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.     

Bayesiam simultaneous estimation for several multinomial distributions

A Bayesian method is proposed for estimating the cell probabilities of several multinomial distributions. Parameters of different distributions are taken to be a priori exchangeable. The prior specification is based upon mixtures of a hierarchical distribution, referred to as the multivariate “Dirichlet-Dirichlet” distribution. The analysis is facilitated by a multinomial approximation relating to the multinomial-Dirichlet distribution. The posterior estimates depend upon measures of entropy for the various distributions and shrink the individual observed proportions towards values obtained by pooling the data across the distributions. As well as incorporating prior information they are particularly useful when some of the cell frequencies are zero. We use them to investigate a numerical classification of males of various vocations, according to cause of death.     

Adaptive Bayesian Procedures Using Random Series Priors

We consider a general class of prior distributions for nonparametric Bayesian estimation which uses finite random series with a random number of terms. A prior is constructed through distributions on the number of basis functions and the associated coefficients. We derive a general result on adaptive posterior contraction rates for all smoothness levels of the target function in the true model by constructing an appropriate ‘sieve’ and applying the general theory of posterior contraction rates. We apply this general result on several statistical problems such as density estimation, various nonparametric regressions, classification, spectral density estimation and functional regression. The prior can be viewed as an alternative to the commonly used Gaussian process prior, but properties of the posterior distribution can be analysed by relatively simpler techniques. An interesting approximation property of B‐spline basis expansion established in this paper allows a canonical choice of prior on coefficients in a random series and allows a simple computational approach without using Markov chain Monte Carlo methods. A simulation study is conducted to show that the accuracy of the Bayesian estimators based on the random series prior and the Gaussian process prior are comparable. We apply the method on Tecator data using functional regression models.     

Influential Observations in the Functional Measurement Error Model

In this work we propose Bayesian measures to quantify the influence of observations on the structural parameters of the simple measurement error model (MEM). Different influence measures, like those based on q-divergence between posterior distributions and Bayes risk, are studied to evaluate the influence. A strategy based on the perturbation function and MCMC samples is used to compute these measures. The samples from the posterior distributions are obtained by using the Metropolis-Hastings algorithm and assuming specific proper prior distributions. The results are illustrated with an application to a real example modeled with MEM in the literature.     

Bayesian analysis of panel data using an MTAR model

Bayesian analysis of panel data using a class of momentum threshold autoregressive (MTAR) models is considered. Posterior estimation of parameters of the MTAR models is done by using a simple Markov Chain Monte Carlo (MCMC) algorithm. Selection of appropriate differenced variables, test for asymmetry and unit roots are recast as model selections and a simple way of computing posterior probabilities of the candidate models is proposed. The proposed method is applied to the yearly unemployment rates of 51 US states and the results show strong evidence of stationarity and asymmetry.     

Bayesian inference for the variance components in general mixed linear models

The general mixed linear model, containing both the fixed and random effects, is considered. Using gamma priors for the variance components, the conditional posterior distributions of the fixed effects and the variance components, conditional on the random effects, are obtained. Using the normal approximation for the multiple t distribution, approximations are obtained for the posterior distributions of the variance components in infinite series form. The same approximation Is used to obtain closed expressions for the moments of the variance components. An example is considered to illustrate the procedure and a numerical study examines the closeness of the approximations.     

Applied state space modelling of non-Gaussian time series using integration-based Kalman filtering

The main topic of the paper is on-line filtering for non-Gaussian dynamic (state space) models by approximate computation of the first two posterior moments using efficient numerical integration. Based on approximating the prior of the state vector by a normal density, we prove that the posterior moments of the state vector are related to the posterior moments of the linear predictor in a simple way. For the linear predictor Gauss-Hermite integration is carried out with automatic reparametrization based on an approximate posterior mode filter. We illustrate how further topics in applied state space modelling, such as estimating hyperparameters, computing model likelihoods and predictive residuals, are managed by integration-based Kalman-filtering. The methodology derived in the paper is applied to on-line monitoring of ecological time series and filtering for small count data.     

Bayesian analysis of outlier problems using divergence measures

A Bayesian approach is presented for detecting influential observations using general divergence measures on the posterior distributions. A sampling-based approach using a Gibbs or Metropolis-within-Gibbs method is used to compute the posterior divergence measures. Four specific measures are proposed, which convey the effects of a single observation or covariate on the posterior. The technique is applied to a generalized linear model with binary response data, an overdispersed model and a nonlinear model. An asymptotic approximation using Laplace method to obtain the posterior divergence is also briefly discussed.     

Bayesian inference for smooth transition autoregressive (STAR) model: A prior sensitivity analysis

The main aim of this paper is to perform sensitivity analysis to the specification of prior distributions in a Bayesian analysis setting of STAR models. To achieve this aim, the joint posterior distribution of model order, coefficient, and implicit parameters in the logistic STAR model is first being presented. The conditional posterior distributions are then shown, followed by the design of a posterior simulator using a combination of Metropolis-Hastings, Gibbs Sampler, RJMCMC, and Multiple Try Metropolis algorithms, respectively. Following this, simulation studies and a case study on the prior sensitivity for the implicit parameters are being detailed at the end.     

Multivariate Bayesian variable selection and prediction

The multivariate regression model is considered with p regressors. A latent vector with p binary entries serves to identify one of two types of regression coefficients: those close to 0 and those not. Specializing our general distributional setting to the linear model with Gaussian errors and using natural conjugate prior distributions, we derive the marginal posterior distribution of the binary latent vector. Fast algorithms aid its direct computation, and in high dimensions these are supplemented by a Markov chain Monte Carlo approach to sampling from the known posterior distribution. Problems with hundreds of regressor variables become quite feasible. We give a simple method of assigning the hyperparameters of the prior distribution. The posterior predictive distribution is derived and the approach illustrated on compositional analysis of data involving three sugars with 160 near infrared absorbances as regressors.     

Disaggregation and Forecasting: A Bayesian Analysis

The problem of temporal disaggregation of time series is analyzed by means of Bayesian methods. The disaggregated values are obtained through a posterior distribution derived by using a diffuse prior on the parameters. Further analysis is carried out assuming alternative conjugate priors. The means of the different posterior distributions are shown to be equivalent to some sampling theory results. Bayesian prediction intervals are obtained. Forecasts for future disaggregated values are derived assuming a conjugate prior for the future aggregated value.     

Reparameterization strategies for hidden Markov models and Bayesian approaches to maximum likelihood estimation

This paper synthesizes a global approach to both Bayesian and likelihood treatments of the estimation of the parameters of a hidden Markov model in the cases of normal and Poisson distributions. The first step of this global method is to construct a non-informative prior based on a reparameterization of the model; this prior is to be considered as a penalizing and bounding factor from a likelihood point of view. The second step takes advantage of the special structure of the posterior distribution to build up a simple Gibbs algorithm. The maximum likelihood estimator is then obtained by an iterative procedure replicating the original sample until the corresponding Bayes posterior expectation stabilizes on a local maximum of the original likelihood function.     

Higher-order Bayesian Approximations for Pseudo-posterior Distributions

The theory of higher-order asymptotics provides accurate approximations to posterior distributions for a scalar parameter of interest, and to the corresponding tail area, for practical use in Bayesian analysis. The aim of this article is to extend these approximations to pseudo-posterior distributions, e.g., posterior distributions based on a pseudo-likelihood function and a suitable prior, which are proved to be particularly useful when the full likelihood is analytically or computationally infeasible. In particular, from a theoretical point of view, we derive the Laplace approximation for a pseudo-posterior distribution, and for the corresponding tail area, for a scalar parameter of interest, also in the presence of nuisance parameters. From a computational point of view, starting from these higher-order approximations, we discuss the higher-order tail area (HOTA) algorithm useful to approximate marginal posterior distributions, and related quantities. Compared to standard Markov chain Monte Carlo methods, the main advantage of the HOTA algorithm is that it gives independent samples at a negligible computational cost. The relevant computations are illustrated by two examples.     

Bayesian estimates in a one-way ANOVA random effects model

Bayesian estimators of variance components are developed, based on posterior mean and posterior mode, respectively, in a one-way ANOVA random effects model with independent prior distributions. The formulas for the proposed estimators are simple. The estimators give sensible results for 'badly-behaved' datasets, where the standard unbiased estimates are negative. They are markedly robust as compared to the existing estimators such as the maximum likelihood estimators and the maximum posterior density estimators.     

Bayesian robustness in meta‐analysis for studies with zero responses

Statistical meta‐analysis is mostly carried out with the help of the random effect normal model, including the case of discrete random variables. We argue that the normal approximation is not always able to adequately capture the underlying uncertainty of the original discrete data. Furthermore, when we examine the influence of the prior distributions considered, in the presence of rare events, the results from this approximation can be very poor. In order to assess the robustness of the quantities of interest in meta‐analysis with respect to the choice of priors, this paper proposes an alternative Bayesian model for binomial random variables with several zero responses. Particular attention is paid to the coherence between the prior distributions of the study model parameters and the meta‐parameter. Thus, our method introduces a simple way to examine the sensitivity of these quantities to the structure dependence selected for study. For illustrative purposes, an example with real data is analysed, using the proposed Bayesian meta‐analysis model for binomial sparse data. Copyright © 2016 John Wiley & Sons, Ltd.     

Bayesian analysis for a change in the intercept of simple linear regression

A Bayesian approach is considered to detect a change-point in the intercept of simple linear regression. The Jeffreys noninformative prior is employed and compared with the uniform prior in Bayesian analysis. The marginal posterior distributions of the change-point, the amount of shift and the slope are derived. Mean square errors, mean absolute errors and mean biases of some Bayesian estimates are considered by Monte Carlo methad and some numerical results are also shown.     

Data augmentation,frequentist estimation,and the Bayesian analysis of multinomial logit models

This article describes a convenient method of selecting Metropolis– Hastings proposal distributions for multinomial logit models. There are two key ideas involved. The first is that multinomial logit models have a latent variable representation similar to that exploited by Albert and Chib (J Am Stat Assoc 88:669–679, 1993) for probit regression. Augmenting the latent variables replaces the multinomial logit likelihood function with the complete data likelihood for a linear model with extreme value errors. While no conjugate prior is available for this model, a least squares estimate of the parameters is easily obtained. The asymptotic sampling distribution of the least squares estimate is Gaussian with known variance. The second key idea in this paper is to generate a Metropolis–Hastings proposal distribution by conditioning on the estimator instead of the full data set. The resulting sampler has many of the benefits of so-called tailored or approximation Metropolis–Hastings samplers. However, because the proposal distributions are available in closed form they can be implemented without numerical methods for exploring the posterior distribution. The algorithm converges geometrically ergodically, its computational burden is minor, and it requires minimal user input. Improvements to the sampler’s mixing rate are investigated. The algorithm is also applied to partial credit models describing ordinal item response data from the 1998 National Assessment of Educational Progress. Its application to hierarchical models and Poisson regression are briefly discussed.     

A Bayesian analysis for the multivariate point null testing problem

A Bayesian test for the point null testing problem in the multivariate case is developed. A procedure to get the mixed distribution using the prior density is suggested. For comparisons between the Bayesian and classical approaches, lower bounds on posterior probabilities of the null hypothesis, over some reasonable classes of prior distributions, are computed and compared with the p-value of the classical test. With our procedure, a better approximation is obtained because the p-value is in the range of the Bayesian measures of evidence.     

A prior for the variance in hierarchical models

The choice of prior distributions for the variances can be important and quite difficult in Bayesian hierarchical and variance component models. For situations where little prior information is available, a ‘nonin-formative’ type prior is usually chosen. ‘Noninformative’ priors have been discussed by many authors and used in many contexts. However, care must be taken using these prior distributions as many are improper and thus, can lead to improper posterior distributions. Additionally, in small samples, these priors can be ‘informative’. In this paper, we investigate a proper ‘vague’ prior, the uniform shrinkage prior (Strawder-man 1971; Christiansen & Morris 1997). We discuss its properties and show how posterior distributions for common hierarchical models using this prior lead to proper posterior distributions. We also illustrate the attractive frequentist properties of this prior for a normal hierarchical model including testing and estimation. To conclude, we generalize this prior to the multivariate situation of a covariance matrix.     

Bayesian semiparametric modeling for stochastic precedence,with applications in epidemiology and survival analysis

We propose a prior probability model for two distributions that are ordered according to a stochastic precedence constraint, a weaker restriction than the more commonly utilized stochastic order constraint. The modeling approach is based on structured Dirichlet process mixtures of normal distributions. Full inference for functionals of the stochastic precedence constrained mixture distributions is obtained through a Markov chain Monte Carlo posterior simulation method. A motivating application involves study of the discriminatory ability of continuous diagnostic tests in epidemiologic research. Here, stochastic precedence provides a natural restriction for the distributions of test scores corresponding to the non-infected and infected groups. Inference under the model is illustrated with data from a diagnostic test for Johne’s disease in dairy cattle. We also apply the methodology to the comparison of survival distributions associated with two distinct conditions, and illustrate with analysis of data on survival time after bone marrow transplantation for treatment of leukemia.