摘要:

Unlike the development of more accurate prior distributions for use in Bayesian imaging, the design of more sensible estimators through loss functions has been neglected in the literature. We discuss the design of loss functions with a local structure that depend only on a binary misclassification vector. The proposed approach is similar to modeling with a Markov random field. The Bayes estimate is calculated in a two-step algorithm using Markov chain Monte Carlo and simulated annealing algorithms. We present simulation experiments with the Ising model, where the observations are corrupted with Gaussian and flip noise.

ISSN:0162-1459    

DOI:10.1080/01621459.1995.10476589

出版商:Taylor & Francis Group    

年代:1995

数据来源: Taylor

摘要:

Markov chain Monte Carlo (MCMC; the Metropolis-Hastings algorithm) has been used for many statistical problems, including Bayesian inference, likelihood inference, and tests of significance. Though the method generally works well, doubts about convergence often remain. Here we propose MCMC methods distantly related to simulated annealing. Our samplers mix rapidly enough to be usable for problems in which other methods would require eons of computing time. They simulate realizations from a sequence of distributions, allowing the distribution being simulated to vary randomly over time. If the sequence of distributions is well chosen, then the sampler will mix well and produce accurate answers for all the distributions. Even when there is only one distribution of interest, these annealing-like samplers may be the only known way to get a rapidly mixing sampler. These methods are essential for attacking very hard problems, which arise in areas such as statistical genetics. We illustrate the methods with an application that is much harder than any problem previously done by MCMC, involving ancestral inference on a very large genealogy (7 generations, 2,024 individuals). The problem is to find, conditional on data on living individuals, the probabilities of each individual having been a carrier of cystic fibrosis. Exact calculation of these conditional probabilities is infeasible. Moreover, a Gibbs sampler for the problem would not mix in a reasonable time, even on the fastest imaginable computers. Our annealing-like samplers have mixing times of a few hours. We also give examples of samplers for the “witch's hat” distribution and the conditional Strauss process.

ISSN:0162-1459    

DOI:10.1080/01621459.1995.10476590

出版商:Taylor & Francis Group    

年代:1995

数据来源: Taylor

摘要:

This article presents new operational convergence criteria for the Gibbs sampler (GS) and related procedures that are useful for determining whether they not only have converged but also have converged to provide reliable results. Three GS convergence criteria are presented and applied: the difference convergence criterion (DC2), the ratio convergence criterion (RC2), and the anchored ratio convergence criterion (ARC2). Their uses and properties are discussed and examples are analyzed to illustrate their application.

ISSN:0162-1459    

DOI:10.1080/01621459.1995.10476591

出版商:Taylor & Francis Group    

年代:1995

数据来源: Taylor

摘要:

To compute a Bayes factor for testingH0: ψ = ψ0in the presence of a nuisance parameter β, priors under the null and alternative hypotheses must be chosen. As in Bayesian estimation, an important problem has been to define automatic, or “reference,” methods for determining priors based only on the structure of the model. In this article we apply the heuristic device of taking the amount of information in the prior on ψ equal to the amount of information in a single observation. Then, after transforming β to be “null orthogonal” to ψ, we take the marginal priors on β to be equal under the null and alternative hypotheses. Doing so, and taking the prior on ψ to be Normal, we find that the log of the Bayes factor may be approximated by the Schwarz criterion with an error of orderOp(n−½), rather than the usual error of orderOp(1). This result suggests the Schwarz criterion should provide sensible approximate solutions to Bayesian testing problems, at least when the hypotheses are nested. When instead the prior on ψ is elliptically Cauchy, a constant correction term must be added to the Schwarz criterion; the result then becomes a multidimensional generalization of Jeffreys's method.

ISSN:0162-1459    

DOI:10.1080/01621459.1995.10476592

出版商:Taylor & Francis Group    

年代:1995

数据来源: Taylor

摘要:

The problem of combining information related toIbinomial experiments, each having a distinct probability of success θi, is considered. Instead of using a standard exchangeable prior forθ; = (θ1, …, θI), we propose a more flexible distribution that takes into account various degrees of similarity among the θi's. Using ideas developed by Malec and Sedransk, we consider a partitiongof the experiments and take the θi's belonging to the same partition subset to be exchangeable and the θi's belonging to distinct subsets to be independent. Next we perform Bayesian inference onθ; conditional ong. Of course, one is typically uncertain about which partition to use, and so a prior distribution is assigned on a set of plausible partitionsg. The final inference onθ; is obtained by combining the conditional inferences according to the posterior distribution ofg. The methodology adopted in this article offers a wide flexibility in structuring the dependence among the θi's. This allows the estimate of θito borrow strength from all other experiments according to an adaptive process governed by the data themselves. The method may be usefully applied to the analysis of binary response variables in the presence of categorical covariates. The latter are used to identify a collection of suitable partitionsg, representing factor main effects and interactions, whose relevance will be summarized in the posterior distribution ofg. Besides providing novel interpretations on the role played by the various factors, the procedure will also produce parameter estimates that may differ, sometimes in an appreciable manner, from those obtained using more traditional techniques. Finally, three real data sets are used to illustrate the methodology and compare it with other approaches, such as empirical Bayes (both parametric and nonparametric), logistic regression, and multiple shrinkage estimators.

ISSN:0162-1459    

DOI:10.1080/01621459.1995.10476593

出版商:Taylor & Francis Group    

年代:1995

数据来源: Taylor

摘要:

This article presents a method for detecting multivariate outliers that might be distorting the estimation of a transformation to normality. It also proposes a robust estimator of the transformation parameter.

ISSN:0162-1459    

DOI:10.1080/01621459.1995.10476594

出版商:Taylor & Francis Group    

年代:1995

数据来源: Taylor

摘要:

Some new diagnostic measures in discriminant analysis are proposed. They can be expressed in terms of the two fundamental influence statistics in discriminant analysis:di2and ψi. A theorem on the asymptotic distributions of the fundamental statistics is derived. Based on the theorem, the proposed measures can be shown to be asymptotically distributed as functions of independent chi-squared and standard normal random variables. Critical values and expected quantiles of the measures can then be constructed. Hence influential observations are detected using Q-Q plots and significance tests. Two measures have analogous forms in regression. The theorem is also useful for getting the asymptotic distributions of existing measures that are functions ofdi2and ψi. A comparison of the diagnostics in linear discriminant analysis, linear regression, and linear logistic regression (discriminant) analysis is made. Although discriminant coefficients can be determined under a regression model, regression diagnostic measures are shown to be inappropriate for detecting influential observations in linear discriminant analysis. The temptation of applying regression diagnostic measures in linear discriminant analysis must be resisted.

ISSN:0162-1459    

DOI:10.1080/01621459.1995.10476595

出版商:Taylor & Francis Group    

年代:1995

数据来源: Taylor

摘要:

Two new approximations for multivariate normal probabilities for rectangular regions, based on conditional expectations and regression with binary variables, are proposed. One is a second-order approximation that is much more accurate but also more numerically time-consuming than the first-order approximation. A third approximation, based on the moment-generating function of a truncated multivariate normal distribution, is proposed for orthant probabilities only. Its accuracy is between the first- and second-order approximations when the dimension is less than seven and the correlations are not large. All of the approximations get worse as correlations get larger. These new approximations offer substantial improvements on previous approximations. They also compare favorably with the methods of Genz for numerical evaluation of the multivariate normal integral. The approximation methods should be especially useful within a quasi-Newton routine for parameter estimation in discrete models that involve the multivariate normal distribution.

ISSN:0162-1459    

DOI:10.1080/01621459.1995.10476596

出版商:Taylor & Francis Group    

年代:1995

数据来源: Taylor

摘要:

We apply general covariance identities and inequalities to some functions of multivariate normal variables. We recover, in particular, a recent covariance identity due to Siegel and provide simple estimates on the variance of order statistics. We also present some computations.

ISSN:0162-1459    

DOI:10.1080/01621459.1995.10476597

出版商:Taylor & Francis Group    

年代:1995

数据来源: Taylor

摘要:

Starting in the early 1970s, the decline in column ozone over much of the earth has received much attention. Satellite ozone data, with the advantage of global coverage, now play an important role in assessing long-term trends in ozone distributions. We consider a class of space-time regression models for the analysis of satellite data on a fixed latitude, which take into account temporal and longitudinal dependence of the observations. The models can be used to test the uniformity of long-term trends in different longitudinal ozone series. Using the property of circular matrices, explicit expressions of the likelihood functions are obtained. Asymptotic properties of the parameter estimates are briefly discussed. A diagnostic method is proposed to tentatively select the orders in the noise terms of the models. The space-time regression models are applied to the total ozone mapping spectrometer (TOMS) data for trend assessment.

ISSN:0162-1459    

DOI:10.1080/01621459.1995.10476598

出版商:Taylor & Francis Group    

年代:1995

数据来源: Taylor