On progressively first failure censored Lindley distribution

This article deals with the progressively first failure censored Lindley distribution. Maximum likelihood and Bayes estimators of the parameter and reliability characteristics of Lindley distribution based on progressively first failure censored samples are derived. Asymptotic confidence intervals based on observed Fisher information and bootstrap confidence intervals of the parameter are constructed. Bayes estimators using non-informative and gamma informative priors are derived using importance sampling procedure and Metropolis–Hastings (MH) algorithm under squared error loss function. Also, HPD credible intervals based on importance sampling procedure and MH algorithm for the parameter are constructed. To study the performance of various estimators discussed in this article, a Monte Carlo simulation study is conducted. Finally, a real data set is studied for illustration purposes.     

Markov chain Monte Carlo Using an Approximation

This article presents a method for generating samples from an unnormalized posterior distribution f(·) using Markov chain Monte Carlo (MCMC) in which the evaluation of f(·) is very difficult or computationally demanding. Commonly, a less computationally demanding, perhaps local, approximation to f(·) is available, say f^**_x(·). An algorithm is proposed to generate an MCMC that uses such an approximation to calculate acceptance probabilities at each step of a modified Metropolis–Hastings algorithm. Once a proposal is accepted using the approximation, f(·) is calculated with full precision ensuring convergence to the desired distribution. We give sufficient conditions for the algorithm to converge to f(·) and give both theoretical and practical justifications for its usage. Typical applications are in inverse problems using physical data models where computing time is dominated by complex model simulation. We outline Bayesian inference and computing for inverse problems. A stylized example is given of recovering resistor values in a network from electrical measurements made at the boundary. Although this inverse problem has appeared in studies of underground reservoirs, it has primarily been chosen for pedagogical value because model simulation has precisely the same computational structure as a finite element method solution of the complete electrode model used in conductivity imaging, or “electrical impedance tomography.” This example shows a dramatic decrease in CPU time, compared to a standard Metropolis–Hastings algorithm.     

Bayesian cylindrical data modeling using Abe–Ley mixtures

This paper proposes a Metropolis–Hastings algorithm based on Markov chain Monte Carlo sampling, to estimate the parameters of the Abe–Ley distribution, which is a recently proposed Weibull-Sine-Skewed-von Mises mixture model, for bivariate circular-linear data. Current literature estimates the parameters of these mixture models using the expectation-maximization method, but we will show that this exhibits a few shortcomings for the considered mixture model. First, standard expectation-maximization does not guarantee convergence to a global optimum, because the likelihood is multi-modal, which results from the high dimensionality of the mixture’s likelihood. Second, given that expectation-maximization provides point estimates of the parameters only, the uncertainties of the estimates (e.g., confidence intervals) are not directly available in these methods. Hence, extra calculations are needed to quantify such uncertainty. We propose a Metropolis–Hastings based algorithm that avoids both shortcomings of expectation-maximization. Indeed, Metropolis–Hastings provides an approximation to the complete (posterior) distribution, given that it samples from the joint posterior of the mixture parameters. This facilitates direct inference (e.g., about uncertainty, multi-modality) from the estimation. In developing the algorithm, we tackle various challenges including convergence speed, label switching and selecting the optimum number of mixture components. We then (i) verify the effectiveness of the proposed algorithm on sample datasets with known true parameters, and further (ii) validate our methodology on an environmental dataset (a traditional application domain of Abe–Ley mixtures where measurements are function of direction). Finally, we (iii) demonstrate the usefulness of our approach in an application domain where the circular measurement is periodic in time.     

MCMC Estimation of Restricted Covariance Matrices

This article is motivated by the difficulty of applying standard simulation techniques when identification constraints or theoretical considerations induce covariance restrictions in multivariate models. To deal with this difficulty, we build upon a decomposition of positive definite matrices and show that it leads to straightforward Markov chain Monte Carlo samplers for restricted covariance matrices. We introduce the approach by reviewing results for multivariate Gaussian models without restrictions, where standard conjugate priors on the elements of the decomposition induce the usual Wishart distribution on the precision matrix and vice versa. The unrestricted case provides guidance for constructing efficient Metropolis–Hastings and accept-reject Metropolis–Hastings samplers in more complex settings, and we describe in detail how simulation can be performed under several important constraints. The proposed approach is illustrated in a simulation study and two applications in economics. Supplemental materials for this article (appendixes, data, and computer code) are available online.     

Bayesian estimation of generalized gamma shared frailty model

Multivariate survival analysis comprises of event times that are generally grouped together in clusters. Observations in each of these clusters relate to data belonging to the same individual or individuals with a common factor. Frailty models can be used when there is unaccounted association between survival times of a cluster. The frailty variable describes the heterogeneity in the data caused by unknown covariates or randomness in the data. In this article, we use the generalized gamma distribution to describe the frailty variable and discuss the Bayesian method of estimation for the parameters of the model. The baseline hazard function is assumed to follow the two parameter Weibull distribution. Data is simulated from the given model and the Metropolis–Hastings MCMC algorithm is used to obtain parameter estimates. It is shown that increasing the size of the dataset improves estimates. It is also shown that high heterogeneity within clusters does not affect the estimates of treatment effects significantly. The model is also applied to a real life dataset.     

Structured Markov Chain Monte Carlo

Abstract

This article introduces a general method for Bayesian computing in richly parameterized models, structured Markov chain Monte Carlo (SMCMC), that is based on a blocked hybrid of the Gibbs sampling and Metropolis—Hastings algorithms. SMCMC speeds algorithm convergence by using the structure that is present in the problem to suggest an appropriate Metropolis—Hastings candidate distribution. Although the approach is easiest to describe for hierarchical normal linear models, we show that its extension to both nonnormal and nonlinear cases is straightforward. After describing the method in detail we compare its performance (in terms of run time and autocorrelation in the samples) to other existing methods, including the single-site updating Gibbs sampler available in the popular BUGS software package. Our results suggest significant improvements in convergence for many problems using SMCMC, as well as broad applicability of the method, including previously intractable hierarchical nonlinear model settings.     

Markov Chain Sampling Methods for Dirichlet Process Mixture Models

Abstract

This article reviews Markov chain methods for sampling from the posterior distribution of a Dirichlet process mixture model and presents two new classes of methods. One new approach is to make Metropolis—Hastings updates of the indicators specifying which mixture component is associated with each observation, perhaps supplemented with a partial form of Gibbs sampling. The other new approach extends Gibbs sampling for these indicators by using a set of auxiliary parameters. These methods are simple to implement and are more efficient than previous ways of handling general Dirichlet process mixture models with non-conjugate priors.     

Perturbation bounds for Monte Carlo within Metropolis via restricted approximations

The Monte Carlo within Metropolis (MCwM) algorithm, interpreted as a perturbed Metropolis–Hastings (MH) algorithm, provides an approach for approximate sampling when the target distribution is intractable. Assuming the unperturbed Markov chain is geometrically ergodic, we show explicit estimates of the difference between the $n$th step distributions of the perturbed MCwM and the unperturbed MH chains. These bounds are based on novel perturbation results for Markov chains which are of interest beyond the MCwM setting. To apply the bounds, we need to control the difference between the transition probabilities of the two chains and to verify stability of the perturbed chain.     

Parameter estimation in stochastic differential equations with Markov chain Monte Carlo and non-linear Kalman filtering

This paper is concerned with parameter estimation in linear and non-linear Itô type stochastic differential equations using Markov chain Monte Carlo (MCMC) methods. The MCMC methods studied in this paper are the Metropolis–Hastings and Hamiltonian Monte Carlo (HMC) algorithms. In these kind of models, the computation of the energy function gradient needed by HMC and gradient based optimization methods is non-trivial, and here we show how the gradient can be computed with a linear or non-linear Kalman filter-like recursion. We shall also show how in the linear case the differential equations in the gradient recursion equations can be solved using the matrix fraction decomposition. Numerical results for simulated examples are presented and discussed in detail.     

Adaptive,Delayed-Acceptance MCMC for Targets With Expensive Likelihoods

When conducting Bayesian inference, delayed-acceptance (DA) Metropolis–Hastings (MH) algorithms and DA pseudo-marginal MH algorithms can be applied when it is computationally expensive to calculate the true posterior or an unbiased estimate thereof, but a computationally cheap approximation is available. A first accept-reject stage is applied, with the cheap approximation substituted for the true posterior in the MH acceptance ratio. Only for those proposals that pass through the first stage is the computationally expensive true posterior (or unbiased estimate thereof) evaluated, with a second accept-reject stage ensuring that detailed balance is satisfied with respect to the intended true posterior. In some scenarios, there is no obvious computationally cheap approximation. A weighted average of previous evaluations of the computationally expensive posterior provides a generic approximation to the posterior. If only the k-nearest neighbors have nonzero weights then evaluation of the approximate posterior can be made computationally cheap provided that the points at which the posterior has been evaluated are stored in a multi-dimensional binary tree, known as a KD-tree. The contents of the KD-tree are potentially updated after every computationally intensive evaluation. The resulting adaptive, delayed-acceptance [pseudo-marginal] Metropolis–Hastings algorithm is justified both theoretically and empirically. Guidance on tuning parameters is provided and the methodology is applied to a discretely observed Markov jump process characterizing predator–prey interactions and an ODE system describing the dynamics of an autoregulatory gene network. Supplementary material for this article is available online.     

Adaptive Mixture Modeling Metropolis Methods for Bayesian Analysis of Nonlinear State-Space Models

We describe a strategy for Markov chain Monte Carlo analysis of nonlinear, non-Gaussian state-space models involving batch analysis for inference on dynamic, latent state variables and fixed model parameters. The key innovation is a Metropolis–Hastings method for the time series of state variables based on sequential approximation of filtering and smoothing densities using normal mixtures. These mixtures are propagated through the nonlinearities using an accurate, local mixture approximation method, and we use a regenerating procedure to deal with potential degeneracy of mixture components. This provides accurate, direct approximations to sequential filtering and retrospective smoothing distributions, and hence a useful construction of global Metropolis proposal distributions for simulation of posteriors for the set of states. This analysis is embedded within a Gibbs sampler to include uncertain fixed parameters. We give an example motivated by an application in systems biology. Supplemental materials provide an example based on a stochastic volatility model as well as MATLAB code.     

Adaptive Component-Wise Multiple-Try Metropolis Sampling

One of the most widely used samplers in practice is the component-wise Metropolis–Hastings (CMH) sampler that updates in turn the components of a vector-valued Markov chain using accept–reject moves generated from a proposal distribution. When the target distribution of a Markov chain is irregularly shaped, a “good” proposal distribution for one region of the state–space might be a “poor” one for another region. We consider a component-wise multiple-try Metropolis (CMTM) algorithm that chooses from a set of candidate moves sampled from different distributions. The computational efficiency is increased using an adaptation rule for the CMTM algorithm that dynamically builds a better set of proposal distributions as the Markov chain runs. The ergodicity of the adaptive chain is demonstrated theoretically. The performance is studied via simulations and real data examples. Supplementary material for this article is available online.     

Generalized Dynamic Factor Models for Mixed-Measurement Time Series

In this article, we propose generalized Bayesian dynamic factor models for jointly modeling mixed-measurement time series. The framework allows mixed-scale measurements associated with each time series, with different measurements having different distributions in the exponential family conditionally on time-varying latent factor(s). Efficient Bayesian computational algorithms are developed for posterior inference on both the latent factors and model parameters, based on a Metropolis–Hastings algorithm with adaptive proposals. The algorithm relies on a Greedy Density Kernel Approximation and parameter expansion with latent factor normalization. We tested the framework and algorithms in simulated studies and applied them to the analysis of intertwined credit and recovery risk for Moody’s rated firms from 1982 to 2008, illustrating the importance of jointly modeling mixed-measurement time series. The article has supplementary materials available online.     

Optimal scaling of random-walk metropolis algorithms on general target distributions

One main limitation of the existing optimal scaling results for Metropolis–Hastings algorithms is that the assumptions on the target distribution are unrealistic. In this paper, we consider optimal scaling of random-walk Metropolis algorithms on general target distributions in high dimensions arising from practical MCMC models from Bayesian statistics. For optimal scaling by maximizing expected squared jumping distance (ESJD), we show the asymptotically optimal acceptance rate 0.234 can be obtained under general realistic sufficient conditions on the target distribution. The new sufficient conditions are easy to be verified and may hold for some general classes of MCMC models arising from Bayesian statistics applications, which substantially generalize the product i.i.d. condition required in most existing literature of optimal scaling. Furthermore, we show one-dimensional diffusion limits can be obtained under slightly stronger conditions, which still allow dependent coordinates of the target distribution. We also connect the new diffusion limit results to complexity bounds of Metropolis algorithms in high dimensions.     

The Shadow Prior

In this article we consider posterior simulation in models with constrained parameter or sampling spaces. Constraints on the support of sampling and prior distributions give rise to a normalization constant in the complete conditional posterior distribution for the (hyper-) parameters of the respective distribution, complicating posterior simulation.

To mitigate the problem of evaluating normalization constants, we propose a computational approach based on model augmentation. We include an additional level in the probability model to separate the (hyper-) parameter from the constrained probability model, and we refer to this additional level in the probability model as a shadow prior. This approach can significantly reduce the overall computational burden if the original (hyper-) prior includes a complicated structure, but a simple form is chosen for the shadow prior, for example, if the original prior includes a mixture model or multivariate distribution, and the shadow prior defines a set of shadow parameters that are iid given the (hyper-) parameters. Although introducing the shadow prior changes the posterior inference on the original parameters, we argue that by appropriate choices of the shadow prior, the change is minimal and posterior simulation in the augmented probability model provides a meaningful approximation to the desired inference. Data used in this article are available online.     

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.     

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 the Relationship Between Markov chain Monte Carlo Methods for Model Uncertainty

This article considers Markov chain computational methods for incorporating uncertainty about the dimension of a parameter when performing inference within a Bayesian setting. A general class of methods is proposed for performing such computations, based upon a product space representation of the problem which is similar to that of Carlin and Chib. It is shown that all of the existing algorithms for incorporation of model uncertainty into Markov chain Monte Carlo (MCMC) can be derived as special cases of this general class of methods. In particular, we show that the popular reversible jump method is obtained when a special form of Metropolis–Hastings (M–H) algorithm is applied to the product space. Furthermore, the Gibbs sampling method and the variable selection method are shown to derive straightforwardly from the general framework. We believe that these new relationships between methods, which were until now seen as diverse procedures, are an important aid to the understanding of MCMC model selection procedures and may assist in the future development of improved procedures. Our discussion also sheds some light upon the important issues of “pseudo-prior” selection in the case of the Carlin and Chib sampler and choice of proposal distribution in the case of reversible jump. Finally, we propose efficient reversible jump proposal schemes that take advantage of any analytic structure that may be present in the model. These proposal schemes are compared with a standard reversible jump scheme for the problem of model order uncertainty in autoregressive time series, demonstrating the improvements which can be achieved through careful choice of proposals.     

Noisy Hamiltonian Monte Carlo for Doubly Intractable Distributions

Hamiltonian Monte Carlo (HMC) has been progressively incorporated within the statistician’s toolbox as an alternative sampling method in settings when standard Metropolis–Hastings is inefficient. HMC generates a Markov chain on an augmented state space with transitions based on a deterministic differential flow derived from Hamiltonian mechanics. In practice, the evolution of Hamiltonian systems cannot be solved analytically, requiring numerical integration schemes. Under numerical integration, the resulting approximate solution no longer preserves the measure of the target distribution, therefore an accept–reject step is used to correct the bias. For doubly intractable distributions—such as posterior distributions based on Gibbs random fields—HMC suffers from some computational difficulties: computation of gradients in the differential flow and computation of the accept–reject proposals poses difficulty. In this article, we study the behavior of HMC when these quantities are replaced by Monte Carlo estimates. Supplemental codes for implementing methods used in the article are available online.     

Asymptotically Optimal Quickest Change Detection in Multistream Data—Part 1: General Stochastic Models

Assume that there are multiple data streams (channels, sensors) and in each stream the process of interest produces generally dependent and non-identically distributed observations. When the process is in a normal mode (in-control), the (pre-change) distribution is known, but when the process becomes abnormal there is a parametric uncertainty, i.e., the post-change (out-of-control) distribution is known only partially up to a parameter. Both the change point and the post-change parameter are unknown. Moreover, the change affects an unknown subset of streams, so that the number of affected streams and their location are unknown in advance. A good changepoint detection procedure should detect the change as soon as possible after its occurrence while controlling for a risk of false alarms. We consider a Bayesian setup with a given prior distribution of the change point and propose two sequential mixture-based change detection rules, one mixes a Shiryaev-type statistic over both the unknown subset of affected streams and the unknown post-change parameter and another mixes a Shiryaev–Roberts-type statistic. These rules generalize the mixture detection procedures studied by Tartakovsky (IEEE Trans Inf Theory 65(3):1413–1429, 2019) in a single-stream case. We provide sufficient conditions under which the proposed multistream change detection procedures are first-order asymptotically optimal with respect to moments of the delay to detection as the probability of false alarm approaches zero.