1,721,007 research outputs found
Recent advances on Bayesian inference for P(X min Y )
Full text linkWe address the statistical problem of evaluating R = P(X < Y ), where X and Y are two independent random variables. Bayesian parametric inference about R, based on the marginal posterior density of R, has been widely discussed under various distributional assumptions on X and Y . This classical approach requires both elicitation of a prior on the complete parameter and numerical integration in order to derive the marginal distribution of R. In this paper, we discuss and apply recent advances in Bayesian inference based on higher-order asymptotics and on pseudo-likelihoods, and related matching priors, which allow to perform accurate inference on the parameter of interest only. The proposed approach has the advantages of avoiding the elicitation on the nuisance parameters and the computation of multidimensional integrals.
The accuracy of the proposed methodology is illustrated both by numerical studies and by real-life data concerning clinical studie
A note on approximate Bayesian credible sets based on modified loglikelihood ratios
Full text linkAsymptotic arguments are widely used in Bayesian inference, and in recent years there has been considerable developments of the so-called higher-order asymptotics. This theory provides very accurate approximations to posterior distributions, and to related quantities, in a variety of parametric statistical problems, even for small sample sizes.
The aim of this contribution is to discuss recent advances in approximate Bayesian computations based on the asymptotic theory of modified loglikelihood ratios, both from theoretical and practical point of views. Results on third-order approximations for univariate posterior distributions, also in the presence of nuisance parameters, are reviewed and a new formula for a vector parameter of interest is presented.
All these approximations may routinely be applied in practice for Bayesian inference, since they require little more than standard likelihood quantities for their implementation, and hence they may be available at little additional computational cost over simple first-order approximations. Moreover, these approximations give rise to a simple simulation scheme, alternative to MCMC, for Bayesian computation of marginal posterior distributions for a scalar parameter of interest. In addition, they can be used for testing precise null hypothesis and to define accurate Bayesian credible sets. Some illustrative examples are discussed, with particular attention to the use of matching priors
On the use of pseudo-likelihoods in Bayesian variable selection.
Full text linkIn the presence of nuisance parameters, we discuss a one-parameter Bayesian analysis based on a pseudo-likelihood assuming a default prior distribution for the parameter of interest only. Although this way to proceed cannot always be considered as orthodox in the Bayesian perspective, it is of interest to evaluate whether the use of suitable pseudo-likelihoods may be proposed for Bayesian inference. Attention is focused in the context of regression models, in particular on inference about a scalar regression coefficient in various multiple regression models, i.e. scale and regression models with non-normal errors, non-linear normal heteroscedastic regression models, and log-linear models for count data with overdispersion. Some interesting conclusions emerge
Higher-order asymptotic computation of Bayesian significance tests for precise null hypotheses in the presence of nuisance parameters
No full textThe full Bayesian significance test (FBST) was introduced by Pereira and Stern for measuring the evidence of a precise null hypothesis. The FBST requires both numerical optimization and multidimensional integration, whose computational cost may be heavy when testing a precise null hypothesis on a scalar parameter of interest in the presence of a large number of nuisance parameters. In this paper we propose a higher order approximation of the measure of evidence for the FBST, based on tail area expansions of the marginal posterior of the parameter of interest. When in particular focus is on matching priors, further results are highlighted. Numerical illustrations are discussed
Default prior distributions from quasi- and quasi-profile likelihoods
No full textn some problems of practical interest, a standard Bayesian analysis can be difficult to perform. This is true, for example, when the class of sampling parametric models is unknown or if robustness with respect to data or to model misspecifications is required. These situations can be usefully handled by using a posterior distribution for the parameter of interest which is based on a pseudo-likelihood function derived from estimating equations, i.e. on a quasi-likelihood, and on a suitable prior distribution.
The aim of this paper is to propose and discuss the construction of a default prior distribution for a scalar parameter of interest to be used together with a quasi- likelihood function. We show that the proposed default prior can be interpreted as a Jeffreys-type prior, since it is proportional to the square-root of the expected information derived from the quasi-likelihood. The frequentist coverage of the credible regions, based on the proposed procedure, is studied through Monte Carlo simulations in the context of robustness theory and of generalized linear models with over- dispersion
A non-parametric bootstrap test for the equality of coefficients of variation
No full textThe problem of testing the equality of coefficients of variation for two different populations leads to various levels of difficulty depending on the possible assumptions on models and parameters for the two populations under study. Simulation techniques appear to be the only feasible way in the case where the available information came only from data and when these do not allow one to make any assumption on the models. In this work we propose a nonparametric bootstrap procedure, both to build the test statistic and also to approximate the p-value. The properties of the test and its critical aspects are illustrated and discussed by means of an application to a real data set of anthropometric measures for the study of the sexual dimorphism
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