A bayesian alternative to the analysis of matched categorical responces

This study investigates the Bayesian appeoach to the analysis of parired responess when the responses are categorical. Using resampling and analytical procedures, inferences for homogeneity and agreement are develped. The posterior analysis is based on the Dirichlet distribution from which repeated samples can be geneated with a random number generator. Resampling and analytical techniques are employed to make Bayesian inferences, and when it is not appropriate to use analytical procedures, resampling techniques are easily implemented. Bayesian methodoloogy is illustrated with several examples and the results show that they are exacr-small sample procedures that can easily solve inference problems for matched designs.     

On a bayesian approach for the shiftpoint problem

Using a direct resampling process, a Bayesian approach is developed for the analysis of the shiftpoint problem. In many problems it is straight forward to isolate the marginal posterior distribution of the shift-point parameter and the conditional distribution of some of the parameters given the shift point and the other remaining parameters. When this is possible, a direct sampling approach is easily implemented whereby standard random number generators can be used to generate samples from the joint posterior distribution of aii the parameters in the model. This technique is illustrated with examples involving one shift for Poisson processes and regression models.     

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.     

Bayesians Should Not Resample a Prior Sample to Learn about the Posterior

It is argued in a Bayesian context in which a set of parameter values is simulated from a prior distribution, that the sampling importance resampling algorithm should not be used to resample these values so as to obtain an approximate sample from the posterior. Rather, the whole set of prior values, along with their appropriate weights, can be more gainfully employed.     

Quasi-random resampling for the bootstrap

Quasi-random sequences are known to give efficient numerical integration rules in many Bayesian statistical problems where the posterior distribution can be transformed into periodic functions on then-dimensional hypercube. From this idea we develop a quasi-random approach to the generation of resamples used for Monte Carlo approximations to bootstrap estimates of bias, variance and distribution functions. We demonstrate a major difference between quasi-random bootstrap resamples, which are generated by deterministic algorithms and have no true randomness, and the usual pseudo-random bootstrap resamples generated by the classical bootstrap approach. Various quasi-random approaches are considered and are shown via a simulation study to result in approximants that are competitive in terms of efficiency when compared with other bootstrap Monte Carlo procedures such as balanced and antithetic resampling.     

Non-parametric Bayesian Inference for Integrals with respect to an Unknown Finite Measure

Abstract.  We consider the problem of estimating a collection of integrals with respect to an unknown finite measure μ from noisy observations of some of the integrals. A new method to carry out Bayesian inference for the integrals is proposed. We use a Dirichlet or Gamma process as a prior for μ , and construct an approximation to the posterior distribution of the integrals using the sampling importance resampling algorithm and samples from a new multidimensional version of a Markov chain by Feigin and Tweedie. We prove that the Markov chain is positive Harris recurrent, and that the approximating distribution converges weakly to the posterior as the sample size increases, under a mild integrability condition. Applications to polymer chemistry and mathematical finance are given.     

Bayesian Analysis of a Generalized Poisson Distribution

A generalized form of the Poisson Distribution with two parameters will be estimated by the Bayesian technique. When one of the parameters is known, several important parametric functions will be estimated and a numerical comparison with estimates obtained by the methods of maximum likelihood and unbiased minimum variance will be drawn. The simplicity of the posterior distribution of the unknown parameter enables us to construct exact probability intervals, and to devise a statistic to test the homogeneity of several populations. When the two parameters are unknown, dependent priors are being considered. Although the posterior distributions are sensitive to the choice of the prior, the posterior estimates are very stable and we use the Pearson system of curves to construct approximate posterior confidence limits for the parameters.     

Detection and Localization in Test Accuracy: A Bayesian Perspective

In assessing the area under the ROC curve for the accuracy of a diagnostic test, it is imperative to detect and locate multiple abnormalities per image. This approach takes that into account by adopting a statistical model that allows for correlation between the reader scores of several regions of interest (ROI).

The ROI method of partitioning the image is taken. The readers give a score to each ROI in the image and the statistical model takes into account the correlation between the scores of the ROI's of an image in estimating test accuracy. The test accuracy is given by Pr[Y > Z] + (1/2)Pr[Y = Z], where Y is an ordinal diagnostic measurement of an affected ROI, and Z is the diagnostic measurement of an unaffected ROI. This way of measuring test accuracy is equivalent to the area under the ROC curve. The parameters are the parameters of a multinomial distribution, then based on the multinomial distribution, a Bayesian method of inference is adopted for estimating the test accuracy.

Using a multinomial model for the test results, a Bayesian method based on the predictive distribution of future diagnostic scores is employed to find the test accuracy. By resampling from the posterior distribution of the model parameters, samples from the posterior distribution of test accuracy are also generated. Using these samples, the posterior mean, standard deviation, and credible intervals are calculated in order to estimate the area under the ROC curve. This approach is illustrated by estimating the area under the ROC curve for a study of the diagnostic accuracy of magnetic resonance angiography for diagnosis of arterial atherosclerotic stenosis. A generalization to multiple readers and/or modalities is proposed.

A Bayesian way to estimate test accuracy is easy to perform with standard software packages and has the advantage of employing the efficient inclusion of information from prior related imaging studies.     

Single-parameter inference based on partial prior information

Partial specification of a prior distribution can be appealing to an analyst, but there is no conventional way to update a partial prior. In this paper, we show how a framework for Bayesian updating with data can be based on the Dirichlet(a) process. Within this framework, partial information predictors generalize standard minimax predictors and have interesting multiple-point shrinkage properties. Approximations to partial-information estimators for squared error loss are defined straightforwardly, and an estimate of the mean shrinks the sample mean. The proposed updating of the partial prior is a consequence of four natural requirements when the Dirichlet parameter a is continuous. Namely, the updated partial posterior should be calculable from knowledge of only the data and partial prior, it should be faithful to the full posterior distribution, it should assign positive probability to every observed event {X,}, and it should not assign probability to unobserved events not included in the partial prior specification.     

Bayesian estimation of the efficient frontier

In this paper, we consider the estimation of the three determining parameters of the efficient frontier, the expected return, and the variance of the global minimum variance portfolio and the slope parameter, from a Bayesian perspective. Their posterior distribution is derived by assigning the diffuse and the conjugate priors to the mean vector and the covariance matrix of the asset returns and is presented in terms of a stochastic representation. Furthermore, Bayesian estimates together with the standard uncertainties for all three parameters are provided, and their asymptotic distributions are established. All obtained findings are applied to real data, consisting of the returns on assets included into the S&P 500. The empirical properties of the efficient frontier are then examined in detail.     

Bayesian estimation for percolation models of disease spread in plant populations

Statistical methods are formulated for fitting and testing percolation-based, spatio-temporal models that are generally applicable to biological or physical processes that evolve in spatially distributed populations. The approach is developed and illustrated in the context of the spread of Rhizoctonia solani, a fungal pathogen, in radish but is readily generalized to other scenarios. The particular model considered represents processes of primary and secondary infection between nearest-neighbour hosts in a lattice, and time-varying susceptibility of the hosts. Bayesian methods for fitting the model to observations of disease spread through space and time in replicate populations are developed. These use Markov chain Monte Carlo methods to overcome the problems associated with partial observation of the process. We also consider how model testing can be achieved by embedding classical methods within the Bayesian analysis. In particular we show how a residual process, with known sampling distribution, can be defined. Model fit is then examined by generating samples from the posterior distribution of the residual process, to which a classical test for consistency with the known distribution is applied, enabling the posterior distribution of the P-value of the test used to be estimated. For the Rhizoctonia-radish system the methods confirm the findings of earlier non-spatial analyses regarding the dynamics of disease transmission and yield new evidence of environmental heterogeneity in the replicate experiments.     

A Bayesian method for classification and discrimination

We discuss Bayesian analyses of traditional normal-mixture models for classification and discrimination. The development involves application of an iterative resampling approach to Monte Carlo inference, commonly called Gibbs sampling, and demonstrates routine application. We stress the benefits of exact analyses over traditional classification and discrimination techniques, including the ease with which such analyses may be performed in a quite general setting, with possibly several normal-mixture components having different covariance matrices, the computation of exact posterior classification probabilities for observed data and for future cases to be classified, and posterior distributions for these probabilities that allow for assessment of second-level uncertainties in classification.     

Sampling the Future: A Bayesian Approach to Forecasting From Univariate Time Series Models

The Box–Jenkins methodology for modeling and forecasting from univariate time series models has long been considered a standard to which other forecasting techniques have been compared. To a Bayesian statistician, however, the method lacks an important facet—a provision for modeling uncertainty about parameter estimates. We present a technique called sampling the future for including this feature in both the estimation and forecasting stages. Although it is relatively easy to use Bayesian methods to estimate the parameters in an autoregressive integrated moving average (ARIMA) model, there are severe difficulties in producing forecasts from such a model. The multiperiod predictive density does not have a convenient closed form, so approximations are needed. In this article, exact Bayesian forecasting is approximated by simulating the joint predictive distribution. First, parameter sets are randomly generated from the joint posterior distribution. These are then used to simulate future paths of the time series. This bundle of many possible realizations is used to project the future in several ways. Highest probability forecast regions are formed and portrayed with computer graphics. The predictive density's shape is explored. Finally, we discuss a method that allows the analyst to subjectively modify the posterior distribution on the parameters and produce alternate forecasts.     

On semiparametric pivotal bayesian inference for quantiles

In semiparametric inference we distinguish between the parameter of interest which may be a location parameter, and a nuisance parameter that determines the remaining shape of the sampling distribution. As was pointed out by Diaconis and Freedman the main problem in semiparametric Bayesian inference is to obtain a consistent posterior distribution for the parameter of interest. The present paper considers a semiparametric Bayesian method based on a pivotal likelihood function. It is shown that when the parameter of interest is the median, this method produces a consistent posterior distribution and is easily implemented, Numerical comparisons with classical methods and with Bayesian methods based on a Dirichlet prior are provided. It is also shown that in the case of symmetric intervals, the classical confidence coefficients have a Bayesian interpretation as the limiting posterior probability of the interval based on the Dirichlet prior with a parameter that converges to zero.     

Experience with a bayesian bootstrap method incorporating proper prior information

Although bootstrapping has become widely used in statistical analysis, there has been little reported concerning bootstrapped Bayesian analyses, especially when there is proper prior informa-tion concerning the parameter of interest. In this paper, we first propose an operationally implementable definition of a Bayesian bootstrap. Thereafter, in simulated studies of the estimation of means and variances, this Bayesian bootstrap is compared to various parametric procedures. It turns out that little information is lost in using the Bayesian bootstrap even when the sampling distribution is known. On the other hand, the parametric procedures are at times very sensitive to incorrectly specified sampling distributions, implying that the Bayesian bootstrap is a very robust procedure for determining the posterior distribution of the parameter.     

Lower posterior death probabilities from a quick medical response in road traffic accidents

Introduction: We use data from Spain on roads and motorways traffic accidents in May 2004 to quantify the statistical association between quick medical response time and mortality rate. Method: Probit and logit parameters are estimated by a Bayesian method in which samples from the posterior densities are obtained through an MCMC simulation scheme. We provide posterior credible intervals and posterior partial effects of a quick medical response at several time levels over which we express our prior beliefs. Results: A reduction of 5 min, from a 25-min response-time level, is associated with lower posterior probabilities of death in roads and motorways accidents of 24% and 30%, respectively.     

The Bayesian and frequentist approaches to testing a one-sided hypothesis about a multivariate mean

This paper compares the Bayesian and frequentist approaches to testing a one-sided hypothesis about a multivariate mean. First, this paper proposes a simple way to assign a Bayesian posterior probability to one-sided hypotheses about a multivariate mean. The approach is to use (almost) the exact posterior probability under the assumption that the data has multivariate normal distribution, under either a conjugate prior in large samples or under a vague Jeffreys prior. This is also approximately the Bayesian posterior probability of the hypothesis based on a suitably flat Dirichlet process prior over an unknown distribution generating the data. Then, the Bayesian approach and a frequentist approach to testing the one-sided hypothesis are compared, with results that show a major difference between Bayesian reasoning and frequentist reasoning. The Bayesian posterior probability can be substantially smaller than the frequentist p-value. A class of example is given where the Bayesian posterior probability is basically 0, while the frequentist p-value is basically 1. The Bayesian posterior probability in these examples seems to be more reasonable. Other drawbacks of the frequentist p-value as a measure of whether the one-sided hypothesis is true are also discussed.     

An adaptive resampling scheme for cycle estimation

Bayesian dynamic linear models (DLMs) are useful in time series modelling, because of the flexibility that they off er for obtaining a good forecast. They are based on a decomposition of the relevant factors which explain the behaviour of the series through a series of state parameters. Nevertheless, the DLM as developed by West and Harrison depend on additional quantities, such as the variance of the system disturbances, which, in practice, are unknown. These are referred to here as 'hyper-parameters' of the model. In this paper, DLMs with autoregressive components are used to describe time series that show cyclic behaviour. The marginal posterior distribution for state parameters can be obtained by weighting the conditional distribution of state parameters by the marginal distribution of hyper-parameters. In most cases, the joint distribution of the hyperparameters can be obtained analytically but the marginal distributions of the components cannot, so requiring numerical integration. We propose to obtain samples of the hyperparameters by a variant of the sampling importance resampling method. A few applications are shown with simulated and real data sets.     

A Bayesian analysis of the change-point problem for directional data

In this paper, we discuss a simple fully Bayesian analysis of the change-point problem for the directional data in the parametric framework with von Mises or circular normal distribution as the underlying distribution. We first discuss the problem of detecting change in the mean direction of the circular normal distribution using a latent variable approach when the concentration parameter is unknown. Then, a simpler approach, beginning with proper priors for all the unknown parameters – the sampling importance resampling technique – is used to obtain the posterior marginal distribution of the change-point. The method is illustrated using the wind data [E.P. Weijers, A. Van Delden, H.F. Vugts and A.G.C.A. Meesters, The composite horizontal wind field within convective structures of the atmospheric surface layer, J. Atmos. Sci. 52 (1995. 3866–3878]. The method can be adapted for a variety of situations involving both angular and linear data and can be used with profit in the context of statistical process control in Phase I of control charting and also in Phase II in conjunction with control charts.     

Simulation-based Bayesian inferences for two-variance components linear models

We present a Bayesian analysis of variance component models via simulation. In particular, we study the 2-component hierarchical design model under balanced and unbalanced experiments. Also, we consider 2-factor additive random effect models and mixed models in a cross-classified design. We assess the sensitivity of inference to the choice of prior by a sampling/resampling technique. Finally, attention is given to non-normal error distributions such as the heavy-tailed t distribution.