Objective Bayesian Analysis for the Student-t Linear Regression

ABSTRACTReference analysis is an objective Bayesian approach to finding noninformative prior distributions. For various models involving nuisance parameters the reference prior has been shown to be superior to the multiparameter Jeffreys prior. In this article, the performance of the reference prior is evaluated for models that are defined at the extremes of parameter ranges, in which case reference and Jeffreys priors are shown to be potentially informative. Two quantities of interest are analyzed: the ratio of two multinomial parameters and the ratio of two independent binomial parameters, where the latter is just one example of inference based on the common 2 × 2 contingency table. For these two specific examples, we show that reference and/or Jeffreys priors lead to overinformative inference in extreme data situations that would seem common in, for example, many medical contexts when interest is in rare events. It is recommended that in the context of noninformative priors and binary data with extreme...

Objective Bayesian analysis accelerated degradation test based on Wiener process models

. In this study, we conducted an objective Bayesian analysis focused on the ratio of scale parameters within generalized half-normal distributions. To support this analysis, we developed non informative priors, derived probability matching priors, and established reference priors. We successfully identified a second-order matching prior that satisfied all specified matching criteria. Our results demonstrated that Jeffreys’ prior and the two-group reference prior adhered to a first-order matching criterion, while the one-at-a-time reference prior was classified as a second-order matching prior. Additionally, we outlined the conditions required for the propriety of posterior distributions under general priors, including the non informative priors we formulated. To evaluate the effectiveness of the proposed priors, we performed a simulation study, which indicated that the one-at-a-time reference prior corresponds with target coverage probabilities from a frequentist standpoint. We further illustrated our methodology through real data examples.

Bayes theorem provides a formal framework for combining prior and sample information for parameter estimation in the presence of measurement and structural errors. Prior knowledge, however, may not always be available or may be too vague to incorporate into a prior distribution. In such cases, a reference prior must be chosen for an objective analysis. Typically, a uniform density over the possible ranges of parameters is chosen as the reference prior. However, the validity of a uniform prior as a reference prior is seldom questioned. In this study, an information‐theoretic approach is pursued to derive reference priors, and the results are compared to those obtained by using a uniform prior. Examples of estimating saturated hydraulic conductivity are presented. Priors over hydraulic conductivity obtained by using the information‐theoretic approach are transformation‐invariant and typically nonuniform. The choice between information‐theoretic and uniform prior influences the posterior distribution of hydraulic conductivity, when sample information is small. The use of reference prior is also demonstrated through the PDG‐GIUH hydrologic model, and issues of computational tractability are addressed.

When powerful numerical tools like the finite element method encounter their limits for the evaluation of physical systems it is very common to use surrogate models as an approximation. There are many possible choices concerning the model approach, among which Gaussian process models are the most popular ones due to their clear statistical basis. A very desirable attribute of such surrogates is a high flexibility for making them applicable to a great class of underlying problems while obtaining interpretable results. To achieve this Gaussian processes are used as basis functions of an additive model in this work. Another important property of a surrogate is stability, which can be especially challenging when it comes to the estimation of the correlation parameters. To solve this we use a Bayesian approach where a reference prior is assigned to each component of the additive model assuring robust correlation matrices. Due to the additive structure of the model a simplified parameter estimation process is proposed that reduces the usually high-dimensional optimization problem to a few sub-routines of low dimension. Finally, we demonstrate this concept by modeling the magnetic field of a magnetic linear position detection system.

We develop a Bayesian analysis based on two different Jeffreys priors for the Student-t regression model with unknown degrees of freedom. It is typically difficult to estimate the number of degrees of freedom: improper prior distributions may lead to improper posterior distributions, whereas proper prior distributions may dominate the analysis. We show that Bayesian analysis with either of the two considered Jeffreys priors provides a proper posterior distribution. Finally, we show that Bayesian estimators based on Jeffreys analysis compare favourably to other Bayesian estimators based on priors previously proposed in the literature.

The application of the Bayesian approach emerges as a plausible alternative in multivariate statistical analysis within structural equation models (SEM), being more precise, reliable, and robust for parameter estimation, especially in situations where traditional methods may not be adequate due to sample size restrictions, non-compliance with multivariate normality, or high theoretical complexity. The study aimed to determine the relevance of the proposed ISSM-TAM e-learning model on the academic performance of students in the 2023-II cycle at the National Agrarian University La Molina (UNALM). The methodology was quantitative, basic, explanatory, non-experimental, and cross-sectional. A sample of 767 undergraduate students was collected. A questionnaire with 10 latent constructs, comprising 40 items measured on a 1-5 ordinal Likert scale, was used. Various first and second-generation multivariate statistical techniques were applied: a variance-based structural equation model (PLS-SEM), covariance-based model (CB-SEM), and Bayesian model (B-SEM) with non-informative prior distributions; uniform and reference priors, and a specific informative prior distribution. The obtained data were analyzed exploratorily, descriptively, and inferentially, validating the measurement and structural models with both frequentist and Bayesian approaches. It was found that the Bayesian model with informative priors achieved acceptable values in goodness-of-fit indices compared to the others, and it explained the structural relationships between the main endogenous latent variables of the proposed hybrid model. Received: 5 January 2025 / Accepted: 29 August 2025 / Published: 05 September 2025

Noninformative priors for the common shape parameters of Weibull distributions

This article deals with the problem of testing for the equality of the shape parameters in two inverse Weibull distributions. We propose Bayesian hypothesis testing procedures for the equality of the shape parameters under the noninformative prior. The noninformative prior is usually improper which yields a calibration problem that makes the Bayes factor to be defined up to a multiplicative constant. So we propose the default Bayesian hypothesis testing procedures based on the fractional Bayes factor and the intrinsic Bayes factors under the reference priors. Simulation study and an example are provided.

In reliability studies, a stress–strength model is often used to analyze a system that fails whenever the applied stress is greater than the strength. Statistical inference of reliability is widely used in a number of areas, such as engineering, clinical trials, and quality control. In addition to the common stress–strength model with one stress and one strength, the reliability of more complex systems has also been studied. In this study, we consider the reliability of a generalized stress–strength model that consists of a serial system with one stress and multiple strengths. We then develop the probability matching priors and reference priors for a generalized exponential stress–strength model. We demonstrate that the two-group reference prior and Jeffreys prior are not a matching prior. Through a simulation study and real data example, we also demonstrate that the proposed probability matching priors match the target coverage probabilities in a frequentist sense even for a small sample size.

Regulatory framework recommends that novel statistical methodology for analyzing trial results parallels the frequentist strategy, e.g. the new method must protect type-I error and arrive at a similar conclusion. Keeping these in mind, we construct a Bayesian approach for non-inferiority trials with normal response. A non-informative prior is assumed for the mean response of the experimental treatment and Jeffrey's prior for its corresponding variance when it is unknown. The posteriors of the mean response and variance of the treatment in historical trials are then assumed as priors for its corresponding parameters in the current trial, where that treatment serves as the active control. From these priors, a Bayesian decision criterion is derived to determine whether the experimental treatment is non-inferior to the active control. This criterion is evaluated and compared with the frequentist method using simulation studies. Results show that both Bayesian and frequentist approaches perform alike, but the Bayesian approach has a higher power when the variances are unknown. Both methods also arrive at the same conclusion of non-inferiority when applied on two real datasets. A major advantage of the proposed Bayesian approach lies in its ability to provide posterior probabilities for varying effect sizes of the experimental treatment over the active control.

Statistical analysis for the doubly accelerated degradation Wiener model: An objective Bayesian approach

Reference priors are theoretically attractive for the analysis of geostatistical data since they enable automatic Bayesian analysis and have desirable Bayesian and frequentist properties. But their use is hindered by computational hurdles that make their application in practice challenging. In this work, we derive a new class of default priors that approximate reference priors for the parameters of some Gaussian random fields. It is based on an approximation to the integrated likelihood of the covariance parameters derived from the spectral approximation of stationary random fields. This prior depends on the structure of the mean function and the spectral density of the model evaluated at a set of spectral points associated with an auxiliary regular grid. In addition to preserving the desirable Bayesian and frequentist properties, these approximate reference priors are more stable, and their computations are much less onerous than those of exact reference priors. Unlike exact reference priors, the marginal approximate reference prior of correlation parameter is always proper, regardless of the mean function or the smoothness of the correlation function. This property has important consequences for covariance model selection. An illustration comparing default Bayesian analyses is provided with a dataset of lead pollution in Galicia, Spain.

In this paper, we carry out an in-depth theoretical investigation of Bayesian inference for the Cox regression model. We establish necessary and sufficient conditions for posterior propriety of the regression coefficient, β, in Cox's partial likelihood, which can be obtained as the limiting marginal posterior distribution of β through the specification of a gamma process prior for the cumulative baseline hazard and a uniform improper prior for β. We also examine necessary and sufficient conditions for posterior propriety of the regression coefficients, β, using full likelihood Bayesian approaches in which a gamma process prior is specified for the cumulative baseline hazard. We examine characterisation of posterior propriety under completely observed data settings as well as for settings involving missing covariates. Latent variables are introduced to facilitate a straightforward Gibbs sampling scheme in the Bayesian computation. A real dataset is presented to illustrate the proposed methodology.

Bayesian Analysis for Random Coefficient Regression Models Using Noninformative Priors