Objective Bayesian analysis for the Student-t regression model

The choice of the prior distribution is a key aspect of Bayesian analysis. The spatial Student-t regression model poses some challenges when eliciting priors. It is well known that the propriety of the posterior distribution over objective priors is not always guaranteed, whereas the use of proper prior distributions may dominate and bias the posterior analysis. In this paper, we show the conditions under which our proposed reference prior and the two introduced Jeffreys priors yield to a proper posterior distribution. Simulation studies and a real data application are used in order to evaluate the performance of the reference prior.

The Generalized Pareto (GP) and Generalized extreme value (GEV) distributions play an important role in extreme value analyses, as models for threshold excesses and block maxima respectively. For each of these distributions we consider Bayesian inference using "reference" prior distributions (in the general sense of priors constructed using formal rules) for the model parameters, specifically a Jeffreys prior, the maximal data information (MDI) prior and independent uniform priors on separate model parameters. We investigate the important issue of whether these improper priors lead to proper posterior distributions. We show that, in the GP and GEV cases, the MDI prior, unless modified, never yields a proper posterior and that in the GEV case this also applies to the Jeffreys prior. We also show that a sample size of three (four) is sufficient for independent uniform priors to yield a proper posterior distribution in the GP (GEV) case.

Bayesian statistical methods have recently been combined with conventional methods for fisheries stock assessment (e.g. catch-age analysis) to provide a conceptually elegant approach for providing fishery management advice under uncertainty. Uncertainties in the advice provided can be conveyed using posterior probability distributions (or “posteriors”) for the potential outcomes of each policy option. Posteriors can be estimated using data (e.g. catch-age data and relative abundance indices) for the fish population of interest and prior probability distributions for population model parameters (e.g. stock-recruit function parameters) based on data from similar fish populations. Despite growing interest, Bayesian methods remain accessible to relatively few. To increase the accessibility of these methods, the conceptual basis for Bayesian statistical estimation is reviewed and set in the context of fisheries stock assessment. The use of Bayesian methods is illustrated by fitting a logistic model to relative abundance indices for Namibian hake (Merlucius capensis and M. paradoxus) and presenting a decision analysis of alternative harvest policy options. Some alternative approaches are outlined for constructing prior and posterior probability distributions and some recent applications in fisheries. Some of the problems that can be encountered while implementing Bayesian methods are also discussed.

In this paper, objective Bayesian analysis for the Student-t linear regression model with unknown degrees of freedom is studied. The reference priors under all the possible group orderings for the parameters in the model are derived. The posterior propriety under each reference prior is validated by considering a larger class of priors. Simulation studies are carried out to investigate the frequentist properties of Bayesian estimators based on the reference priors. Finally, the Bayesian approach is applied to two real data sets.

Bayesian Analysis of the Pragmatic Airway Resuscitation Trial

The authors consider Bayesian analysis for continuous‐time Markov chain models based on a conditional reference prior. For such models, inference of the elapsed time between chain observations depends heavily on the rate of decay of the prior as the elapsed time increases. Moreover, improper priors on the elapsed time may lead to improper posterior distributions. In addition, an infinitesimal rate matrix also characterizes this class of models. Experts often have good prior knowledge about the parameters of this matrix. The authors show that the use of a proper prior for the rate matrix parameters together with the conditional reference prior for the elapsed time yields a proper posterior distribution. The authors also demonstrate that, when compared to analyses based on priors previously proposed in the literature, a Bayesian analysis on the elapsed time based on the conditional reference prior possesses better frequentist properties. The type of prior thus represents a better default prior choice for estimation software.

Bayesian inference plays an essential role in the development of mathematical theory, as an important component of statistical methods. The prior distribution, the likelihood function and the posterior distribution are used in Bayesian inference. A combination of the prior distribution and the likelihood function will represent the posterior distribution, where the background information decides the prior distribution, and the background information and the observed data together form the likelihood function. This paper presents the background of Bayesian inference, the main developments, and the challenges of computing the posterior distribution. It focuses on six MCMC-based methods, such as the Metropolis-Hastings algorithm, Gibbs sampler, Reversible Jump MCMC, Hamiltonian Monte Carlo, Adaptive Metropolis, and preconditioned Crank-Nicolson. The advantages, limitations and applications of each algorithm are also briefly described.

Bayesian system reliability assessment method with maximum entropy as prior distribution is developed in the present analysis. The system reliability assessment model is established, and the method and process of system reliability assessment are put forward. The method is based on the theory of information entropy. The probability density function and distribution function of system reliability parameters are given by using maximum entropy method. Prior distribution of Bayesian analysis method is the distribution function. Likelihood function is given by a small amount of field test data. According to Bayesian methods, posterior distribution is determined by using prior distribution and likelihood function. The point estimation and interval estimation of system reliability parameters are calculated in order to evaluate the system reliability. The method solves better the problem of system reliability assessment for small sample cases, and is an optimum method for reliability assessment of large and complex systems.

BackgroundBayesian phylogenetic inference holds promise as an alternative to maximum likelihood, particularly for large molecular-sequence data sets. We have investigated the performance of Bayesian inference with empirical and simulated protein-sequence data under conditions of relative branch-length differences and model violation.ResultsWith empirical protein-sequence data, Bayesian posterior probabilities provide more-generous estimates of subtree reliability than does the nonparametric bootstrap combined with maximum likelihood inference, reaching 100% posterior probability at bootstrap proportions around 80%. With simulated 7-taxon protein-sequence datasets, Bayesian posterior probabilities are somewhat more generous than bootstrap proportions, but do not saturate. Compared with likelihood, Bayesian phylogenetic inference can be as or more robust to relative branch-length differences for datasets of this size, particularly when among-sites rate variation is modeled using a gamma distribution. When the (known) correct model was used to infer trees, Bayesian inference recovered the (known) correct tree in 100% of instances in which one or two branches were up to 20-fold longer than the others. At ratios more extreme than 20-fold, topological accuracy of reconstruction degraded only slowly when only one branch was of relatively greater length, but more rapidly when there were two such branches. Under an incorrect model of sequence change, inaccurate trees were sometimes observed at less extreme branch-length ratios, and (particularly for trees with single long branches) such trees tended to be more inaccurate. The effect of model violation on accuracy of reconstruction for trees with two long branches was more variable, but gamma-corrected Bayesian inference nonetheless yielded more-accurate trees than did either maximum likelihood or uncorrected Bayesian inference across the range of conditions we examined. Assuming an exponential Bayesian prior on branch lengths did not improve, and under certain extreme conditions significantly diminished, performance. The two topology-comparison metrics we employed, edit distance and Robinson-Foulds symmetric distance, yielded different but highly complementary measures of performance.ConclusionsOur results demonstrate that Bayesian inference can be relatively robust against biologically reasonable levels of relative branch-length differences and model violation, and thus may provide a promising alternative to maximum likelihood for inference of phylogenetic trees from protein-sequence data.

Previous research showed that employing results from meta-analyses of relevant previous fMRI studies can improve the performance of voxelwise Bayesian second-level fMRI analysis. In this process, prior distributions for Bayesian analysis can be determined by information acquired from the meta-analyses. However, only image-based meta-analysis, which is not widely accessible to fMRI researchers due to the lack of shared statistical images, was tested in the previous study, so the applicability of the prior determination method proposed by the previous study might be limited. In the present study, whether determining prior distributions based on coordinate-based meta-analysis, which is widely accessible to researchers, can also improve the performance of Bayesian analysis, was examined. Three different types of coordinate-based meta-analyses, BrainMap and Ginger ALE, and NeuroQuery, were tested as information sources for prior determination. Five different datasets addressing three task conditions, i.e., working memory, speech, and face processing, were analyzed via Bayesian analysis with a meta-analysis informed prior distribution, Bayesian analysis with a default Cauchy prior adjusted for multiple comparisons, and frequentist analysis with familywise error correction. The findings from the aforementioned analyses suggest that use of coordinate-based meta-analysis also significantly enhanced performance of Bayesian analysis as did image-based meta-analysis.

Equality and inequality constrained multivariate linear models: Objective model selection using constrained posterior priors

BackgroundIn epidemiological studies explanatory variables are frequently subject to measurement error. The aim of this paper is to develop a Bayesian method to correct for measurement error in multiple continuous exposures in individually matched case-control studies. This is a topic that has not been widely investigated. The new method is illustrated using data from an individually matched case-control study of the association between thyroid hormone levels during pregnancy and exposure to perfluorinated acids. The objective of the motivating study was to examine the risk of maternal hypothyroxinemia due to exposure to three perfluorinated acids measured on a continuous scale. Results from the proposed method are compared with those obtained from a naive analysis.MethodsUsing a Bayesian approach, the developed method considers a classical measurement error model for the exposures, as well as the conditional logistic regression likelihood as the disease model, together with a random-effect exposure model. Proper and diffuse prior distributions are assigned, and results from a quality control experiment are used to estimate the perfluorinated acids' measurement error variability. As a result, posterior distributions and 95% credible intervals of the odds ratios are computed. A sensitivity analysis of method's performance in this particular application with different measurement error variability was performed.ResultsThe proposed Bayesian method to correct for measurement error is feasible and can be implemented using statistical software. For the study on perfluorinated acids, a comparison of the inferences which are corrected for measurement error to those which ignore it indicates that little adjustment is manifested for the level of measurement error actually exhibited in the exposures. Nevertheless, a sensitivity analysis shows that more substantial adjustments arise if larger measurement errors are assumed.ConclusionsIn individually matched case-control studies, the use of conditional logistic regression likelihood as a disease model in the presence of measurement error in multiple continuous exposures can be justified by having a random-effect exposure model. The proposed method can be successfully implemented in WinBUGS to correct individually matched case-control studies for several mismeasured continuous exposures under a classical measurement error model.

Beta distributions with both parameters equal to 0, ½, or 1 are the usual choices for “noninformative” priors for Bayesian estimation of the binomial parameter. However, as illustrated by two examples from the Bayesian literature, care needs to be taken with parameter values below 1, both for noninformative and informative priors, as such priors concentrate their mass close to 0 and/or 1 and can suppress the importance of the observed data. These examples concern the case of no successes (or failures) and illustrate the informativeness of the Jeffreys prior usually recommended as the “consensus prior.” In particular, the second example suggests that when the binomial parameter is known to be very small, an informative prior from the beta (1, b) family (b > 1) seems appropriate, while a beta (a, b) with a < 1 can be too informative. It is thus argued that sensitivity analysis of an informative prior should be based on a consensus posterior corresponding to the Bayes–Laplace prior rather than the Jeffreys prior.

A fundamental assumption underlying latent class analysis (LCA) is that class indicators are conditionally independent of each other, given latent class membership. Bayesian LCA enables researchers to detect and accommodate violations of this assumption by estimating any number of correlations among indicators with proper prior distributions. However, little is known about how the choice of prior may affect the performance of Bayesian LCA. This article presents a Monte Carlo simulation study that investigates (1) the utility of priors in a range of prior variances (i.e., strongly non-informative to strongly informative priors) in terms of Type I error and power for detecting conditional dependence and (2) the influence of imposing approximate independence on model fit of Bayesian LCA. Simulation results favored the use of a weakly informative prior with large variance–model fit (posterior predictive p–value) was always satisfactory when the class indicators were either independent or dependent. Based on the current findings and the additional literature, this article offers methodological guidelines and suggestions for applied researchers.

Multiply imputing missing values in data sets with mixed measurement scales using a sequence of generalised linear models