC) using optimization methods, such as expectation-maximization algorithms, and therefore the new labelling method can be implemented by an online algorithm, which can reduce the storage requirements and save much computation time. Using the Acid data set and Galaxy data set, we demonstrate the success of the proposed labelling method for removing the labelling switching in the raw MCMC samples.     

Bayesian curve fitting and clustering with Dirichlet process mixture models for microarray data

In the field of molecular biology, it is often of interest to analyze microarray data for clustering genes based on similar profiles of gene expression to identify genes that are differentially expressed under multiple biological conditions. One of the notable characteristics of a gene expression profile is that it shows a cyclic curve over a course of time. To group sequences of similar molecular functions, we propose a Bayesian Dirichlet process mixture of linear regression models with a Fourier series for the regression coefficients, for each of which a spike and slab prior is assumed. A full Gibbs-sampling algorithm is developed for an efficient Markov chain Monte Carlo (MCMC) posterior computation. Due to the so-called “label-switching” problem and different numbers of clusters during the MCMC computation, a post-process approach of Fritsch and Ickstadt (2009) is additionally applied to MCMC samples for an optimal single clustering estimate by maximizing the posterior expected adjusted Rand index with the posterior probabilities of two observations being clustered together. The proposed method is illustrated with two simulated data and one real data of the physiological response of fibroblasts to serum of Iyer et al. (1999).     

Robust mixture modeling using multivariate skew <Emphasis Type="Italic">t</Emphasis> distributions

This paper presents a robust mixture modeling framework using the multivariate skew t distributions, an extension of the multivariate Student’s t family with additional shape parameters to regulate skewness. The proposed model results in a very complicated likelihood. Two variants of Monte Carlo EM algorithms are developed to carry out maximum likelihood estimation of mixture parameters. In addition, we offer a general information-based method for obtaining the asymptotic covariance matrix of maximum likelihood estimates. Some practical issues including the selection of starting values as well as the stopping criterion are also discussed. The proposed methodology is applied to a subset of the Australian Institute of Sport data for illustration.     

Ascent-based Monte Carlo expectation– maximization

Summary.  The expectation–maximization (EM) algorithm is a popular tool for maximizing likelihood functions in the presence of missing data. Unfortunately, EM often requires the evaluation of analytically intractable and high dimensional integrals. The Monte Carlo EM (MCEM) algorithm is the natural extension of EM that employs Monte Carlo methods to estimate the relevant integrals. Typically, a very large Monte Carlo sample size is required to estimate these integrals within an acceptable tolerance when the algorithm is near convergence. Even if this sample size were known at the onset of implementation of MCEM, its use throughout all iterations is wasteful, especially when accurate starting values are not available. We propose a data-driven strategy for controlling Monte Carlo resources in MCEM. The algorithm proposed improves on similar existing methods by recovering EM's ascent (i.e. likelihood increasing) property with high probability, being more robust to the effect of user-defined inputs and handling classical Monte Carlo and Markov chain Monte Carlo methods within a common framework. Because of the first of these properties we refer to the algorithm as 'ascent-based MCEM'. We apply ascent-based MCEM to a variety of examples, including one where it is used to accelerate the convergence of deterministic EM dramatically.     

Probabilistic relabelling strategies for the label switching problem in Bayesian mixture models

The label switching problem is caused by the likelihood of a Bayesian mixture model being invariant to permutations of the labels. The permutation can change multiple times between Markov Chain Monte Carlo (MCMC) iterations making it difficult to infer component-specific parameters of the model. Various so-called ‘relabelling’ strategies exist with the goal to ‘undo’ the label switches that have occurred to enable estimation of functions that depend on component-specific parameters. Existing deterministic relabelling algorithms rely upon specifying a loss function, and relabelling by minimising its posterior expected loss. In this paper we develop probabilistic approaches to relabelling that allow for estimation and incorporation of the uncertainty in the relabelling process. Variants of the probabilistic relabelling algorithm are introduced and compared to existing deterministic relabelling algorithms. We demonstrate that the idea of probabilistic relabelling can be expressed in a rigorous framework based on the EM algorithm.     

A computational framework for empirical Bayes inference

In empirical Bayes inference one is typically interested in sampling from the posterior distribution of a parameter with a hyper-parameter set to its maximum likelihood estimate. This is often problematic particularly when the likelihood function of the hyper-parameter is not available in closed form and the posterior distribution is intractable. Previous works have dealt with this problem using a multi-step approach based on the EM algorithm and Markov Chain Monte Carlo (MCMC). We propose a framework based on recent developments in adaptive MCMC, where this problem is addressed more efficiently using a single Monte Carlo run. We discuss the convergence of the algorithm and its connection with the EM algorithm. We apply our algorithm to the Bayesian Lasso of Park and Casella (J. Am. Stat. Assoc. 103:681–686, 2008) and on the empirical Bayes variable selection of George and Foster (J. Am. Stat. Assoc. 87:731–747, 2000).     

Computing Critical Values of Exact Tests by Incorporating Monte Carlo Simulations Combined with Statistical Tables

Various exact tests for statistical inference are available for powerful and accurate decision rules provided that corresponding critical values are tabulated or evaluated via Monte Carlo methods. This article introduces a novel hybrid method for computing p‐values of exact tests by combining Monte Carlo simulations and statistical tables generated a priori. To use the data from Monte Carlo generations and tabulated critical values jointly, we employ kernel density estimation within Bayesian‐type procedures. The p‐values are linked to the posterior means of quantiles. In this framework, we present relevant information from the Monte Carlo experiments via likelihood‐type functions, whereas tabulated critical values are used to reflect prior distributions. The local maximum likelihood technique is employed to compute functional forms of prior distributions from statistical tables. Empirical likelihood functions are proposed to replace parametric likelihood functions within the structure of the posterior mean calculations to provide a Bayesian‐type procedure with a distribution‐free set of assumptions. We derive the asymptotic properties of the proposed nonparametric posterior means of quantiles process. Using the theoretical propositions, we calculate the minimum number of needed Monte Carlo resamples for desired level of accuracy on the basis of distances between actual data characteristics (e.g. sample sizes) and characteristics of data used to present corresponding critical values in a table. The proposed approach makes practical applications of exact tests simple and rapid. Implementations of the proposed technique are easily carried out via the recently developed STATA and R statistical packages.     

Relabelling in Bayesian mixture models by pivotal units

Label switching is a well-known and fundamental problem in Bayesian estimation of finite mixture models. It arises when exploring complex posterior distributions by Markov Chain Monte Carlo (MCMC) algorithms, because the likelihood of the model is invariant to the relabelling of mixture components. If the MCMC sampler randomly switches labels, then it is unsuitable for exploring the posterior distributions for component-related parameters. In this paper, a new procedure based on the post-MCMC relabelling of the chains is proposed. The main idea of the method is to perform a clustering technique on the similarity matrix, obtained through the MCMC sample, whose elements are the probabilities that any two units in the observed sample are drawn from the same component. Although it cannot be generalized to any situation, it may be handy in many applications because of its simplicity and very low computational burden.     

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.     

Maximum Likelihood Inference for Multivariate Frailty Models Using an Automated Monte Carlo EM Algorithm

We present a maximum likelihood estimation procedure for the multivariate frailty model. The estimation is based on a Monte Carlo EM algorithm. The expectation step is approximated by averaging over random samples drawn from the posterior distribution of the frailties using rejection sampling. The maximization step reduces to a standard partial likelihood maximization. We also propose a simple rule based on the relative change in the parameter estimates to decide on sample size in each iteration and a stopping time for the algorithm. An important new concept is acquiring absolute convergence of the algorithm through sample size determination and an efficient sampling technique. The method is illustrated using a rat carcinogenesis dataset and data on vase lifetimes of cut roses. The estimation results are compared with approximate inference based on penalized partial likelihood using these two examples. Unlike the penalized partial likelihood estimation, the proposed full maximum likelihood estimation method accounts for all the uncertainty while estimating standard errors for the parameters.     

Estimation for the scaled half-logistic distribution under Type-I progressively hybrid censoring scheme

This paper addresses the estimation for the unknown scale parameter of the half-logistic distribution based on a Type-I progressively hybrid censoring scheme. We evaluate the maximum likelihood estimate (MLE) via numerical method, and EM algorithm, and also the approximate maximum likelihood estimate (AMLE). We use a modified acceptance rejection method to obtain the Bayes estimate and corresponding highest posterior confidence intervals. We perform Monte Carlo simulations to compare the performances of the different methods, and we analyze one dataset for illustrative purposes.     

Inference in mixed proportional hazard models with <Emphasis Type="Italic">K</Emphasis> random effects

A general formulation of mixed proportional hazard models with K random effects is provided. It enables to account for a population stratified at K different levels. I then show how to approximate the partial maximum likelihood estimator using an EM algorithm. In a Monte Carlo study, the behavior of the estimator is assessed and I provide an application to the ratification of ILO conventions. Compared to other procedures, the results indicate an important decrease in computing time, as well as improved convergence and stability.     

Bayesian analysis of mixture modelling using the multivariate t distribution

A finite mixture model using the multivariate t distribution has been shown as a robust extension of normal mixtures. In this paper, we present a Bayesian approach for inference about parameters of t-mixture models. The specifications of prior distributions are weakly informative to avoid causing nonintegrable posterior distributions. We present two efficient EM-type algorithms for computing the joint posterior mode with the observed data and an incomplete future vector as the sample. Markov chain Monte Carlo sampling schemes are also developed to obtain the target posterior distribution of parameters. The advantages of Bayesian approach over the maximum likelihood method are demonstrated via a set of real data.     

A SEMIPARAMETRIC MIXTURE MODEL FOR THE ANALYSIS OF COMPETING RISKS DATA

This paper deals with the regression analysis of failure time data when there are censoring and multiple types of failures. We propose a semiparametric generalization of a parametric mixture model of Larson & Dinse (1985), for which the marginal probabilities of the various failure types are logistic functions of the covariates. Given the type of failure, the conditional distribution of the time to failure follows a proportional hazards model. A marginal like lihood approach to estimating regression parameters is suggested, whereby the baseline hazard functions are eliminated as nuisance parameters. The Monte Carlo method is used to approximate the marginal likelihood; the resulting function is maximized easily using existing software. Some guidelines for choosing the number of Monte Carlo replications are given. Fixing the regression parameters at their estimated values, the full likelihood is maximized via an EM algorithm to estimate the baseline survivor functions. The methods suggested are illustrated using the Stanford heart transplant data.     

Parameter Estimation of Type-II Hybrid Censored Weighted Exponential Distribution

A hybrid censoring scheme is a mixture of Type-I and Type-II censoring schemes. We study the estimation of parameters of weighted exponential distribution based on Type-II hybrid censored data. By applying the EM algorithm, maximum likelihood estimators are evaluated. Using Fisher information matrix, asymptotic confidence intervals are provided. By applying Markov chain Monte Carlo techniques, Bayes estimators, and corresponding highest posterior density confidence intervals of parameters are obtained. Monte Carlo simulations are performed to compare the performances of the different methods, and one dataset is analyzed for illustrative purposes.     

Burr-XII Distribution Parametric Estimation and Estimation of Reliability of Multicomponent Stress-Strength

In this paper, we estimate multicomponent stress-strength reliability by assuming Burr-XII distribution. The research methodology adopted here is to estimate the parameter using maximum likelihood estimation. Reliability is estimated using the maximum likelihood method of estimation and results are compared using the Monte Carlo simulation for small samples. Using real data sets we illustrate the procedure clearly.     

Local asymptotic normality for Student-Lévy processes under high-frequency sampling

There is considerable interest in parameter estimation in Lévy models. The maximum likelihood estimator is widely used because under certain conditions it enjoys asymptotic efficiency properties. The toolkit for Lévy processes is the local asymptotic normality which guarantees these conditions. Although the likelihood function is not known explicitly, we prove local asymptotic normality for the location and scale parameters of the Student-Lévy process assuming high-frequency data. In addition, we propose a numerical method to make maximum likelihood estimates feasible based on the Monte Carlo expectation-maximization algorithm. A simulation study verifies the theoretical results.     

Automatic choice of driving values in Monte Carlo likelihood approximation via posterior simulations

For models with random effects or missing data, the likelihood function is sometimes intractable analytically but amenable to Monte Carlo approximation. To get a good approximation, the parameter value that drives the simulations should be sufficiently close to the maximum likelihood estimate (MLE) which unfortunately is unknown. Introducing a working prior distribution, we express the likelihood function as a posterior expectation and approximate it using posterior simulations. If the sample size is large, the sample information is likely to outweigh the prior specification and the posterior simulations will be concentrated around the MLE automatically, leading to good approximation of the likelihood near the MLE. For smaller samples, we propose to use the current posterior as the next prior distribution to make the posterior simulations closer to the MLE and hence improve the likelihood approximation. By using the technique of data duplication, we can simulate from the sharpened posterior distribution without actually updating the prior distribution. The suggested method works well in several test cases. A more complex example involving censored spatial data is also discussed.