Objective Bayesian analysis of spatial data with uncertain nugget and range parameters

The authors develop default priors for the Gaussian random field model that includes a nugget parameter accounting for the effects of microscale variations and measurement errors. They present the independence Jeffreys prior, the Jeffreys‐rule prior and a reference prior and study posterior propriety of these and related priors. They show that the uniform prior for the correlation parameters yields an improper posterior. In case of known regression and variance parameters, they derive the Jeffreys prior for the correlation parameters. They prove posterior propriety and obtain that the predictive distributions at ungauged locations have finite variance. Moreover, they show that the proposed priors have good frequentist properties, except for those based on the marginal Jeffreys‐rule prior for the correlation parameters, and illustrate their approach by analyzing a dataset of zinc concentrations along the river Meuse. The Canadian Journal of Statistics 40: 304–327; 2012 © 2012 Statistical Society of Canada     

Objective priors for causal AR(p) with partial autocorrelations

We consider the problem of deriving formal objective priors for the causal/stationary autoregressive model of order p. We compare the frequentist behaviour of the most common default priors, namely the uniform (over the stationarity region) prior, the Jeffreys’ prior and the reference prior.     

Objective Priors for Parameters in a Normal Linear Regression with Measurement Error

We investigate certain objective priors for the parameters in a normal linear regression models with one of the explanatory variables subject to measurement error. We first show that the use of the standard non informative prior for normal linear regression without measurement error leads to an improper posterior in the measurement error model. We then derive the Jeffreys prior and reference priors, and show that they lead to proper posteriors. We use simulation study to compare the frequentist performance of the estimates derived using these priors, and the MLE.     

Conservative prior distributions for variance parameters in hierarchical models

Bayesian hierarchical models typically involve specifying prior distributions for one or more variance components. This is rather removed from the observed data, so specification based on expert knowledge can be difficult. While there are suggestions for “default” priors in the literature, often a conditionally conjugate inverse‐gamma specification is used, despite documented drawbacks of this choice. The authors suggest “conservative” prior distributions for variance components, which deliberately give more weight to smaller values. These are appropriate for investigators who are skeptical about the presence of variability in the second‐stage parameters (random effects) and want to particularly guard against inferring more structure than is really present. The suggested priors readily adapt to various hierarchical modelling settings, such as fitting smooth curves, modelling spatial variation and combining data from multiple sites.     

Bayesian analysis of the generalized gamma distribution using non-informative priors

The Generalized gamma (GG) distribution plays an important role in statistical analysis. For this distribution, we derive non-informative priors using formal rules, such as Jeffreys prior, maximal data information prior and reference priors. We have shown that these most popular formal rules with natural ordering of parameters, lead to priors with improper posteriors. This problem is overcome by considering a prior averaging approach discussed in Berger et al. [Overall objective priors. Bayesian Analysis. 2015;10(1):189–221]. The obtained hybrid Jeffreys-reference prior is invariant under one-to-one transformations and yields a proper posterior distribution. We obtained good frequentist properties of the proposed prior using a detailed simulation study. Finally, an analysis of the maximum annual discharge of the river Rhine at Lobith is presented.     

Reference Priors for Poisson Processes Involving Nuisance Parameters

A Bayesian reference analysis for determining the posterior distribution of the strength of a radiation source is performed. The only pieces of information available are the numbers of counts gathered in a gross and a background measurement along with the respective counting times and a state-of-knowledge distribution for the efficiency. This situation is addressed by combining the calculations of a “one-at-a-time” reference prior and a reference prior with partial information. The posterior distribution of the source strength obtained with the reference prior leads to credible intervals that have better frequentist coverage than corresponding intervals founded on uniform or Jeffreys’ priors.     

Natural (Non‐)Informative Priors for Skew‐symmetric Distributions

In this paper, we present an innovative method for constructing proper priors for the skewness (shape) parameter in the skew‐symmetric family of distributions. The proposed method is based on assigning a prior distribution on the perturbation effect of the shape parameter, which is quantified in terms of the total variation distance. We discuss strategies to translate prior beliefs about the asymmetry of the data into an informative prior distribution of this class. We show via a Monte Carlo simulation study that our non‐informative priors induce posterior distributions with good frequentist properties, similar to those of the Jeffreys prior. Our informative priors yield better results than their competitors from the literature. We also propose a scale‐invariant and location‐invariant prior structure for models with unknown location and scale parameters and provide sufficient conditions for the propriety of the corresponding posterior distribution. Illustrative examples are presented using simulated and real data.     

Propriety of posteriors in structured additive regression models: Theory and empirical evidence

Structured additive regression comprises many semiparametric regression models such as generalized additive (mixed) models, geoadditive models, and hazard regression models within a unified framework. In a Bayesian formulation, non-parametric functions, spatial effects and further model components are specified in terms of multivariate Gaussian priors for high-dimensional vectors of regression coefficients. For several model terms, such as penalized splines or Markov random fields, these Gaussian prior distributions involve rank-deficient precision matrices, yielding partially improper priors. Moreover, hyperpriors for the variances (corresponding to inverse smoothing parameters) may also be specified as improper, e.g. corresponding to Jeffreys prior or a flat prior for the standard deviation. Hence, propriety of the joint posterior is a crucial issue for full Bayesian inference in particular if based on Markov chain Monte Carlo simulations. We establish theoretical results providing sufficient (and sometimes necessary) conditions for propriety and provide empirical evidence through several accompanying simulation studies.     

Noninformative priors for the between-group variance in the unbalanced one-way random effects model with heterogeneous error variances

For the unbalanced one-way random effects model with heterogeneous error variances, we propose the non-informative priors for the between-group variance and develop the first- and second-order matching priors. It turns out that the second-order matching priors do not exist and the reference prior and Jeffreys prior do not satisfy a first-order matching criterion. We also show that the first-order matching prior meets the frequentist target coverage probabilities much better than the Jeffreys prior and reference prior through simulation study, and the Bayesian credible intervals based on the matching prior and reference prior give shorter intervals than the existing confidence intervals by examples.     

Erratum to the paper: a note on reference prior for the scaler skew-normal distribution

Liseo and Loperfido [A note on reference priors for the scalar skew-normal distribution. J Statist Plann Inference. 2006;136(2):373–389] studied some peculiar features of default Bayes analysis of the scalar skew-normal model. In particular, they showed that, by considering the simplest model with a single unknown parameter λ of skewness, the reference – or Jeffreys’ – prior for this parameter is proper. They proved that tails of Jeffreys’ prior are of order O(λ^?3/2). But they made a mistake in their proof. In this note, we will modify their proof.     

Intrinsic Bayes factor approach to a test for the power law process

The versatile new criterion called the intrinsic Bayes factor (IBF), introduced by Berger and Pericchi [J. Amer. Statist. Assoc. 91 (1996) 109–122], has made it possible to perform model selection and hypotheses testing using standard (improper) noninformative priors in a variety of situations. In this paper, we use their methodology to test several hypotheses regarding the shape parameter of the power law process, which has been widely used to model failure times of repairable systems. Assuming that we have data from the process according to the time-truncation sampling scheme, we derive the arithmetic IBFs using four default priors, including the reference and Jeffreys priors. We establish the frequentist probability matching properties of these priors. We also identify two priors that are justifiable under both time-truncation and failure-truncation schemes, so that the IBFs for both schemes can be unified. Deducing the intrinsic priors of a certain canonical form, as the time of truncation tends to infinity, we show that the arithmetic IBFs correspond asymptotically to actual Bayes factors. We also discuss the expected IBFs, which are useful with small samples. We then use these results to analyze an actual data set on the interruption times of a transmission line, summarizing our results under the default priors.     

Inference for the Hyperparameters of Structural Models Under Classical and Bayesian Perspectives: A Comparison Study

Structural models—or dynamic linear models as they are known in the Bayesian literature—have been widely used to model and predict time series using a decomposition in non observable components. Due to the direct interpretation of the parameters, structural models are a powerful and simple methodology to analyze time series in several areas, such as economy, climatology, environmental sciences, among others. The parameters of such models can be estimated either using maximum likelihood or Bayesian procedures, generally implemented using conjugate priors, and there are plenty of works in the literature employing both methods. But are there situations where one of these approaches should be preferred? In this work, instead of conjugate priors for the hyperparameters, the Jeffreys prior is used in the Bayesian approach, along with the uniform prior, and the results are compared to the maximum likelihood method, in an extensive Monte Carlo study. Interval estimation is also evaluated and, to this purpose, bootstrap confidence intervals are introduced in the context of structural models and their performance is compared to the asymptotic and credibility intervals. A real time series of a Brazilian electric company is used as illustration.     

Bayes estimation of reliability for the two-parameter lognormal distribution

Bayes estimators of reliability for the lognormal failure distribution with two parameters (M,∑) are obtained both for informative priors of normal-gamma type and for the vague prior of Jeffreys. The estimators are in terms of the t-distribution function. The Bayes estimators are compared with the maximum likelihood and minimum variance unbiased estimators of reliabil-ity using Monte Carlo simulations.     

Non-informative priors for the inverse Weibull distribution

In this paper, we develop the non-informative priors for the inverse Weibull model when the parameters of interest are the scale and the shape parameters. We develop the first-order and the second-order matching priors for both parameters. For the scale parameter, we reveal that the second-order matching prior is not a highest posterior density (HPD) matching prior, does not match the alternative coverage probabilities up to the second order and is not a cumulative distribution function (CDF) matching prior. Also for the shape parameter, we reveal that the second-order matching prior is an HPD matching prior and a CDF matching prior and also matches the alternative coverage probabilities up to the second order. For both parameters, we reveal that the one-at-a-time reference prior is the second-order matching prior, but Jeffreys’ prior is not the first-order and the second-order matching prior. A simulation study is performed to compare the target coverage probabilities and a real example is given.     

A matching prior for the product of normal means based on the modified profile likelihood

In this paper, we develop a matching prior for the product of means in several normal distributions with unrestricted means and unknown variances. For this problem, properly assigning priors for the product of normal means has been issued because of the presence of nuisance parameters. Matching priors, which are priors matching the posterior probabilities of certain regions with their frequentist coverage probabilities, are commonly used but difficult to derive in this problem. We developed the first order probability matching priors for this problem; however, the developed matching priors are unproper. Thus, we apply an alternative method and derive a matching prior based on a modification of the profile likelihood. Simulation studies show that the derived matching prior performs better than the uniform prior and Jeffreys’ prior in meeting the target coverage probabilities, and meets well the target coverage probabilities even for the small sample sizes. In addition, to evaluate the validity of the proposed matching prior, Bayesian credible interval for the product of normal means using the matching prior is compared to Bayesian credible intervals using the uniform prior and Jeffrey’s prior, and the confidence interval using the method of Yfantis and Flatman.     

Objective Bayesian Analysis for Log-logistic Distribution

In this article, Bayesian approach is applied to estimate the parameters of Log-logistic distribution under reference prior and Jeffreys’ prior. The reference prior is derived and it is found that the reference prior is also a second-order matching priors as for the case of any parameter of interest. The Bayesian estimators cannot be obtained in explicit forms. Metropolis within Gibbs sampling algorithm is used to obtain the Bayesian estimators. The Bayesian estimates are compared with the maximum likelihood estimates via simulation study. A real dataset is considered for illustrative purposes.     

An objective Bayesian analysis of the two-parameter half-logistic distribution based on progressively type-II censored samples

This paper develops an objective Bayesian analysis method for estimating unknown parameters of the half-logistic distribution when a sample is available from the progressively Type-II censoring scheme. Noninformative priors such as Jeffreys and reference priors are derived. In addition, derived priors are checked to determine whether they satisfy probability-matching criteria. The Metropolis–Hasting algorithm is applied to generate Markov chain Monte Carlo samples from these posterior density functions because marginal posterior density functions of each parameter cannot be expressed in an explicit form. Monte Carlo simulations are conducted to investigate frequentist properties of estimated models under noninformative priors. For illustration purposes, a real data set is presented, and the quality of models under noninformative priors is evaluated through posterior predictive checking.     

The use of Jeffreys priors for the Student-t distribution

The Jeffreys-rule prior and the marginal independence Jeffreys prior are recently proposed in Fonseca et al. [Objective Bayesian analysis for the Student-t regression model, Biometrika 95 (2008), pp. 325–333] as objective priors for the Student-t regression model. The authors showed that the priors provide proper posterior distributions and perform favourably in parameter estimation. Motivated by a practical financial risk management application, we compare the performance of the two Jeffreys priors with other priors proposed in the literature in a problem of estimating high quantiles for the Student-t model with unknown degrees of freedom. Through an asymptotic analysis and a simulation study, we show that both Jeffreys priors perform better in using a specific quantile of the Bayesian predictive distribution to approximate the true quantile.     

Non-informative priors for the common mean in the one-way random effects model with heterogeneous error variances

The paper develops some objective priors for the common mean in the one-way random effects model with heterogeneous error variances. We derive the first and second order matching priors and reference priors. It turns out that the second order matching prior matches the alternative coverage probabilities up to the second order, and is also an HPD matching prior. However, derived reference priors just satisfy a first order matching criterion. Our simulation studies indicate that the second order matching prior performs better than the reference prior and the Jeffreys prior in terms of matching the target coverage probabilities in a frequentist sense. We also illustrate our results using real data.     

The conflict between improper priors and robustness

Most methods for assessing the sensitivity of the posterior to the prior do not work if the prior is improper. We characterize the neighborhoods of priors that permit non-trivial sensitivity analysis for improper priors. We show that these neighborhoods are not “tail rich”, i.e. they do not contain measures with a variety of tail behavior. Thus there are no neighborhoods that are simultaneously non-trivial and rich. We also show that using the formal structure of finitely additive priors does not solve the problem. We suggest some directions for addressing the problem. In particular, we consider replacing the improper prior with a sequence of asymptotically well-behaved data-dependent priors.