Based on expert opinions, informative prior elicitation for the common Weibull lifetime distribution usually presents some difficulties since it requires to elicit a two-dimensional joint prior. We consider here a reliability framework where the available expert information states directly in terms of prior predictive values (lifetimes) and not parameter values, which are less intuitive. The novelty of our procedure is to weigh the expert information by the size m of a virtual sample yielding a similar information, the prior being seen as a reference posterior. Thus, the prior calibration by the Bayesian analyst, who has to moderate the subjective information with respect to the data information, is made simple. A main result is the full tractability of the prior under mild conditions, despite the conjugation issues encountered with the Weibull distribution. Besides, m is a practical focus point for discussion between analysts and experts, and a helpful parameter for leading sensitivity studies and reducing the potential imbalance in posterior selection between Bayesian Weibull models, which can be due to favoring arbitrarily a prior. The calibration of m is discussed and a real example is treated along the paper.
Subjective Bayesian prior distributions elicited from experts can be aggregated together to form group priors. This paper compares aggregated priors formed by Equal Weight Aggregation, the Classical Method, and the Sheffield Elicitation Framework to each other and individual expert priors, using an expert elicitation carried out for a clinical trial. Aggregation methods and individual expert prior distributions are compared using proper scoring rules to compare the informativeness and calibration of the distributions. The three aggregation methods outperform the individual experts, and the She…
This paper presents a complete methodology for Bayesian inference on a semi-Markov process, from the elicitation of the prior distribution, to the computation of posterior summaries, including a guidance for its JAGS implementation. The holding times (conditional on the transition between two given states) are assumed to be Weibull-distributed. We examine the elicitation of the joint prior density of the shape and scale parameters of the Weibull distributions, deriving a specific class of priors in a natural way, along with a method for the determination of hyperparameters based on ``learning …
In this note, the practical use of priors for Bayes estimators, of the two parameters of the Weibull reliability model, is discussed in a technological context. The meaning of the priors as expression of virtual data samples is analyzed. The implications of physics of failures are also highlighted. The whole analysis shows a rational way to convert a really available technological knowledge into prior information, effectively and without a long elicitation process. A large Monte Carlo study, on both complete and censored samples of small size, shows the good properties of some estimators, whic…
With the continued increase in the use of Bayesian methods in drug development, there is a need for statisticians to have tools to develop robust and defensible informative prior distributions. Whilst relevant empirical data should, where possible, provide the basis for such priors, it is often the case that limitations in data and/or our understanding may preclude direct construction of a data-based prior. Formal expert elicitation methods are a key technique that can be used to determine priors in these situations. Within GlaxoSmithKline (GSK), we have adopted a structured approach to prior …
The prior distribution for the unknown model parameters plays a crucial role in the process of statistical inference based on Bayesian methods. However, specifying suitable priors is often difficult even when detailed prior knowledge is available in principle. The challenge is to express quantitative information in the form of a probability distribution. Prior elicitation addresses this question by extracting subjective information from an expert and transforming it into a valid prior. Most existing methods, however, require information to be provided on the unobservable parameters, whose effe…
A major issue of extreme value analysis is the determination of the shape parameter $ξ$ common to Generalized Extreme Value (GEV) and Generalized Pareto (GP) distributions, which drives the tail behavior, and is of major impact on the estimation of return levels and periods. Many practitioners make the choice of a Bayesian framework to conduct this assessment for accounting of parametric uncertainties, which are typically high in such analyses characterized by a low number of observations. Nonetheless, such approaches can provide large credibility domains for $ξ$, including negative and positi…
Remote, bivariate expert elicitation to determine the prior probability distribution for sample size calculation in a Bayesian non-inferiority multicenter randomized controlled trial (Croup Dosing Trial)
The Hospital for Sick Children（多伦多儿童医院）；Children's Hospital Research Institute of Manitoba（曼尼托巴儿童医院研究所）；University of Manitoba（曼尼托巴大学）；College of Medicine, University of Saskatchewan（萨斯喀彻温大学医学院）；Jim Pattison Children’s Hospital（吉姆·帕蒂森儿童医院）；Saskatchewan Health Authority（萨斯喀彻温卫生管理局）；University of Ottawa（渥太华大学）；Children’s Hospital of Eastern Ontario（安大略东部儿童医院）
In this paper we study bayesian analysis of Modified Weibull distribution under progressively censored competing risk model. This study is made for progressively censored data. We use deterministic scan Gibbs sampling combined with slice sampling to generate from the posterior distribution. Posterior distribution is formed by taking prior distribution as reference prior. A real life data analysis is shown for illustrative purpose.
This paper presents a hierarchical Bayesian approach to the estimation of components' reliability (survival) using a Weibull model for each of them. The proposed method can be used to estimation with general survival censored data, because the estimation of a component's reliability in a series (parallel) system is equivalent to the estimation of its survival function with right- (left-) censored data. Besides the Weibull parametric model for reliability data, independent gamma distributions are considered at the first hierarchical level for the Weibull parameters and independent uniform distr…
The connection between Bayesian neural networks and Gaussian processes gained a lot of attention in the last few years. Hidden units are proven to follow a Gaussian process limit when the layer width tends to infinity. Recent work has suggested that finite Bayesian neural networks may outperform their infinite counterparts because they adapt their internal representations flexibly. To establish solid ground for future research on finite-width neural networks, our goal is to study the prior induced on hidden units. Our main result is an accurate description of hidden units tails which shows tha…
Estimation of reliability and hazard rate is one of the most important problems raised in many applications especially in engineering studies as well as human lifetime. In this regard, different methods of estimation have been used. Each method exploits various tools and suffers from problems such as complexity of computations, low precision, and so forth. This study is employed the E-Bayesian method, for estimating the parameter and survival functions of the Weibull Generalized Exponential distribution. The estimators are obtained under squared error and LINEX loss functions under progressive…
This paper develops a Bayesian control chart for the percentiles of the Weibull distribution, when both its in-control and out-of-control parameters are unknown. The Bayesian approach enhances parameter estimates for small sample sizes that occur when monitoring rare events as in high-reliability applications or genetic mutations. The chart monitors the parameters of the Weibull distribution directly, instead of transforming the data as most Weibull-based charts do in order to comply with their normality assumption. The chart uses the whole accumulated knowledge resulting from the likelihood o…
Eliciting informative prior distributions for Bayesian inference can often be complex and challenging. While popular methods rely on asking experts probability based questions to quantify uncertainty, these methods are not without their drawbacks and many alternative elicitation methods exist. This paper explores methods for eliciting informative priors categorized by type and briefly discusses their strengths and limitations. Most of the review literature in this field focuses on a particular type of elicitation approach. The primary aim of this work, however, is to provide a more complete ye…
A method for conducting Bayesian elicitation and learning in risk assessment is presented. It assumes that the risk process can be described as a fault tree. This is viewed as a belief network, for which prior distributions on primary event probabilities are elicited by means of a pairwise comparison approach. A Bayesian updating procedure, following observation of some or all of the events in the fault tree, is described. The application is illustrated through the motivating example of risk assessment of spacecraft explosion during controlled re-entry.
University of Edinburgh（爱丁堡大学）；Microsoft Research（微软研究院）；Stanford University（斯坦福大学）
A good clustering can help a data analyst to explore and understand a data set, but what constitutes a good clustering may depend on domain-specific and application-specific criteria. These criteria can be difficult to formalize, even when it is easy for an analyst to know a good clustering when she sees one. We present a new approach to interactive clustering for data exploration, called \ciif, based on a particularly simple feedback mechanism, in which an analyst can choose to reject individual clusters and request new ones. The new clusters should be different from previously rejected clust…
University of Edinburgh（爱丁堡大学）；Microsoft Research（微软研究院）；Massachusetts Institute of Technology（麻省理工学院）；Harvard University（哈佛大学）；Twitter（推特）
A good clustering can help a data analyst to explore and understand a data set, but what constitutes a good clustering may depend on domain-specific and application-specific criteria. These criteria can be difficult to formalize, even when it is easy for an analyst to know a good clustering when they see one. We present a new approach to interactive clustering for data exploration called TINDER, based on a particularly simple feedback mechanism, in which an analyst can reject a given clustering and request a new one, which is chosen to be different from the previous clustering while fitting th…
Background: Pairwise and network meta-analyses using fixed effect and random effects models are commonly applied to synthesise evidence from randomised controlled trials. The models differ in their assumptions and the interpretation of the results. The model choice depends on the objective of the analysis and knowledge of the included studies. Fixed effect models are often used because there are too few studies with which to estimate the between-study standard deviation from the data alone. Objectives: The aim is to propose a framework for eliciting an informative prior distribution for the be…
The aim of the present work is to investigate the performances of a specific Bayesian control chart used to compare two processes. The chart monitors the ratio of the percentiles of a key characteristic associated with the processes. The variability of such a characteristic is modeled via the Weibull distribution and a practical Bayesian approach to deal with Weibull data is adopted. The percentiles of the two monitored processes are assumed to be independent random variables. The Weibull distributions of the key characteristic of both processes are assumed to have the same and stable shape pa…
The hybrid censoring is a mixture of Type I and Type II censoring schemes. This paper presents the statistical inferences of the Inverse Weibull distribution when the data are Type-I hybrid censored. First we consider the maximum likelihood estimators of the unknown parameters. It is observed that the maximum likelihood estimators can not be obtained in closed form. We further obtain the Bayes estimators and the corresponding highest posterior density credible intervals of the unknown parameters under the assumption of independent gamma priors using the importance sampling procedure. We also c…
The aim of this work is to study the problem of prior elicitation for the Mallows model with Spearman's distance, a popular distance-based model for rankings or permutation data. Previous Bayesian inference for such model has been limited to the use of the uniform prior over the space of permutations. We present a novel strategy to elicit subjective prior beliefs on the location parameter of the model, discussing the interpretation of hyper-parameters and the implication of prior choices for the posterior analysis.
Bayesian P-splines assume an intrinsic Gaussian Markov random field prior on the spline coefficients, conditional on a precision hyper-parameter $τ$. Prior elicitation of $τ$ is difficult. To overcome this issue we aim to building priors on an interpretable property of the model, indicating the complexity of the smooth function to be estimated. Following this idea, we propose Penalized Complexity (PC) priors for the number of effective degrees of freedom. We present the general ideas behind the construction of these new PC priors, describe their properties and show how to implement them in P-s…
Discrete data are collected in many application areas and are often characterised by highly skewed and power-lawlike distributions. An example of this, which is considered in this paper, is the number of visits to a specialist, often taken as a measure of demand in healthcare. A discrete Weibull regression model was recently proposed for regression problems with a discrete response and it was shown to possess two important features: the ability to capture over and under-dispersion simultaneously and a closed-form analytical expression of the quantiles of the conditional distribution. In this p…
We propose a Bayesian approach using improper priors for hierarchical linear mixed models with flexible random effects and residual error distributions. The error distribution is modelled using scale mixtures of normals, which can capture tails heavier than those of the normal distribution. This generalisation is useful to produce models that are robust to the presence of outliers. The case of asymmetric residual errors is also studied. We present general results for the propriety of the posterior that also cover cases with censored observations, allowing for the use of these models in the con…
The elicitation of power priors, based on the availability of historical data, is realized by raising the likelihood function of the historical data to a fractional power δ, which quantifies the degree of discounting of the historical information in making inference with the current data. When δ is not pre-specified and is treated as random, it can be estimated from the data using Bayesian updating paradigm. However, in the original form of the joint power prior Bayesian approach, certain positive constants before the likelihood of the historical data could be multiplied when different setting…
Vrije Universiteit Brussel（布鲁塞尔自由大学）；University of Oxford（牛津大学）；Vrije Universiteit Amsterdam（阿姆斯特丹自由大学）；City of Amsterdam, Signal Control Design and Analysis（阿姆斯特丹市信号控制设计与分析部门）；Delft University of Technology（代尔夫特理工大学）
In multi-objective decision planning and learning, much attention is paid to producing optimal solution sets that contain an optimal policy for every possible user preference profile. We argue that the step that follows, i.e, determining which policy to execute by maximising the user's intrinsic utility function over this (possibly infinite) set, is under-studied. This paper aims to fill this gap. We build on previous work on Gaussian processes and pairwise comparisons for preference modelling, extend it to the multi-objective decision support scenario, and propose new ordered preference elici…
We derive the Jeffreys prior for the parameter of the Multivariate Ewens Distribution and study some of its properties. In particular, we show that this prior is proper and has no finite moments. We also investigate the impact of this default prior on the a priori distribution of the number of species and the a priori probability of discovery of a new species, which are usually employed in subjective prior elicitation. The effect of the Jeffreys prior for posterior inference is illustrated using examples arising in the context of inference for species sampling models and Dirichlet process mixt…
This paper develops Bayesian sample size formulae for experiments comparing two groups. We assume the experimental data will be analysed in the Bayesian framework, where pre-experimental information from multiple sources can be represented into robust priors. In particular, such robust priors account for preliminary belief about the pairwise commensurability between parameters that underpin the historical and new experiments, to permit flexible borrowing of information. Averaged over the probability space of the new experimental data, appropriate sample sizes are found according to criteria th…
The use of flat or weakly informative priors is popular due to the objective a priori belief in the absence of strong prior information. In the case of the Weibull model the improper uniform, equal parameter gamma and joint Jeffrey's priors for the shape parameter are popular choices. The effects and behaviors of these priors have yet to be established from a modeling viewpoint, especially their ability to reduce to the simpler exponential model. In this work we propose a new principled prior for the shape parameter of the Weibull model, originating from a prior on the distance function, and a…
In wavelet shrinkage and thresholding, most of the standard techniques do not consider information that wavelet coefficients might be bounded, although information about bounded energy in signals can be readily available. To address this, we present a Bayesian approach for shrinkage of bounded wavelet coefficients in the context of non-parametric regression. We propose the use of a zero-contaminated beta distribution with a support symmetric around zero as the prior distribution for the location parameter in the wavelet domain in models with additive gaussian errors. The hyperparameters of the…
Analysis of competing risks data plays an important role in the lifetime data analysis. Recently Feizjavadian and Hashemi (Computational Statistics and Data Analysis, vol. 82, 19-34, 2015) provided a classical inference of a competing risks data set using four-parameter Marshall-Olkin bivariate Weibull distribution when the failure of an unit at a particular time point can happen due to more than one cause. The aim of this paper is to provide the Bayesian analysis of the same model based on a very flexible Gamma-Dirichlet prior on the scale parameters. It is observed that the Bayesian inferenc…
In reliability engineering, data about failure events is often scarce. To arrive at meaningful estimates for the reliability of a system, it is therefore often necessary to also include expert information in the analysis, which is straightforward in the Bayesian approach by using an informative prior distribution. A problem called prior-data conflict then can arise: observed data seem very surprising from the viewpoint of the prior, i.e., information from data is in conflict with prior assumptions. Models based on conjugate priors can be insensitive to prior-data conflict, in the sense that th…
The power prior is a popular tool for constructing informative prior distributions based on historical data. The method consists of raising the likelihood to a discounting factor in order to control the amount of information borrowed from the historical data. It is customary to perform a sensitivity analysis reporting results for a range of values of the discounting factor. However, one often wishes to assign it a prior distribution and estimate it jointly with the parameters, which in turn necessitates the computation of a normalising constant. In this paper we are concerned with how to recyc…
Across the empirical sciences, few statistical procedures rival the popularity of the frequentist t-test. In contrast, the Bayesian versions of the t-test have languished in obscurity. In recent years, however, the theoretical and practical advantages of the Bayesian t-test have become increasingly apparent and various Bayesian t-tests have been proposed, both objective ones (based on general desiderata) and subjective ones (based on expert knowledge). Here we propose a flexible t-prior for standardized effect size that allows computation of the Bayes factor by evaluating a single numerical in…
Cluster analysis aims at partitioning data into groups or clusters. In applications, it is common to deal with problems where the number of clusters is unknown. Bayesian mixture models employed in such applications usually specify a flexible prior that takes into account the uncertainty with respect to the number of clusters. However, a major empirical challenge involving the use of these models is in the characterisation of the induced prior on the partitions. This work introduces an approach to compute descriptive statistics of the prior on the partitions for three selected Bayesian mixture …
This paper develops a Bayesian approach for assessing equivalence and non-inferiority hypotheses in two-arm trials using relative belief ratios. A relative belief ratio is a measure of statistical evidence and can indicate evidence either for or against a hypothesis. In addition to the relative belief ratio, we also compute a measure of the strength of this evidence as a calibration of the relative belief ratio. Furthermore, we make use of the relative belief ratio as a measure of evidence, to assess whether a given prior induces bias either for or against a hypothesis. Prior elicitation, mode…
In this paper we show that there is a link between approximate Bayesian methods and prior robustness. We show that what is typically recognized as an approximation to the likelihood, either due to the simulated data as in the Approximate Bayesian Computation (ABC) methods or due to the functional approximation to the likelihood, can instead also be viewed upon as an implicit exercise in prior robustness. We first define two new classes of priors for the cases where the sufficient statistics is available, establish their mathematical properties and show, for a simple illustrative example, that …
Bayesian model comparison requires the specification of a prior distribution on the parameter space of each candidate model. In this connection two concerns arise: on the one hand the elicitation task rapidly becomes prohibitive as the number of models increases; on the other hand numerous prior specifications can only exacerbate the well-known sensitivity to prior assignments, thus producing less dependable conclusions. Within the subjective framework, both difficulties can be counteracted by linking priors across models in order to achieve simplification and compatibility; we discuss links w…
We propose a comprehensive Bayesian hierarchical model for monthly maxima of instantaneous flow in river catchments. The Gumbel distribution is used as the probabilistic model for the observations, which are assumed to come from several catchments. Our suggested latent model is Gaussian and designed for monthly maxima, making better use of the data than the standard approach using annual maxima. At the latent level, linear mixed models are used for both the location and scale parameters of the Gumbel distribution, accounting for seasonal dependence and covariates from the catchments. The speci…
Methods are developed for eliciting a Dirichlet prior based upon bounds on the individual probabilities that hold with virtual certainty. This approach to selecting a prior is applied to a contingency table problem where it is demonstrated how to assess the bias in the prior as well as how to check for prior-data conflict. It is shown that the assessment of a hypothesis via relative belief can easily take into account what it means for the falsity of the hypothesis to correspond to a difference of practical importance and provide evidence in favor of a hypothesis.
We propose a cautious Bayesian variable selection routine by investigating the sensitivity of a hierarchical model, where the regression coefficients are specified by spike and slab priors. We exploit the use of latent variables to understand the importance of the co-variates. These latent variables also allow us to obtain the size of the model space which is an important aspect of high dimensional problems. In our approach, instead of fixing a single prior, we adopt a specific type of robust Bayesian analysis, where we consider a set of priors within the same parametric family to specify the …
