In this paper,we introduce a new Bayesian nonparametric model for estimating an unknown function in the presence of Gaussian noise.The proposed model involves a mixture of a point mass and an arbitrary (nonparametric) symmetric and unimodal distribution for modeling wavelet coefficients. Posterior simulation uses slice sampling ideas and the consistency under the proposed model is discussed. In particular, the method is shown to be computationally competitive with some of best Empirical wavelet estimation methods.

The infinitely-many-neutral-alleles model has recently been extended to a class of diffusion processes associated with Gibbs partitions of two-parameter Poisson-Dirichlet type. This paper introduces a family of infinite-dimensional diffusions associated with a different subclass of Gibbs partitions, induced by normalized inverse- Gaussian random probability measures. Such diffusions describe the evolution of the frequencies of infinitely-many types together with the dynamics of the time-varying mutation rate, which is driven by an alpha-diversity diffusion. Constructed as a dynamic version, relative to this framework, of the corresponding notion for Gibbs partitions, the latter is explicitly derived from an underlying population model and shown to coincide, in a special case, with the diffusion approximation of a critical Galton-Watson branching process. The class of infinite-dimensional processes is characterized in terms of its infinitesimal generator on an appropriate domain, and shown to be the limit in distribution of a certain sequence of Feller diffusions with finitelymany types. Moreover, a discrete representation is provided by means of appropriately transformed Moran-type particle processes, where the particles are samples from a normalized inverse-Gaussian random probability measure. The relationship between the limit diffusion and the two-parameter model is also discussed.

This paper is concerned with the construction of prior probability measures for parametric families of densities where the framework is such that only beliefs or knowledge about a single observable data point is required. We pay particular attention to the parameter which minimizes a measure of divergence to the distribution providing the data. The prior distribution reflects this attention and we discuss the application of the Bayes rule from this perspective. Our framework is fundamentally non-parametric and we are able to interpret prior distributions on the parameter space using ideas of matching loss functions, one of which is coming from the data model and the other from the prior.

We propose a more efficient version of the slice sampler for Dirichlet process mixture models described by Walker (Commun. Stat., Simul. Comput. 36:45–54, 2007). This new sampler allows for the fitting of infinite mixture models with a wide-range of prior specifications. To illustrate this flexibility we consider priors defined through infinite sequences of independent positive random variables. Two applications are considered: density estimation using mixture models and hazard function estimation. In each case we show how the slice efficient sampler can be applied to make inference in the models. In the mixture case, two submodels are studied in detail. The first one assumes that the positive random variables are Gamma distributed and the second assumes that they are inverse-Gaussian distributed. Both priors have two hyperparameters and we consider their effect on the prior distribution of the number of occupied clusters in a sample. Extensive computational comparisons with alternative “conditional” simulation techniques for mixture models using the standard Dirichlet process prior and our new priors are made. The properties of the new priors are illustrated on a density estimation problem.

This article describes posterior simulation methods for mixture models whose mixing distribution has a Normalized Random Measure prior. The methods use slice sampling ideas and introduce no truncation error. The approach can be easily applied to both homogeneous and nonhomogeneous Normalized Random Measures and allows the updating of the parameters of the random measure. The methods are illustrated on data examples using both Dirichlet and Normalized Generalized Gamma process priors. In particular, the methods are shown to be computationally competitive with previously developed samplers for Dirichlet process mixture models. Matlab code to implement these methods is available as supplemental material.

Clustering is a common and important issue, and finite mixture models based on the normal distribution are frequently used to address the problem. In this article, we consider a classification model and build a mixture model around it. A good assessment of the allocation of observations and number of clusters is easily obtained from this approach.

The paper proposes two Bayesian approaches to non-parametric monotone function estimation. The first approach uses a hierarchical Bayes framework and a characterization of smooth monotone functions given by Ramsay that allows unconstrained estimation. The second approach uses a Bayesian regression spline model of Smith and Kohn with a mixture distribution of constrained normal distributions as the prior for the regression coefficients to ensure the monotonicity of the resulting function estimate. The small sample properties of the two function estimators across a range of functions are provided via simulation and compared with existing methods. Asymptotic results are also given that show that Bayesian methods provide consistent function estimators for a large class of smooth functions. An example is provided involving economic demand functions that illustrates the application of the constrained regression spline estimator in the context of a multiple-regression model where two functions are constrained to be monotone.

We explore a method for constructing first-order stationary autoregressive-type models with given marginal distributions. We impose the underlying dependence structure in the model using Bayesian non-parametric predictive distributions. This approach allows for nonlinear dependency and at the same time works for any choice of marginal distribution. In particular, we look at the case of discrete-valued models; that is the marginal distributions are supported on the non-negative integers.

Polya trees fix partitions and use random probabilities in order to construct random probability measures. With quantile pyramids we instead fix probabilities and use random partitions. For nonparametric Bayesian inference we use a prior which supports piecewise linear quantile functions, based on the need to work with a finite set of partitions, yet we show that the limiting version of the prior exists. We also discuss and investigate an alternative model based on the so-called substitute likelihood, Both approaches factorize in a convenient way leading to relatively straightforward analysis via MCMC, since analytic summaries of posterior distributions are too complicated. We give conditions securing the existence of an absolute continuous quantile process, and discuss consistency and approximate normality for the sequence of posterior distributions. Illustrations are included.

A Bayesian nonparametric approach to modeling a nonlinear dynamic model is presented. New techniques for sampling infinite mixture models are used. The inference procedure specifically in the case of the logistic model and when the nonparametric component is applied to the additive errors is demonstrated.

Using reinforced processes related to beta-Stacy process and generalized Polya urn scheme jointly with a structure assumption about dependence, a Bayesian nonparametric prior and a predictive estimator for a multivariate survival function are provided. This estimator can be computed through an easy implementation of a Gibbs sampler algorithm. Moreover consistency of the estimator is studied.

A Bayesian nonparametric model is introduced for score equating. It is applicable to all major equating designs, and has advantages over previous equating models. Unlike the previous models, the Bayesian model accounts for positive dependence between distributions of scores from two tests. The Bayesian model and the previous equating models are compared through the analysis of data sets famous in the equating literature. Also, the classical percentile-rank, linear, and mean equating models are each proven to be a special case of a Bayesian model under a highly-informative choice of prior distribution.

In this paper we argue that model selection, as commonly practised in psychometrics, violates certain principles of coherence. On the other hand, we show that Bayesian nonparametrics provides a coherent basis for model selection, through the use of a 'nonparametric' prior distribution that has a large support on the space of sampling distributions. We illustrate model selection under the Bayesian nonparametric approach, through the analysis of real questionnaire data. Also, we present ways to use the Bayesian nonparametric framework to define very flexible psychometric models, through the specification of a nonparametric prior distribution that supports all distribution functions for the inverse link, including the standard logistic distribution functions. The Bayesian nonparametric approach provides a coherent method for model selection that can be applied to any statistical model, including psychometric models. Moreover, under a 'non-informative' choice of nonparametric prior, the Bayesian nonparametric approach is easy to apply, and selects the model that maximizes the log likelihood. Thus, under this choice of prior, the approach can be extended to non-Bayesian settings where the parameters of the competing models are estimated by likelihood maximization, and it can be used with any psychometric software package that routinely reports the model log likelihood.

This paper provides a countable representation for a class of infinite-dimensional diffusions which extends the infinitely-many-neutral-alleles model and is related to the two-parameter Poisson-Dirichlet process. By means of Gibbs sampling procedures, we define a reversible Moran-type population process. The associated process of ranked relative frequencies of types is shown to converge in distribution to the two-parameter family of diffusions, which is stationary and ergodic with respect to the two-parameter Poisson-Dirichlet distribution. The construction provides interpretation for the limiting process in terms of individual dynamics.

We define and investigate a new class ofmeasure-valuedMarkov chains by resorting to ideas formulated in Bayesian nonparametrics related to the Dirichlet process and the Gibbs sampler. Dependent random prob- ability measures in this class are shown to be stationary and ergodic with respect to the law of a Dirichlet process and to converge in distribution to the neutral diffusion model.

In this paper we provide an alternative constructive definition for the Dirichlet process which generalizes the one given by Sethuraman.

Recently, James [L.F. James, Bayesian Poisson process partition calculus with an application to Bayesian Levy moving averages, Ann. Statist. 33 (2005), pp. 1771-1799.] and [L.F. James, Poisson calculus for spatial neutral to the right processes, Ann. Statist. 34 (2006), pp. 416-440.] has derived important results for various models in Bayesian nonparametric inference. In particular, in ref. [L.F. James, Poisson calculus for spatial neutral to the right processes, Ann. Statist. 34 (2006), pp. 416-440.] a spatial version of neutral to the right processes is defined and their posterior distribution derived. Moreover, in ref. [L.F. James, Bayesian Poisson process partition calculus with an application to Bayesian Levy moving averages, Ann. Statist. 33 (2005), pp. 1771-1799.] the posterior distribution for an intensity or hazard rate modelled as a mixture under a general multiplicative intensity model is obtained. His proofs rely on the so-called Bayesian Poisson partition calculus. Here we provide alternative proofs based on a different technique.

For rare diseases the observed disease count may exhibit extra Poisson variability, particularly in areas with low or sparse populations. Hence the variance of the estimates of disease risk, the standardized mortality ratios, may be highly unstable. This overdispersion must be taken into account otherwise subsequent maps based on standardized mortality ratios will be misleading and, rather than displaying the true spatial pattern of disease risk, the most extreme values will be highlighted. Neighbouring areas tend to exhibit spatial correlation as they may share more similarities than non-neighbouring areas. The need to address overdispersion and spatial correlation has led to the proposal of Bayesian approaches for smoothing estimates of disease risk. We propose a new model for investigating the spatial variation of disease risks in conjunction with an alternative specification for estimates of disease risk in geographical areas-the multivariate Poisson-gamma model. The main advantages of this new model lie in its simplicity and ability to account naturally for overdispersion and spatial auto-correlation. Exact expressions for important quantities such as expectations, variances and covariances can be easily derived.

This paper provides a novel approach to constructing bivariate prior distributions. The idea is based on the notion of partial exchangeability. In particular, in a simple extension of the exchangeable sequence, we create two dependent exchangeable sequences via a branching mechanism. This implies the existence of a bivariate prior distribution.

This paper investigates nonparametric priors that induce infinite Gibbs-type partitions; such a feature is desirable both from a conceptual and a mathematical point of view. Recently it has been shown that Gibbs-type priors, with sigma is an element of (0, 1), are equivalent to sigma-stable Poisson-Kingman models. By looking at solutions to a recursive equation arising through Gibbs partitions, we provide an alternative proof of this fundamental result. Since practical implementation of general sigma-stable Poisson-Kingman models is difficult, we focus on a related class of priors, namely normalized random measures with independent increments; these are easily implementable in complex Bayesian models. We establish the result that the only Gibbs-type priors within this class are those based on a generalized gamma random measure.

We consider discrete nonparametric priors which induce Gibbs-type exchangeable random partitions and investigate their posterior behavior in detail. In particular, we deduce conditi onal distributions and the corresponding Bayesian nonparametric estimators, which can be readily exploited for predicting various features of additional samples. The results provide useful tools for genomic applications where prediction of future outcomes is required

This paper provides a construction in the Bayesian framework of the Fleming-Viot measure-valued diffusion with diploid fertility selection, and highlights new connections between Bayesian nonparametrics and population genetics. Via a generalisation of the Blackwell-MacQueen Polya-urn scheme, a Markov particle process is defined such that the associated process of empirical measures converges to the Fleming-Viot diffusion. The stationary distribution, known from Ethier and Kurtz (1994), is then derived through an application of the Dirichlet process mixture model and shown to be the de Finetti measure of the particle process. The Fleming-Viot process with haploid selection is derived as a special case

We consider lower bounds for the largest eigenvalue of a symmetric matrix. In particular we extend a recent approach by Piet Van Mieghem

This paper considers the problem of reporting a "posterior distribution" using a parametric family of distributions while working in a nonparametric framework. This "posterior" is obtained as the solution to a decision problem and can be found via a well-known optimization algorithm

In this paper, we provide a Doob-style consistency theorem for stationary models. Many applications involving Bayesian inference deal with non independent and identically distributed data, in particular, with stationary data. However, for such models, there is still a theoretical gap to be filled regarding the asymptotic properties of Bayesian procedures. The primary goal to be achieved is establishing consistency of the sequence of posterior distributions. Here we provide an answer to the problem. Bayesian methods have recently gained growing popularity in economic modeling, thus implying the timeliness of the present paper. Indeed, we secure Bayesian procedures against possible inconsistencies. No results of such a generality are known up to now.

This paper introduces a new approach to the study of rates of convergence for posterior distributions. It is a natural extension of a recent approach to the study of Bayesian consistency. In particular, we improve on current rates of convergence for models including the mixture of Dirichlet process model and the random Bernstein polynomial model.

We introduce approaches to performing Bayesian nonparametric statistical inference for distribution functions exhibiting a stochastic ordering. We consider Polya tree prior distributions, and Bernstein polynomial prior distributions, and each prior provides an appealing and simple way of introducing the stochastic order. (C) 2007 Elsevier B.V. All rights reserved.

This paper combines two ideas to construct autoregressive processes of arbitrary order. The first idea is the construction of first order stationary processes described in Pitt et al. [(2002). Constructing first order autoregressive models via latent processes. Scand. J. Statist. 29, 657-663] and the second idea is the construction of higher order processes described in Raftery [(1985). A model for high order Markov chains. J. Roy Statist. Soc. B. 47, 528-539]. The resulting models provide appealing alternatives to model non-linear and non-Gaussian time series.

We provide details on the full reconstruction of the dynamic equations from measured time series data, given the general class of the underlying physical process. Our results can be used by researchers in physical modelling and statistical mechanics interested in an efficient estimation of low dimensional models, incorporating dynamic as well as observational noise. Our approach is Bayesian, based on an auxiliary variables algorithm that is fast and accurate, and direct, in the sense that only uniform distributions need to be sampled. This method is simpler than other Bayesian approaches where one has to sample from non-standard-unknown distributions using MCMC methods. (c) 2007 Elsevier B.V. All rights reserved.

Rates of convergence of Bayesian nonparametric procedures are expressed as the maximum between two rates: one is determined via suitable measures of concentration of the prior around the "true" density f(0), and the other is related to the way the mass is spread outside a neighborhood of f(0). Here we provide a lower bound for the former in terms of the usual notion of prior concentration and in terms of an alternative definition of prior concentration. Moreover, we determine the latter for two important classes of priors: the infinite-dimensional exponential family, and the Polya trees.

This paper finds conditions under which the generalized hyperbolic ARCH-type model is strictly stationary. Properties of the model are investigated and in particular an estimation procedure is proposed. The resulting stationary model provides with a robust non-Gaussian ARCH-type alternative.

In this paper we introduce a Bayesian semiparametric model for bivariate and multivariate survival data. The marginal densities are well-known nonparametric survival models and the joint density is constructed via a mixture. Our construction also defines a copula and the properties of this new copula are studied. We also consider the model in the presence of covariates and, in particular, we find a simple generalisation of the widely used frailty model, which is based on a new bivariate gamma distribution.

In this paper we show that particular Gibbs sampler Markov processes can be modified to an autoregressive Markov process. The procedure allows the easy derivation of the innovation variables which provide strictly stationary autoregressive processes with fixed marginals. In particular, we provide the innovation variables for beta, gamma and Dirichlet processes.

We introduce an approach for incorporating dependence between outcomes from a Poisson regression model, with the possibility of incorporating covariate information. In common with other approaches, we use a latent process to induce correlation between outcomes. Previous approaches have modelled the Poisson parameter as a function of a latent process which is assumed to be log-normally distributed. Dependence is introduced by the mean of this normal distribution being a function of previous values, using either an auto-regressive process or random walk process. Instead, we use a gamma distribution for the latent variable with the fundamental difference being that instead of the rate of the Poisson distribution at a particular location (in time or space) being directly associated with the value of the latent variable at that location, the latent variables lie on the boundaries between the locations. The rate for a particular location is then modelled as a combination of the latent variables lying on its boundaries; this combination induces correlation between the rates, and thus the outcomes. The attraction of such an approach is the ease of working with a Poisson-gamma set-up in which exact expressions for expectations, variances and covariances are available.

We provide a new approach to the sampling of the well known mixture of Dirichlet process model. Recent attention has focused on retention of the random distribution function in the model, but sampling algorithms have then suffered from the countably infinite representation these distributions have. The key to the algorithm detailed in this article, which also keeps the random distribution functions, is the introduction of a latent variable which allows a finite number, which is known, of objects to be sampled within each iteration of a Gibbs sampler.

This paper provides a construction of a Fleming-Viot measure valued diffusion process, for which the transition function is known, by extending recent ideas of the Gibbs sampler based Markov processes. In particular, we concentrate on the Chapman-Kolmogorov consistency conditions which allows a simple derivation of such a Fleming-Viot process, once a key and apparently new combinatorial result for Polya-urn sequences has been established

The approximate likelihood function introduced by Whittle has been used to estimate the spectral density and certain parameters of a variety of time series models. In this note we attempt to empirically quantify the loss of efficiency of Whittle's method in nonstandard settings. A recently developed representation of some first-order non-Gaussian stationary autoregressive process allows a direct comparison of the true likelihood function with that of Whittle. The conclusion is that Whittle's likelihood can produce unreliable estimates in the non-Gaussian case, even for moderate sample sizes. Moreover, for small samples, and if the autocorrelation of the process is high, Whittle's approximation is not efficient even in the Gaussian case. While these facts are known to some extent, the present study sheds more light on the degree of efficiency loss incurred by using Whittle's likelihood, in both Gaussian and non-Gaussian cases.

Under the principle of minimum description length, the optimal predictive model maximizes the normalized maximum likelihood (NML). While the Bayesian approach to model selection aims to identify the model that best describes the (unknown) true distribution that generated a set of data, the NML approach to model selection makes no reference to a true distribution, and this is seen as a significant advantage of the latter approach. In contrast, this article shows that, for a specific choice of utility function, the NML approach is equivalent to a Bayesian model selection under the Bayesian boostrap and with a specific penalty function for model complexity. This new characterization uncovers some statistical issues about the NML approach. (c) 2006 Elsevier Inc. All rights reserved.

This paper describes a method for sampling from a non-standard distribution which is important in both population genetics and directional statistics. Current approaches rely on complicated procedures which do not work well, if at all, in high dimensions and usual parameter set-ups. We use a Gibbs sampler which seems necessary in practical situations of high dimensions.

A new simulation method, auxiliary random functions is introduced. When used within a Gibbs sampler, this method enables a unified treatment of exact, right-censored, left-censored, left-truncated and interval censored data, with and without covariates in survival models. The models and methods are exemplified via illustrative analysis.

Let Omega be a space of densities with respect to some sigma-finite measure mu and let Pi be a prior distribution having support Omega with respect to some suitable topology. Conditional on f, let X-n = (X-1 ,..., X-n) be an independent and identically distributed sample of size n from f. This paper introduces a Bayesian non-parametric criterion for sample size determination which is based on the integrated squared distance between posterior predictive densities. An expression for the sample size is obtained when the prior is a Dirichlet mixture of normal densities.

This paper extends recent ideas for constructing classes of stationary autoregressive processes of order 1. A Gibbs sampler representation of such processes is extended in a straightforward way to introduce new processes. These maintain a linear expectation property which provides a simple exponential form for the autocorrelation function.

In this paper, we consider the Bayesian posterior distribution as the solution to a minimization rule, first observed by Zellner (1988). The expression to be minimized is a mixture of two pieces, one piece involving the prior distribution, which is minimized by the prior, and the other piece involves the data, which is minimized by the measure putting all the mass on the maximum likelihood estimator. From this perspective of the posterior distribution, Bayesian model selection and the search for an objective prior distribution, can be viewed in a way which is different from usual Bayesian approaches.

This paper considers parametric statistical decision problems conducted within a Bayesian nonparametric context. Our work was motivated by the realisation that typical parametric model selection procedures are essentially incoherent. We argue that one solution to this problem is to use a flexible enough model in the first place, a model that will not be checked no matter what data arrive. Ideally, one would use a nonparametric model to describe all the uncertainty about the density function generating the data. However, parametric models are the preferred choice for many statisticians, despite the incoherence involved in model checking, incoherence that is quite often ignored for pragmatic reasons. In this paper we show how coherent parametric inference can be carried out via decision theory and Bayesian nonparametrics. None of the ingredients discussed here are new, but our main point only becomes evident when one sees all priors-even parametric ones-as measures on sets of densities as opposed to measures on finite-dimensional parameter spaces.

The past decade has seen a remarkable development in the area of Bayesian nonparametric inference from both theoretical and applied perspectives. As for the latter, the celebrated Dirichlet process has been successfully exploited within Bayesian mixture models, leading to many interesting applications. As for the former, some new discrete nonparametric priors have been recently proposed in the literature that have natural use as alternatives to the Dirichlet process in a Bayesian hierarchical model for density estimation. When using such models for concrete applications, an investigation of their statistical properties is mandatory. Of these properties, a prominent role is to be assigned to consistency. Indeed, strong consistency of Bayesian nonparametric procedures for density estimation has been the focus of a considerable amount of research; in particular, much attention has been devoted to the normal mixture of Dirichlet process. In this article we improve on previous contributions by establishing strong consistency of the mixture of Dirichlet process under fairly general conditions. Besides the usual Kullback-Leibler support condition, consistency is achieved by finiteness of the mean of the base measure of the Dirichlet process and an exponential decay of the prior on the standard deviation. We show that the same conditions are also sufficient for mixtures based on priors more general than the Dirichlet process. This leads to the easy establishment of consistency for many recently proposed mixture models.

An approach to constructing strictly stationary AR(1)-type models with arbitrary stationary distributions and a flexible dependence structure is introduced. Bayesian nonparametric predictive density functions, based on single observations, are used to construct the one-step ahead predictive density. This is a natural and highly flexible way to model a one-step predictive/transition density.

In the presence of covariate information, the proportional hazards model is one of the most popular models. In this paper, in a Bayesian nonparametric framework, we use a Markov (Levy-driven) process to model the baseline hazard rate. Previous Bayesian nonparametric models have been based on neutral to the right processes, which have a number of drawbacks, such as discreteness of the cumulative hazard function. We allow the covariates to be time dependent functions and develop a full posterior analysis via substitution sampling. A detailed illustration is presented.

Here we present a novel method for modeling stationary time series. Our approach is to construct the model with a specified marginal family and build the dependence structure around it. We show that the resulting time series is linear with a simple autocorrelation structure. We construct models that parallel existing structures, namely state-space models, autoregressive conditional heteroscedasticity (ARCH) models, and generalized ARCH models. We use Bayesian techniques to estimate the resulting models. We also demonstrate that the models perform well compared with competing methods for the applications considered, count models and volatility models.

We provide a method for improving bounds for nonmaximal eigenvalues of positive matrices. A numerical example indicates the improvements can be Substantial. (C) 2004 Elsevier Inc. All rights reserved.

We deal with strong consistency for Bayesian density estimation. An awkward consequence of inconsistency is described. It is pointed out that consistency at some density f(0) depends on the prior mass assigned to the 'pathological' set of those densities that are close to f(0), in a weak sense, and far apart from f(0), in a Hellinger sense. An analysis of these sets leads to the identification of the notion of 'data tracking'. Specific examples in which this phenomenon cannot occur are discussed. When it can happen, we show how and where things can go wrong, thus providing more intuition about the sources of inconsistency.

This note establishes a connection between Bayes factors and the use of the logarithmic score utility function for model selection in a Bayesian context. The connection presented provides insights into Bayes factors

We define a reinforced stochastic process of random variables indexed by the vertices of a k-tree and with values in a Polish space. The work presents a natural extension from an exchangeable to a partially exchangeable setting of previous work done by the authors

We extend Doob's well-known result on Bayesian consistency The extension covers the case where the nonparametric prior is fully supported by densities. However, our use of martingales differs from that of Doob. We also consider rates.

We use martingales to study Bayesian consistency. We derive sufficient conditions for both Hettinger and Kullback-Leibler consistency, which do not rely on the use of a sieve. Alternative sufficient conditions for Hellinger consistency are also found and demonstrated on examples.

This paper introduces and studies a new class of nonparametric prior distributions. Random probability distribution functions are constructed via normalization of random measures driven by increasing additive processes. In particular, we present results for the distribution of means under both prior and posterior conditions and, via the use of strategic latent variables, undertake a full Bayesian analysis. Our class of priors includes the well-known and widely used mixture of a Dirichlet process.

In this paper we present and investigate a new class of non-parametric priors for modelling a cumulative distribution function. We take F(t) = 1 - exp{-Z(t)}; where Z(t) = integral(t)/(0) x(s) ds is continuous and x((.)) is a Markov process. This is in contrast to the widely used class of neutral to the right priors (Doksum (1974)) for which Z(.) is discrete and has independent increments. The Markov process allows the modelling of trends in Z(.), not possible with independent increments. We derive posterior distributions and present a, full Bayesian analysis.

The innovation random variable for a non-negative self-decomposable random variable can have a compound Poisson distribution. In this case, we provide the density function for the compounded variable. When it does not have a compound Poisson representation, there is a straightforward and easily available compound Poisson approximation for which the density function of the compounded variable is also available. These results can be used in the simulation of Ornstein-Uhlenbeck type processes with given marginal distributions. Previously, simulation of such processes used the inverse of the corresponding tail Levy measure. We show this approach corresponds to the use of an inverse cdf method of a certain distribution. With knowledge of this distribution and hence density function, the sampling procedure is open to direct sampling methods.

In this paper, we highlight properties of Bayesian models in which the prior puts positive mass on all Kullback-Leibler neighborhoods of all densities. These properties are concerned with model choice via the Bayes factor, density estimation and the maximization of expected utility for decision problems. In four illustrations we focus on the Bayes factor and show that whatever models are being compared, the [log(Bayes factor)]/[sample size] converges to a non-random number which has a nice interpretation. A parametric versus semiparametric model comparison provides a fifth illustration.

A survey of modern Bayesian asymptotics is given. Specific attention is paid to the Hellinger consistency of posterior distributions and the asymptotic study of Bayes factors.

This paper contributes to the theory of Bayesian consistency for a sequence of posterior and predictive distributions arising from an independent and identically distributed sample. A new sufficient condition for posterior Hellinger consistency is presented which provides motivation for recent results appearing in the literature. Such motivation is important since current sufficient conditions are not known to be necessary. It also provides new insights into Bayesian consistency. A new consistency theorem for the sequence of predictive densities is given.

We describe a Bayesian semiparametric (failure time) transformation model for which an unknown monotone transformation of failure times is assumed linearly dependent on observed covariates with an unspecified error distribution. The two unknowns: the monotone transformation and error distribution are assigned prior distributions with large supports. Our class of regression model includes the proportional hazards, accelerated failure time, and frailty models. Numerical examples are presented.

Markov chain sampling methods that adapt to characteristics of the distribution being sampled can be constructed using the principle that one can ample from a distribution by sampling uniformly from the region under the plot of its density function. A Markov chain that converges to this uniform distribution can be constructed by alternating uniform sampling in the vertical direction with uniform sampling from the horizontal "slice" defined by the current vertical position, or more generally, with some update that leaves the uniform distribution over this slice invariant. Such "slice sampling" methods are easily implemented for univariate distributions, and can be used to sample from a multivariate distribution by updating each variable in turn. This approach is often easier to implement than Gibbs sampling and more efficient than simple Metropolis updates, due to the ability of slice sampling to adaptively choose the magnitude of changes made. It is therefore attractive for routine and automated use. Slice sampling methods that update all variables simultaneously are also possible. These methods can adaptively choose the magnitudes of changes made to each variable, based on the local properties of the density function. More ambitiously, such methods could potentially adapt to the dependencies between variables by constructing local quadratic approximations. Another approach is to improve sampling efficiency by suppressing random walks. This can be done for univariate slice sampling by "overrelaxation," and for multivariate slice sampling by "reflection" from the edges of the slice.

A bivariate and unimodal distribution is introduced to model an unconventionally distributed data set collected by the Forensic Science Service. This family of distributions allows for a different kurtosis in each orthogonal direction and has a constructive rather than probability density function definition, making conventional inference impossible. However, the construction and inference work well with a Bayesian Markov chain Monte Carlo analysis.

This paper introduces a bivariate Dirichlet process for modelling a partially exchangeable sequence of observables. The proposed model would be relevant when two distributions are unknown but are thought to be close to each other. For two random distributions with the same marginals, the belief in the degree of closeness is expressed through the correlation between masses assigned to equal sets

This paper is concerned with Cheeger-type bounds for nonmaximal eigenvalues of nonnegative irreducible matrices. It is shown that recent upper bounds found by Nabben can be strictly improved when the matrices are positive, stochastic, and reversible, indicating the Nabben bounds are never sharp in this case.

The paper considers a Bayesian nonparametric decision theoretic approach to sample size calculations, where the ultimate goal is to make a terminal action from a finite set of actions. This terminal action is made via the maximization of expected utility, the maximization being made with respect to a probability measure on the states of nature. The probability measure depends on the amount of information, i.e. the number of samples collected. It is the prior in the case of no samples and the posterior when samples have been taken.

In this paper, it is demonstrated that an extension to the exponential power family allows for robustness characteristics for the normal location parameter problem, previously thought to be restricted to the Student-t and a subclass of the positive stable families. (C) 2003 Elsevier B.V. All rights reserved.

In this paper we present a Bayesian non-parametric analysis of survival time data, involving information from two types of treatment. We present an easy to compute Bayes factor comparing two model assumptions: no treatment difference and treatment difference and use this to model summaries for each of the two treatments, in particular predictive distributions.

This article shows how to perform perfect simulation of a functional of a Dirichlet process, when the values of the functional are bounded. It also gives a procedure for “approximate” simulation in the case of unbounded functional values.

This paper generalizes the discrete time independent increment beta process of Hjort (1990), for modelling discrete failure times, and also generalizes the independent gamma process for modelling piecewise constant hazard rates (Walker and Mallick, 1997). The generalizations are from independent increment to Markov increment prior processes allowing the modelling of smoothness. We derive posterior distributions and undertake a full Bayesian analysis.

A new method for estimating individual variability in the von Bertalanffy growth parameters of fish species is presented. The method uses a nonlinear random effects model, which explicitly assumes that an individual's growth parameters represent samples from a multivariate population of growth parameters characteristic of a species or population. The method was applied to backcalculated length-at-age data from the tropical emperor, Lethrinus mahsena. Individual growth parameter variability estimates were compared with those derived using the current "standard" method, which characterizes the joint distribution of growth parameter estimates obtained by independently fitting a growth curve to each individual data set. Estimates of mean von Bertalanffy growth parameters from the two methods were similar. However, estimated growth parameter variances were much higher using the standard method. Using the random effects model, the estimated correlation between population mean values of L-infinity and K was -0.52 or -0.42, depending on the marginal distribution assumed for K. The latter estimate had a 95% posterior credibility interval of -0.62 to -0.17. These represent the first reliable estimate of this correlation and confirm the view that these parameters are negatively correlated in fish populations; however, the absolute correlation value is somewhat lower than has been assumed.

First order stationary autoregressive (AR(1)) models are introduced for which there exists a linear relation between the expectations of the observations, and where it is readily possible to arrange the marginal distributions to be other than normal.

A topic which is receiving much current attention is the problem of Bayesian model averaging. This paper introduces a decision theoretic approach to the selection of the mixing density in the exchangeable data case. This effectively chooses an optimal averaging of the models according to some well-defined criterion.

This paper proposes a predictive approach to Bayesian model selection based on independent and identically distributed observations. In particular, we generalise the criterion of San Martini and Spezzaferri (J. Roy. Statist. Soc. B 46 (1984) 296–303) to take into account more realistic views as discussed by Bernardo and Smith (Bayesian Theory. Wiley, Chichester, 1994). The former authors only consider what the latter authors name the -closed view; that is, the assumption that one of the competing models is the true model. More realistic is the -open view in which it is believed that none of the competing models is the true model. Our new approach can encompass both of these views and moreover we introduce the -mixture view where the experimenter can express prior opinion concerning his/her belief as to whether one of the competing models is the true model or not. Essentially, we embed the -open view in a larger (nonparametric) -closed view.

We consider the Bayesian analysis of constrained parameter and truncated data problems within a Gibbs sampling framework and concentrate on sampling truncated densities that arise as full conditional densities within the context of the Gibbs sampler. In particular, we restrict attention to the normal, beta, and gamma densities. We demonstrate that, in many instances, it is possible to introduce a latent variable which facilitates an easy solution to the problem. We also discuss a novel approach to sampling truncated densities via a “black-box” algorithm, based on the latent variable idea, valid outside of the context of a Gibbs sampler.

We consider a generalization of ridge regression and demonstrate advantages over ridge regression. We provide an empirical Bayes method for determining the ridge constants, using the Bayesian interpretation of ridge estimators, and show that this coincides with a method based on a generalization of the CP statistic and the non-negative garrote. These provide an automatic variable selection procedure for the canonical variables.

The Gompertz distribution has many applications, particularly in medical and actuarial studies. However, there has been little recent work on the Gompertz in comparison with its early investigation. The problem of finding and analysing a bivariate (or multivariate) Gompertz distribution is of interest and the focus of this paper. A search of the literature suggests there is currently no multivariate or even useful bivariate Gompertz distribution.

We consider a sequence of posterior distributions based on a data-dependent prior (which we shall refer to as a pseudoposterior distribution) and establish simple conditions under which the sequence is Hellinger consistent. It is shown how investigations into these pseudo posteriors assist with the understanding of some true posterior distributions, including Pólya trees, the infinite dimensional exponential family and mixture models.

Non-Gaussian processes of Ornstein–Uhlenbeck (OU) type offer the possibility of capturing important distributional deviations from Gaussianity and for flexible modelling of dependence structures. This paper develops this potential, drawing on and extending powerful results from probability theory for applications in statistical analysis. Their power is illustrated by a sustained application of OU processes within the context of finance and econometrics. We construct continuous time stochastic volatility models for financial assets where the volatility processes are superpositions of positive OU processes, and we study these models in relation to financial data and theory.

Karen Barlow and Robert Minns1 report a one in 4065 chance of a child in Scotland having NAHI by age 1 year. They give the annual incidence of shaken impact syndrome as being 24·6 per 100 000. Even though the study sample size and data structure were not described in detail, these risks are clearly the same. Their analysis treats NAHI and shaken impact syndrome as identical, even though part of their report defines shaken impact syndrome as a subgroup of NAHIs. Intuitively, the syndrome that includes an acute encephalopathy with subdural haemorrhages, cerebral oedema, retinal haemorrhages, and fractures, occurring in the context of an inappropriate or inconsistent history, commonly with evidence of other impact or malicious injuries, is an extreme form of non-accidental head injury. We think that the risk estimates should be different. The effects of variation in the data are not described adequately by the cited binomial CI. In their earlier, retrospective study of children younger than 5 years, Barlow and Minns showed appreciable variation beyond that which would be explained by a binomial model.2 For example, in 1990 they identified four cases of NAHI and, in 1994, 26 cases. With such variation, and even with fuller reporting, it is difficult to justify basing an annual incidence on a study of 18 months' duration. We are not convinced that the new term shaken impact syndrome is required. Caffey3 used the term whiplash shaken baby syndrome, and Duhaime and colleagues4 the simpler term shaken baby syndrome. We think that the best term would be one that makes few or no assumptions about mechanisms of injury, which are contentious. Barlow and Minns did not assess how many children had injuries with these clinical features that were classified as accidental. This information is essential to study regional variations in incidence and to decide whether these arise because of differences in diagnostic classification. Changes in classification might be affected by time trends and substantially alter assessments of new policies. The investigators say that their measurement of incidence for NAHI in infants younger than 1 year is precise and will help in assessing the impact of new policies. We encourage the government to consider policies that, although unpopular, might lead to a lower incidence of disabling non-accidental injuries in children. We suggest that estimates from several sources are used when assessing the effect of new laws.