Hybrid Samplers for Ill-Posed Inverse Problems

Abstract.  In the Bayesian approach to ill-posed inverse problems, regularization is imposed by specifying a prior distribution on the parameters of interest and Markov chain Monte Carlo samplers are used to extract information about its posterior distribution. The aim of this paper is to investigate the convergence properties of the random-scan random-walk Metropolis (RSM) algorithm for posterior distributions in ill-posed inverse problems. We provide an accessible set of sufficient conditions, in terms of the observational model and the prior, to ensure geometric ergodicity of RSM samplers of the posterior distribution. We illustrate how these conditions can be checked in an application to the inversion of oceanographic tracer data.     

Statistical inference based on generalized Lindley record values

This paper addresses the problems of frequentist and Bayesian estimation for the unknown parameters of generalized Lindley distribution based on lower record values. We first derive the exact explicit expressions for the single and product moments of lower record values, and then use these results to compute the means, variances and covariance between two lower record values. We next obtain the maximum likelihood estimators and associated asymptotic confidence intervals. Furthermore, we obtain Bayes estimators under the assumption of gamma priors on both the shape and the scale parameters of the generalized Lindley distribution, and associated the highest posterior density interval estimates. The Bayesian estimation is studied with respect to both symmetric (squared error) and asymmetric (linear-exponential (LINEX)) loss functions. Finally, we compute Bayesian predictive estimates and predictive interval estimates for the future record values. To illustrate the findings, one real data set is analyzed, and Monte Carlo simulations are performed to compare the performances of the proposed methods of estimation and prediction.     

Bayesian inference for joint location and scale nonlinear models with skew-normal errors

A regression model with skew-normal errors provides a useful extension for ordinary normal regression models when the dataset under consideration involves asymmetric outcomes. In this article, we explore the use of Markov Chain Monte Carlo (MCMC) methods to develop a Bayesian analysis for joint location and scale nonlinear models with skew-normal errors, which relax the normality assumption and include the normal one as a special case. The main advantage of these class of distributions is that they have a nice hierarchical representation that allows the implementation of MCMC methods to simulate samples from the joint posterior distribution. Finally, simulation studies and a real example are used to illustrate the proposed methodology.     

Manipulating and summarizing posterior simulations using random variable objects

Practical Bayesian data analysis involves manipulating and summarizing simulations from the posterior distribution of the unknown parameters. By manipulation we mean computing posterior distributions of functions of the unknowns, and generating posterior predictive distributions. The results need to be summarized both numerically and graphically. We introduce, and implement in R, an object-oriented programming paradigm based on a random variable object type that is implicitly represented by simulations. This makes it possible to define vector and array objects that may contain both random and deterministic quantities, and syntax rules that allow to treat these objects like any numeric vectors or arrays, providing a solution to various problems encountered in Bayesian computing involving posterior simulations. We illustrate the use of this new programming environment with examples of Bayesian computing, demonstrating missing-value imputation, nonlinear summary of regression predictions, and posterior predictive checking.     

Nonparametric Bayesian survival analysis using mixtures of Weibull distributions

Bayesian nonparametric methods have been applied to survival analysis problems since the emergence of the area of Bayesian nonparametrics. However, the use of the flexible class of Dirichlet process mixture models has been rather limited in this context. This is, arguably, to a large extent, due to the standard way of fitting such models that precludes full posterior inference for many functionals of interest in survival analysis applications. To overcome this difficulty, we provide a computational approach to obtain the posterior distribution of general functionals of a Dirichlet process mixture. We model the survival distribution employing a flexible Dirichlet process mixture, with a Weibull kernel, that yields rich inference for several important functionals. In the process, a method for hazard function estimation emerges. Methods for simulation-based model fitting, in the presence of censoring, and for prior specification are provided. We illustrate the modeling approach with simulated and real data.     

A data-driven selection of the number of clusters in the Dirichlet allocation model via Bayesian mixture modelling

In this paper, we consider a Bayesian mixture model that allows us to integrate out the weights of the mixture in order to obtain a procedure in which the number of clusters is an unknown quantity. To determine clusters and estimate parameters of interest, we develop an MCMC algorithm denominated by sequential data-driven allocation sampler. In this algorithm, a single observation has a non-null probability to create a new cluster and a set of observations may create a new cluster through the split-merge movements. The split-merge movements are developed using a sequential allocation procedure based in allocation probabilities that are calculated according to the Kullback–Leibler divergence between the posterior distribution using the observations previously allocated and the posterior distribution including a ‘new’ observation. We verified the performance of the proposed algorithm on the simulated data and then we illustrate its use on three publicly available real data sets.     

Perfect samplers for mixtures of distributions

Summary. We consider the construction of perfect samplers for posterior distributions associated with mixtures of exponential families and conjugate priors, starting with a perfect slice sampler in the spirit of Mira and co-workers. The methods rely on a marginalization akin to Rao–Blackwellization and illustrate the duality principle of Diebolt and Robert. A first approximation embeds the finite support distribution on the latent variables within a continuous support distribution that is easier to simulate by slice sampling, but we later demonstrate that the approximation can be very poor. We conclude by showing that an alternative perfect sampler based on a single backward chain can be constructed. This alternative can handle much larger sample sizes than the slice sampler first proposed.     

A note on nonexistence of posterior moments

Bayesian analyses often take for granted the assumption that the posterior distribution has at least a first moment. They often include computed or estimated posterior means. In this note, the authors show an example of a Weibull distribution parameter where the theoretical posterior mean fails to exist for commonly used proper semi–conjugate priors. They also show that posterior moments can fail to exist with commonly used noninformative priors including Jeffreys, reference and matching priors, despite the fact that the posteriors are proper. Moreover, within a broad class of priors, the predictive distribution also has no mean. The authors illustrate the problem with a simulated example. Their results demonstrate that the unwitting use of estimated posterior means may yield unjustified conclusions.     

Exact filtering for partially observed continuous time models

Summary.  The forward–backward algorithm is an exact filtering algorithm which can efficiently calculate likelihoods, and which can be used to simulate from posterior distributions. Using a simple result which relates gamma random variables with different rates, we show how the forward–backward algorithm can be used to calculate the distribution of a sum of gamma random variables, and to simulate from their joint distribution given their sum. One application is to calculating the density of the time of a specific event in a Markov process, as this time is the sum of exponentially distributed interevent times. This enables us to apply the forward–backward algorithm to a range of new problems. We demonstrate our method on three problems: calculating likelihoods and simulating allele frequencies under a non-neutral population genetic model, analysing a stochastic epidemic model and simulating speciation times in phylogenetics.     

Monte Carlo approximation of likelihood function in spatial GLMMs through an empirical Bayes method

In spatial generalized linear mixed models (SGLMMs), statistical inference encounters problems, since random effects in the model imply high-dimensional integrals to calculate the marginal likelihood function. In this article, we temporarily treat parameters as random variables and express the marginal likelihood function as a posterior expectation. Hence, the marginal likelihood function is approximated using the obtained samples from the posterior density of the latent variables and parameters given the data. However, in this setting, misspecification of prior distribution of correlation function parameter and problems associated with convergence of Markov chain Monte Carlo (MCMC) methods could have an unpleasant influence on the likelihood approximation. To avoid these challenges, we utilize an empirical Bayes approach to estimate prior hyperparameters. We also use a computationally efficient hybrid algorithm by combining inverse Bayes formula (IBF) and Gibbs sampler procedures. A simulation study is conducted to assess the performance of our method. Finally, we illustrate the method applying a dataset of standard penetration test of soil in an area in south of Iran.     

Sampling the Future: A Bayesian Approach to Forecasting From Univariate Time Series Models

The Box–Jenkins methodology for modeling and forecasting from univariate time series models has long been considered a standard to which other forecasting techniques have been compared. To a Bayesian statistician, however, the method lacks an important facet—a provision for modeling uncertainty about parameter estimates. We present a technique called sampling the future for including this feature in both the estimation and forecasting stages. Although it is relatively easy to use Bayesian methods to estimate the parameters in an autoregressive integrated moving average (ARIMA) model, there are severe difficulties in producing forecasts from such a model. The multiperiod predictive density does not have a convenient closed form, so approximations are needed. In this article, exact Bayesian forecasting is approximated by simulating the joint predictive distribution. First, parameter sets are randomly generated from the joint posterior distribution. These are then used to simulate future paths of the time series. This bundle of many possible realizations is used to project the future in several ways. Highest probability forecast regions are formed and portrayed with computer graphics. The predictive density's shape is explored. Finally, we discuss a method that allows the analyst to subjectively modify the posterior distribution on the parameters and produce alternate forecasts.     

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.     

An objective stepwise Bayes approach to small area estimation

The term ‘small area’ or ‘small domain’ is commonly used to denote a small geographical area that has a small subpopulation of people within a large area. Small area estimation is an important area in survey sampling because of the growing demand for better statistical inference for small areas in public or private surveys. In small area estimation problems the focus is on how to borrow strength across areas in order to develop a reliable estimator and which makes use of available auxiliary information. Some traditional methods for small area problems such as empirical best linear unbiased prediction borrow strength through linear models that provide links to related areas, which may not be appropriate for some survey data. In this article, we propose a stepwise Bayes approach which borrows strength through an objective posterior distribution. This approach results in a generalized constrained Dirichlet posterior estimator when auxiliary information is available for small areas. The objective posterior distribution is based only on the assumption of exchangeability across related areas and does not make any explicit model assumptions. The form of our posterior distribution allows us to assign a weight to each member of the sample. These weights can then be used in a straight forward fashion to make inferences about the small area means. Theoretically, the stepwise Bayes character of the posterior allows one to prove the admissibility of the point estimators suggesting that inferential procedures based on this approach will tend to have good frequentist properties. Numerically, we demonstrate in simulations that the proposed stepwise Bayes approach can have substantial strengths compared to traditional methods.     

基于MCMC模拟和伪似然估计法的交叉分类信度模型费率厘定简

针对传统交叉分类信度模型计算复杂且在结构参数先验信息不足的情况下不能得到参数无偏后验估计的问题，利用MCMC模拟和GLMM方法，对交叉分类信度模型进行实证分析证明模型的有效性。结果表明：基于MCMC方法能够动态模拟参数的后验分布，并可提高模型估计的精度；基于GLMM能大大简化计算过程且操作方便，可利用图形和其它诊断工具选择模型，并对模型实用性做出评价。     

Exact Bayesian inference for off-line change-point detection in tree-structured graphical models

We consider the problem of change-point detection in multivariate time-series. The multivariate distribution of the observations is supposed to follow a graphical model, whose graph and parameters are affected by abrupt changes throughout time. We demonstrate that it is possible to perform exact Bayesian inference whenever one considers a simple class of undirected graphs called spanning trees as possible structures. We are then able to integrate on the graph and segmentation spaces at the same time by combining classical dynamic programming with algebraic results pertaining to spanning trees. In particular, we show that quantities such as posterior distributions for change-points or posterior edge probabilities over time can efficiently be obtained. We illustrate our results on both synthetic and experimental data arising from biology and neuroscience.     

Exact posterior distributions and model selection criteria for multiple change-point detection problems

In segmentation problems, inference on change-point position and model selection are two difficult issues due to the discrete nature of change-points. In a Bayesian context, we derive exact, explicit and tractable formulae for the posterior distribution of variables such as the number of change-points or their positions. We also demonstrate that several classical Bayesian model selection criteria can be computed exactly. All these results are based on an efficient strategy to explore the whole segmentation space, which is very large. We illustrate our methodology on both simulated data and a comparative genomic hybridization profile.     

Quasi Bayesian likelihood

If the form of the distribution of data is unknown, the Bayesian method fails in the parametric inference because there is no posterior distribution of the parameter. In this paper, a theoretical framework of Bayesian likelihood is introduced via the Hilbert space method, which is free of the distributions of data and the parameter. The posterior distribution and posterior score function based on given inner products are defined and, consequently, the quasi posterior distribution and quasi posterior score function are derived, respectively, as the projections of the posterior distribution and posterior score function onto the space spanned by given estimating functions. In the space spanned by data, particularly, an explicit representation for the quasi posterior score function is obtained, which can be derived as a projection of the true posterior score function onto this space. The methods of constructing conservative quasi posterior score and quasi posterior log-likelihood are proposed. Some examples are given to illustrate the theoretical results. As an application, the quasi posterior distribution functions are used to select variables for generalized linear models. It is proved that, for linear models, the variable selections via quasi posterior distribution functions are equivalent to the variable selections via the penalized residual sum of squares or regression sum of squares.     

Optimal collapsing of mixture distributions in robust recursive estimation

Several authors have discussed Kalman filtering procedures using a mixture of normals as a model for the distributions of the noise in the observation and/or the state space equations. Under this model, resulting posteriors involve a mixture of normal distributions, and a “collapsing method” must be found in order to keep the recursive procedure simple. We prove that the Kullback-Leibler distance between the mixture posterior and that of a single normal distribution is minimized when we choose the mean and variance of the single normal distribution to be the mean and variance of the mixture posterior. Hence, “collapsing by moments” is optimal in this sense. We then develop the resulting optimal algorithm for “Kalman filtering” for this situation, and illustrate its performance with an example.     

Criticism of the predictive distribution

The predictive distribution is a mixture of the original distribution model and is used for predicting a future observation. Therein, the mixing distribution is the posterior distribution of the distribution parameters in the Bayesian inference. The mixture can also be computed for the frequentist inference because the Bayesian posterior distribution has the same meaning as a frequentist confidence interval. I present arguments against the concept of predictive distribution. Examples illustrate these. The most important argument is that the predictive distribution can depend on the parameterization. An improvement of the theory of the predictive distribution is recommended.     

State estimation for aoristic models

Aoristic data can be described by a marked point process in time in which the points cannot be observed directly but are known to lie in observable intervals, the marks. We consider Bayesian state estimation for the latent points when the marks are modeled in terms of an alternating renewal process in equilibrium and the prior is a Markov point process. We derive the posterior distribution, estimate its parameters and present some examples that illustrate the influence of the prior distribution. The model is then used to estimate times of occurrence of interval censored crimes.