Rates of Convergence for a Bayesian Level Set Estimation

Abstract.  We are interested in estimating level sets using a Bayesian non-parametric approach, from an independent and identically distributed sample drawn from an unknown distribution. Under fairly general conditions on the prior, we provide an upper bound on the rate of convergence of the Bayesian level set estimate, via the rate at which the posterior distribution concentrates around the true level set. We then consider, as an application, the log-spline prior in the two-dimensional unit cube. Assuming that the true distribution belongs to a class of Hölder, we provide an upper bound on the rate of convergence of the Bayesian level set estimates. We compare our results with existing rates of convergence in the frequentist non-parametric literature: the Bayesian level set estimator proves to be competitive and is also easy to compute, which is of no small importance. A simulation study is given as an illustration.     

A note on Bayesian nonparametric regression function estimation

In this note the problem of nonparametric regression function estimation in a random design regression model with Gaussian errors is considered from the Bayesian perspective. It is assumed that the regression function belongs to a class of functions with a known degree of smoothness. A prior distribution on the given class can be induced by a prior on the coefficients in a series expansion of the regression function through an orthonormal system. The rate of convergence of the resulting posterior distribution is employed to provide a measure of the accuracy of the Bayesian estimation procedure defined by the posterior expected regression function. We show that the Bayes’ estimator achieves the optimal minimax rate of convergence under mean integrated squared error over the involved class of regression functions, thus being comparable to other popular frequentist regression estimators.     

On Rates of Convergence for Bayesian Density Estimation

Abstract.  We consider the problem of estimating a compactly supported density taking a Bayesian nonparametric approach. We define a Dirichlet mixture prior that, while selecting piecewise constant densities, has full support on the Hellinger metric space of all commonly dominated probability measures on a known bounded interval. We derive pointwise rates of convergence for the posterior expected density by studying the speed at which the posterior mass accumulates on shrinking Hellinger neighbourhoods of the sampling density. If the data are sampled from a strictly positive, α -Hölderian density, with α  ∈ ( 0,1] , then the optimal convergence rate n^{− α / (2 α +1)} is obtained up to a logarithmic factor. Smoothing histograms by polygons, a continuous piecewise linear estimator is obtained that for twice continuously differentiable, strictly positive densities satisfying boundary conditions attains a rate comparable up to a logarithmic factor to the convergence rate n ^−4/5 for integrated mean squared error of kernel type density estimators.     

Convergence properties of sequential Bayesian D-optimal designs

We establish convergence properties of sequential Bayesian optimal designs. In particular, for sequential D-optimality under a general nonlinear location-scale model for binary experiments, we establish posterior consistency, consistency of the design measure, and the asymptotic normality of posterior following the design. We illustrate our results in the context of a particular application in the design of phase I clinical trials, namely a sequential design of Haines et al. [2003. Bayesian optimal designs for phase I clinical trials. Biometrics 59, 591–600] that incorporates an ethical constraint on overdosing.     

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.     

Bayesian analysis of complementary Poisson rate parameters with data subject to misclassification

We formulate closed-form Bayesian estimators for two complementary Poisson rate parameters using double sampling with data subject to misclassification and error free data. We also derive closed-form Bayesian estimators for two misclassification parameters in the modified Poisson model we assume. We use our results to determine credible sets for the rate and misclassification parameters. Additionally, we use MCMC methods to determine Bayesian estimators for three or more rate parameters and the misclassification parameters. We also perform a limited Monte Carlo simulation to examine the characteristics of these estimators. We demonstrate the efficacy of the new Bayesian estimators and highest posterior density regions with examples using two real data sets.     

A random weighting approach for posterior distributions

ABSTRACT

In Bayesian theory, calculating a posterior probability distribution is highly important but typically difficult. Therefore, some methods have been proposed to deal with such problem, among which, the most popular one is the asymptotic expansions of posterior distributions. In this paper, we propose an alternative approach, named a random weighting method, for scaled posterior distributions, and give an ideal convergence rate, o(n^{( ? 1/2)}), which serves as the theoretical guarantee for methods of numerical simulations.     

Assessing convergence diagnostic tests for Bayesian Cox regression

The Markov chain Monte Carlo (MCMC) method generates samples from the posterior distribution and uses these samples to approximate expectations of quantities of interest. For the process, researchers have to decide whether the Markov chain has reached the desired posterior distribution. Using convergence diagnostic tests are very important to decide whether the Markov chain has reached the target distribution. Our interest in this study was to compare the performances of convergence diagnostic tests for all parameters of Bayesian Cox regression model with different number of iterations by using a simulation and a real lung cancer dataset.     

Nonparametric hierarchical Bayes analysis of binomial data via Bernstein polynomial priors

For binomial data analysis, many methods based on empirical Bayes interpretations have been developed, in which a variance‐stabilizing transformation and a normality assumption are usually required. To achieve the greatest model flexibility, we conduct nonparametric Bayesian inference for binomial data and employ a special nonparametric Bayesian prior—the Bernstein–Dirichlet process (BDP)—in the hierarchical Bayes model for the data. The BDP is a special Dirichlet process (DP) mixture based on beta distributions, and the posterior distribution resulting from it has a smooth density defined on [0, 1]. We examine two Markov chain Monte Carlo procedures for simulating from the resulting posterior distribution, and compare their convergence rates and computational efficiency. In contrast to existing results for posterior consistency based on direct observations, the posterior consistency of the BDP, given indirect binomial data, is established. We study shrinkage effects and the robustness of the BDP‐based posterior estimators in comparison with several other empirical and hierarchical Bayes estimators, and we illustrate through examples that the BDP‐based nonparametric Bayesian estimate is more robust to the sample variation and tends to have a smaller estimation error than those based on the DP prior. In certain settings, the new estimator can also beat Stein's estimator, Efron and Morris's limited‐translation estimator, and many other existing empirical Bayes estimators. The Canadian Journal of Statistics 40: 328–344; 2012 © 2012 Statistical Society of Canada     

Bayesian Inference of Odds Ratios in Misclassified Binary Data with a Validation Substudy

We propose a fully Bayesian model with a non-informative prior for analyzing misclassified binary data with a validation substudy. In addition, we derive a closed-form algorithm for drawing all parameters from the posterior distribution and making statistical inference on odds ratios. Our algorithm draws each parameter from a beta distribution, avoids the specification of initial values, and does not have convergence issues. We apply the algorithm to a data set and compare the results with those obtained by other methods. Finally, the performance of our algorithm is assessed using simulation studies.     

Convergence of posterior odds

The problem of approximating an interval null or imprecise hypothesis test by a point null or precise hypothesis test under a Bayesian framework is considered. In the literature, some of the methods for solving this problem have used the Bayes factor for testing a point null and justified it as an approximation to the interval null. However, many authors recommend evaluating tests through the posterior odds, a Bayesian measure of evidence against the null hypothesis. It is of interest then to determine whether similar results hold when using the posterior odds as the primary measure of evidence. For the prior distributions under which the approximation holds with respect to the Bayes factor, it is shown that the posterior odds for testing the point null hypothesis does not approximate the posterior odds for testing the interval null hypothesis. In fact, in order to obtain convergence of the posterior odds, a number of restrictive conditions need to be placed on the prior structure. Furthermore, under a non-symmetrical prior setup, neither the Bayes factor nor the posterior odds for testing the imprecise hypothesis converges to the Bayes factor or posterior odds respectively for testing the precise hypothesis. To rectify this dilemma, it is shown that constraints need to be placed on the priors. In both situations, the class of priors constructed to ensure convergence of the posterior odds are not practically useful, thus questioning, from a Bayesian perspective, the appropriateness of point null testing in a problem better represented by an interval null. The theories developed are also applied to an epidemiological data set from White et al. (Can. Veterinary J. 30 (1989) 147–149.) in order to illustrate and study priors for which the point null hypothesis test approximates the interval null hypothesis test. AMS Classification: Primary 62F15; Secondary 62A15     

Bayesian inference for comparing two Poisson rates using data subject to underreporting and validation data

We derive a new Bayesian credible interval estimator for comparing two Poisson rates when counts are underreported and an additional validation data set is available. We provide a closed-form posterior density for the difference between the two rates that yields insightful information on which prior parameters influence the posterior the most. We also apply the new interval estimator to a real-data example, investigate the performance of the credible interval, and examine the impact of informative priors on the rate difference posterior via Monte Carlo simulations.     

Estimation of the intensity function of an inhomogeneous Poisson process with a change‐point

Recent work on point processes includes studying posterior convergence rates of estimating a continuous intensity function. In this article, convergence rates for estimating the intensity function and change‐point are derived for the more general case of a piecewise continuous intensity function. We study the problem of estimating the intensity function of an inhomogeneous Poisson process with a change‐point using non‐parametric Bayesian methods. An Markov Chain Monte Carlo (MCMC) algorithm is proposed to obtain estimates of the intensity function and the change‐point which is illustrated using simulation studies and applications. The Canadian Journal of Statistics 47: 604–618; 2019 © 2019 Statistical Society of Canada     

A permutation-based Bayesian approach for inverse covariance estimation

Abstract

Covariance estimation and selection for multivariate datasets in a high-dimensional regime is a fundamental problem in modern statistics. Gaussian graphical models are a popular class of models used for this purpose. Current Bayesian methods for inverse covariance matrix estimation under Gaussian graphical models require the underlying graph and hence the ordering of variables to be known. However, in practice, such information on the true underlying model is often unavailable. We therefore propose a novel permutation-based Bayesian approach to tackle the unknown variable ordering issue. In particular, we utilize multiple maximum a posteriori estimates under the DAG-Wishart prior for each permutation, and subsequently construct the final estimate of the inverse covariance matrix. The proposed estimator has smaller variability and yields order-invariant property. We establish posterior convergence rates under mild assumptions and illustrate that our method outperforms existing approaches in estimating the inverse covariance matrices via simulation studies.     

Latent variable models with ordinal categorical covariates

We propose a general latent variable model for multivariate ordinal categorical variables, in which both the responses and the covariates are ordinal, to assess the effect of the covariates on the responses and to model the covariance structure of the response variables. A?fully Bayesian approach is employed to analyze the model. The Gibbs sampler is used to simulate the joint posterior distribution of the latent variables and the parameters, and the parameter expansion and reparameterization techniques are used to speed up the convergence procedure. The proposed model and method are demonstrated by simulation studies and a real data example.     

Bayesian variable selection in Poisson change-point regression analysis

In this article, we develop a Bayesian variable selection method that concerns selection of covariates in the Poisson change-point regression model with both discrete and continuous candidate covariates. Ranging from a null model with no selected covariates to a full model including all covariates, the Bayesian variable selection method searches the entire model space, estimates posterior inclusion probabilities of covariates, and obtains model averaged estimates on coefficients to covariates, while simultaneously estimating a time-varying baseline rate due to change-points. For posterior computation, the Metropolis-Hastings within partially collapsed Gibbs sampler is developed to efficiently fit the Poisson change-point regression model with variable selection. We illustrate the proposed method using simulated and real datasets.     

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.     

Sequential classification on partially ordered sets

Summary. A general theorem on the asymptotically optimal sequential selection of experiments is presented and applied to a Bayesian classification problem when the parameter space is a finite partially ordered set. The main results include establishing conditions under which the posterior probability of the true state converges to 1 almost surely and determining optimal rates of convergence. Properties of a class of experiment selection rules are explored.     

Posterior sampling in two classes of multivariate fractionally integrated models: corrigendum to Ravishanker,N. and B. K. Ray (1997) Australian Journal of Statistics 39 (3), 295–311

We discuss posterior sampling for two distinct multivariate generalisations of the univariate autoregressive integrated moving average (ARIMA) model with fractional integration. The existing approach to Bayesian estimation, introduced by Ravishanker & Ray, claims to provide a posterior‐sampling algorithm for fractionally integrated vector autoregressive moving averages (FIVARMAs). We show that this algorithm produces posterior draws for vector autoregressive fractionally integrated moving averages (VARFIMAs), a model of independent interest that has not previously received attention in the Bayesian literature.     

Bayesian quadrature with non-normal approximating functions

We consider an efficient Bayesian approach to estimating integration-based posterior summaries from a separate Bayesian application. In Bayesian quadrature we model an intractable posterior density function f(·) as a Gaussian process, using an approximating function g(·), and find a posterior distribution for the integral of f(·), conditional on a few evaluations of f (·) at selected design points. Bayesian quadrature using normal g (·) is called Bayes-Hermite quadrature. We extend this theory by allowing g(·) to be chosen from two wider classes of functions. One is a family of skew densities and the other is the family of finite mixtures of normal densities. For the family of skew densities we describe an iterative updating procedure to select the most suitable approximation and apply the method to two simulated posterior density functions.