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Ecological modelling often involves addressing challenges such as dependence in responses, e.g., spatial and/or temporal correlation, heterogeneity of variance, and hierarchical structures inherent in ecological processes and data. A constant challenge is the inadequacy of the data to well address the questions of interest. What is observable may not be sufficiently informative. What has been observed may not have been well designed. Carefully conceived modelling can take us to improved inference compared with adopting standard inference tools like basic regression and analysis of variance. Hierarchical modelling techniques provide a powerful framework for capturing these complexities by explicitly modelling the multi-level structure of ecological systems. In this paper we focus on good modelling practice in the hierarchical Bayesian framework. We discuss good modelling practice in the form of 10 steps, elaborated with a running example, to aid quantitative ecologists interested in employing Bayesian modelling. Two of the authors are statisticians who view themselves primarily as stochastic modellers. The other two are quantitative ecologists who view such stochastic tools as essential for advancing ecological knowledge. Hence, implicitly, we pay careful attention to good modelling practice. Further, we argue the benefits that accrue to working in a Bayesian framework as the paper is developed. It is worth noting that the steps we propose, though presented in the context of ecological modelling, are appropriate for effective model building across most fields of application. [ABSTRACT FROM AUTHOR]

Background: As levels of anthropogenic noise in the marine environment rise, it is crucial to quantify potential associated effects on marine mammals. Yet measuring responses is challenging because most species spend the majority of their time submerged. Consequently, much of their sub-surface behavior is difficult or impossible to observe and it can be difficult to determine if—during or following an exposure to sound—an observed dive differs from previously recorded dives. We propose a method for initial assessment of potential behavioral responses observed during controlled exposure experiments (CEEs), in which animals are intentionally exposed to anthropogenic sound sources. To identify possible behavioral responses in dive data collected from satellite-linked time–depth recorders, and to inform the selection and parameters for subsequent individual and population-level response analyses, we propose to use kernel density estimates of conditional distributions for quantitative comparison of pre- and post-exposure behavior. Results: We apply the proposed method to nine Cuvier's beaked whales (Ziphius cavirostris) exposed to a lower-amplitude simulation of Mid-Frequency Active Sonar within the context of a CEE. The exploratory procedure provides evidence that exposure to sound causes animals to change their diving behavior. Nearly all animals tended to dive deep immediately following exposure, potentially indicating avoidance behavior. Following the initial deep dive after exposure, the procedure provides evidence that animals either avoided deep dives entirely or initiated deep dives at unusual times relative to their pre-exposure, baseline behavior patterns. The procedure also provides some evidence that animals exposed as a group may tend to respond as a group. Conclusions: The exploratory approach we propose can identify potential behavioral responses across a range of diving parameters observed during CEEs. The method is particularly useful for analyzing data collected from animals for which neither the baseline, unexposed patterns in dive behavior nor the potential types or duration of behavioral responses is well characterized in the literature. The method is able to be applied in settings where little a priori knowledge is known because the statistical analyses employ kernel density estimates of conditional distributions, which are flexible non-parametric techniques. The kernel density estimates allow researchers to initially assess potential behavioral responses without making strong, model-based assumptions about the data. [ABSTRACT FROM AUTHOR]

The distinction between an overlap in species daily activity patterns and proximate co-occurrence of species for a location and time due to behavioral attraction or avoidance is critical when addressing the question of species co-occurrence. We use data from a dense grid of camera traps in a forest in central North Carolina to inform about proximate co-occurrence. Camera trigger times are recorded when animals pass in front of the camera’s field of vision. We view the data as a point pattern over time for each species and model the intensities driving these patterns. These species-specific intensities are modeled jointly in linear time to preserve the notion of co-occurrence. We show that a multivariate log-Gaussian Cox process incorporating both circular and linear time provides a preferred choice for modeling occurrence of forest mammals based on daily activity rhythms. Model inference is obtained under a hierarchical Bayesian framework with an efficient Markov chain Monte Carlo sampling algorithm. After model fitting, we account for imperfect detection of individuals by the camera traps by incorporating species-specific detection probabilities that adjust estimates of occurrence and co-occurrence. We obtain rich inference including assessment of the probability of presence of one species in a particular time interval given presence of another species in the same or adjacent interval, enabling probabilities of proximate co-occurrence. Our results describe the ecology and interactions of four common mammals within this suburban forest including their daily rhythms, responses to temperature and rainfall, and effects of the presence of predator species. Supplementary materials accompanying this paper appear online. [ABSTRACT FROM AUTHOR]

It is often of interest to model the incidence and duration of threshold exceedance events for an environmental variable over a set of monitoring locations. Such data arrive over continuous time and can be considered as observations of a two-state process yielding, sequentially, a length of time in the below threshold state followed by a length of time in the above threshold state, then returning to the below threshold state, etc. We have a two-state continuous time Markov process, often referred to as an alternating renewal process. The process is observed over a truncated time window and, within this window, duration in each state is modeled using a distinct cumulative intensity specification. Initially, we model each intensity over the window using a parametric regression specification. We extend the regression specification adding temporal random effects to enrich the model using a realization of a log Gaussian process over time. With only one type of renewal, this specification is referred to as a Gaussian process modulated renewal process. Here, we introduce Gaussian process modulation to the intensity for each state. Model fitting is done within a Bayesian framework. We clarify that fitting with a customary log Gaussian process specification over a lengthy time window is computationally infeasible. The nearest neighbor Gaussian process, which supplies sparse covariance structure, is adopted to enable tractable computation. We propose methods for both generating data under our models and for conducting model comparison. The model is applied to hourly ozone data for four monitoring sites at different locations across the United States for the ozone season of 2014. For each site, we obtain estimated profiles of up-crossing and down-crossing intensity functions through time. In addition, we obtain inference regarding the number of exceedances, the distribution of the duration of exceedance events, and the proportion of time in the above and below threshold state for any time interval. [ABSTRACT FROM AUTHOR]

In many applications involving spatial point patterns, we find evidence of inhibition or repulsion. The most commonly used class of models for such settings are the Gibbs point processes. A recent alternative, at least to the statistical community, is the determinantal point process. Here, we examine model fitting and inference for both of these classes of processes in a Bayesian framework. While usual MCMC model fitting can be available, the algorithms are complex and are not always well behaved. We propose using approximate Bayesian computation (ABC) for such fitting. This approach becomes attractive because, though likelihoods are very challenging to work with for these processes, generation of realizations given parameter values is relatively straightforward. As a result, the ABC fitting approach is well-suited for these models. In addition, such simulation makes them well-suited for posterior predictive inference as well as for model assessment. We provide details for all of the above along with some simulation investigation and an illustrative analysis of a point pattern of tree data exhibiting repulsion. R code and datasets are included in the supplementary material. [ABSTRACT FROM PUBLISHER]

This paper develops methodology for local sensitivity analysis based on directional derivatives associated with spatial processes. Formal gradient analysis for spatial processes was elaborated in previous papers, focusing on distribution theory for directional derivatives associated with a response variable assumed to follow a Gaussian process model. In the current work, these ideas are extended to additionally accommodate a continuous covariate whose directional derivatives are also of interest and to relate the behavior of the directional derivatives of the response surface to those of the covariate surface. It is of interest to assess whether, in some sense, the gradients of the response follow those of the explanatory variable. The joint Gaussian structure of all variables, including the directional derivatives, allows for explicit distribution theory and, hence, kriging across the spatial region using multivariate normal theory. Working within a Bayesian hierarchical modeling framework, posterior samples enable all gradient analysis to occur post model fitting. As a proof of concept, we show how our methodology can be applied to a standard geostatistical modeling setting using a simulation example. For a real data illustration, we work with point pattern data, deferring our gradient analysis to the intensity surface, adopting a log-Gaussian Cox process model. In particular, we relate elevation data to point patterns associated with several tree species in Duke Forest. [ABSTRACT FROM AUTHOR]

In spatial analysis, typically we specify a region of interest and consider a spatial surface over the region. It is often of interest to ascertain where the surface is changing rapidly. Identifying locations or curves where there is rapid change is referred to as wombling. The surface may arise continuously over the region or discretely, in which case values are provided for a collection of areal units. In either setting, algorithmic strategies are available to attempt to identify so-called wombling boundaries. In this study, the surfaces of interest are all assumed to be random, realizations of a Gaussian process in the continuous case, of a Markov random field in the discrete case. With specifications given as stochastic models, we discuss Bayesian approaches to implement desired boundary analysis. We refer to this as Bayesian wombling and show how fully model-based inference can be carried out, including assessment of uncertainty. The approach for the continuous case is more theoretically demanding (expected with an uncountable set of locations) but yields elegant distribution theory. The discrete case is more straightforward. Each case is illustrated with a brief example. WIREs Comput Stat 2015, 7:307-315. doi: 10.1002/wics.1360 For further resources related to this article, please visit the . [ABSTRACT FROM AUTHOR]

Directional data naturally arise in many scientific fields, such as oceanography (wave direction), meteorology (wind direction), and biology (animal movement direction). Our contribution is to develop a fully model-based approach to capture structured spatial dependence for modeling directional data at different spatial locations. We build a projected Gaussian spatial process, induced from an inline bivariate Gaussian spatial process. We discuss the properties of the projected Gaussian process and show how to fit this process as a model for data, using suitable latent variables, with Markov chain Monte Carlo methods. We also show how to implement spatial interpolation and conduct model comparison in this setting. Simulated examples are provided as proof of concept. A data application arises for modeling wave direction data in the Adriatic sea, off the coast of Italy. In fact, this directional data is available across time, requiring a spatio-temporal model for its analysis. We discuss and illustrate this extension. [ABSTRACT FROM AUTHOR]

A frequent challenge encountered with ecological data is how to interpret, analyze, or model data having a high proportion of zeros. Much attention has been given to zero-inflated count data, whereas models for non-negative continuous data with an abundance of 0s are much fewer. We consider zero-inflated data on the unit interval and provide modeling to capture two types of 0s in the context of a Beta regression model. We model 0s due to missing by chance through left-censoring of a latent regression and 0s due to unsuitability using an independent Bernoulli specification. We extend the model by introducing spatial random effects. We specify models hierarchically, employing latent variables, and fit them within a Bayesian framework. Our motivating dataset consists of percent cover abundance of two plant families at a collection of sites in the Cape Floristic Region of South Africa. We find that environmental features enable learning about both types of 0s as well as positive percent cover. We also show that the spatial random effects model improves predictive performance. The proposed modeling enables ecologists to extract a better understanding of an organism's absence due to unsuitability vs. missingness by chance, as well as abundance behavior when present. Supplementary materials accompanying this paper appear online. [ABSTRACT FROM AUTHOR]

We provide methods that can be used to obtain more accurate environmental exposure assessment. In particular, we propose two modeling approaches to combine monitoring data at point level with numerical model output at grid cell level, yielding improved prediction of ambient exposure at point level. Extending our earlier downscaler model (Berrocal, V. J., Gelfand, A. E., and Holland, D. M. (2010b). A spatio-temporal downscaler for outputs from numerical models. Journal of Agricultural, Biological and Environmental Statistics 15, 176-197), these new models are intended to address two potential concerns with the model output. One recognizes that there may be useful information in the outputs for grid cells that are neighbors of the one in which the location lies. The second acknowledges potential spatial misalignment between a station and its putatively associated grid cell. The first model is a Gaussian Markov random field smoothed downscaler that relates monitoring station data and computer model output via the introduction of a latent Gaussian Markov random field linked to both sources of data. The second model is a smoothed downscaler with spatially varying random weights defined through a latent Gaussian process and an exponential kernel function, that yields, at each site, a new variable on which the monitoring station data is regressed with a spatial linear model. We applied both methods to daily ozone concentration data for the Eastern US during the summer months of June, July and August 2001, obtaining, respectively, a 5% and a 15% predictive gain in overall predictive mean square error over our earlier downscaler model (Berrocal et al., 2010b). Perhaps more importantly, the predictive gain is greater at hold-out sites that are far from monitoring sites. [ABSTRACT FROM AUTHOR]

Many applications involve count data from a process that yields an excess number of zeros. Zero-inflated count models, in particular, zero-inflated Poisson (ZIP) and zero-inflated negative binomial (ZINB) models, along with Poisson hurdle models, are commonly used to address this problem. However, these models struggle to explain extreme incidence of zeros (say more than 80%), especially to find important covariates. In fact, the ZIP may struggle even when the proportion is not extreme. To redress this problem we propose the class of k-ZIG models. These models allow more flexible modeling of both the zero-inflation and the nonzero counts, allowing interplay between these two components. We develop the properties of this new class of models, including reparameterization to a natural link function. The models are straightforwardly fitted within a Bayesian framework. The methodology is illustrated with simulated data examples as well as a forest seedling dataset obtained from the USDA Forest Service's Forest Inventory and Analysis program. [ABSTRACT FROM AUTHOR]

Traditional spatial linear regression models assume that the mean of the response is a linear combination of predictors measured at the same location as the response. In spatial applications, however, it seems plausible that neighboring predictors can also inform about the response. This article proposes using unobserved kernel averaged predictors in such regressions. The kernels are parametric introducing additional parameters that are estimated with the data. Properties and challenges of using kernel averaged predictors within a regression model are detailed in the simple case of a univariate response and a single predictor. Additionally, extensions to multiple predictors and generalized linear models are discussed. The methods are demonstrated using a data set of dew duration and shrub density. Supplemental materials are available online. [ABSTRACT FROM AUTHOR]

Birthweight and gestational age are closely related and represent important indicators of a healthy pregnancy. Customary modeling for birthweight is conditional on gestational age. However, joint modeling directly addresses the relationship between gestational age and birthweight, and provides increased flexibility and interpretation as well as a strategy to avoid using gestational age as an intermediate variable. Previous proposals have utilized finite mixtures of bivariate regression models to incorporate well-established risk factors into analysis (e.g. sex and birth order of the baby, maternal age, race, and tobacco use) while examining the non-Gaussian shape of the joint birthweight and gestational age distribution. We build on this approach by demonstrating the inferential (prognostic) benefits of joint modeling (e.g. investigation of 'age inappropriate' outcomes like small for gestational age) and hence re-emphasize the importance of capturing the non-Gaussian distributional shapes. We additionally extend current models through a latent specification which admits interval-censored gestational age. We work within a Bayesian framework which enables inference beyond customary parameter estimation and prediction as well as exact uncertainty assessment. The model is applied to a portion of the 2003-2006 North Carolina Detailed Birth Record data ( n=336129) available through the Children's Environmental Health Initiative and is fitted using the Bayesian methodology and Markov chain Monte Carlo approaches. Copyright © 2010 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

Sensitivity and specificity are two customary performance measures associated with medical diagnostic tests. Typically, they are modeled independently as a function of risk factors using logistic regression, which provides estimated functions for these probabilities. Change in these probabilities across levels of risk factors is of primary interest and the indirect relationship is often displayed using a receiver operating characteristic curve. We refer to this as analysis of 'first-order' behavior. Here, we consider what we refer to as 'second-order' behavior where we examine the stochastic dependence between the (random) estimates of sensitivity and specificity. To do so, we argue that a model for the four cell probabilities that determine the joint distribution of screening test result and outcome result is needed. Such a modeling induces sensitivity and specificity as functions of these cell probabilities. In turn, this raises the issue of a coherent specification for these cell probabilities, given risk factors, i.e. a specification that ensures that all probabilities calculated under it fall between 0 and 1. This leads to the question of how to provide models that are coherent and mechanistically appropriate as well as computationally feasible to fit, particularly with large data sets. The goal of this article is to illuminate these issues both algebraically and through analysis of a real data set. Copyright © 2008 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

This article proposes a space-time model for daily 8-hour maximum ozone levels to provide input for regulatory activities: detection, evaluation, and analysis of spatial patterns and temporal trend in ozone summaries. The model is applied to the analysis of data from the state of Ohio that contains a mix of urban, suburban, and rural ozone monitoring sites. The proposed space-time model is autoregressive and incorporates the most important meteorological variables observed at a collection of ozone monitoring sites as well as at several weather stations where ozone levels have not been observed. This misalignment is handled through spatial modeling. In so doing we adopt a computationally convenient approach based on the successive daily increments in meteorological variables. The resulting hierarchical model is specified within a Bayesian framework and is fitted using Markov chain Monte Carlo techniques. Full inference with regard to model unknowns as well as for predictions in time and space, evaluation of annual summaries, and assessment of trends are presented. [ABSTRACT FROM AUTHOR]

Modélisation de I'erreur associée à la position d'un objet sur une carte en vue de sa localisation Les auteurs s'intéressent à l'erreur associée à la position d'un objet sur une carte, c'est-à-dire à la différence entre sa position telle que représentée par ses coordonnées spatiales (c'est-à-dire sur une carte) et sa véritable localisation inconnue dans l'espace. Ils proposent un modèle tenant compte de différents aspects des procédés de cartographie tels que mis en œuvre dans les systèmes d'information à référence géographique, y compris la correction géométrique par membrane élastique. Ils expliquent comment estimer l'erreur associée au relevé et donc la position réelle d'un objet. Us montrent qu'en opérant une moyenne sur différents modèles et en s'aidant de cartes de précision variée et de données de contrǒle au sol, on peut arriver à déterminer la véritable position de l'objet. [ABSTRACT FROM AUTHOR]

Large-scale inference for random spatial surfaces over a region using spatial process models has been well studied. Under such models, local analysis of the surface (e.g., gradients at given points) has received recent attention. A more ambitious objective is to move from points to curves, to attempt to assign a meaningful gradient to a curve. For a point, if the gradient in a particular direction is large (positive or negative), then the surface is rapidly increasing or decreasing in that direction. For a curve, if the gradients in the direction orthogonal to the curve tend to be large, then the curve tracks a path through the region where the surface is rapidly changing. In the literature, learning about where the surface exhibits rapid change is called wombling, and a curve such as we have described is called a wombling boundary. Existing wombling methods have focused mostly on identifying points and then connecting these points using an ad hoc algorithm to create curvilinear wombling boundaries. Such methods are not easily incorporated into a statistical modeling setting. The contribution of this article is to formalize the notion of a curvilinear wombling boundary in a vector analytic framework using parametric curves and to develop a comprehensive statistical framework for curvilinear boundary analysis based on spatial process models for point-referenced data. For a given curve that may represent a natural feature (e.g., a mountain, a river, or a political boundary), we address the issue of testing or assessing whether it is a wombling boundary. Our approach is applicable to both spatial response surfaces and, often more appropriately, spatial residual surfaces. We illustrate our methodology with a simulation study, a weather dataset for the state of Colorado, and a species presence/absence dataset from Connecticut. [ABSTRACT FROM AUTHOR]

Many biomedical data sets are concerned with relating the result of screening procedure(s) for a clinical event to the occurrence of that event. The effect of risk factors on measures of accuracy such as positive predictive value and negative predictive value is of great interest for clinicians. In this paper we propose a generic approach to estimate these measures of accuracy in the setting where an explanatory model has been fitted to the joint screening and event outcome data but information on one or more risk factors in the model is not available. We refer to these as conditional rates, i.e. rates conditioned on only a subset of risk factors. We argue that, based upon the joint distribution of the event outcome, the screening result and the risk factor occurrence, a formal expression for such a rate can be obtained. This expression is a function of model parameters and thus can be estimated once the model has been fitted. Inference within the Bayesian framework is particularly attractive since simulation based model fitting straightforwardly yields samples from the posterior distribution of any conditional rate of interest. We perform a simulation study to compare these estimated conditional rates with frequently used ad hoc estimates. Differences can be substantial. We also illustrate the proposed methodology to compute conditional positive predictive value for a screening mammography data set. The proposed approach is also applicable when there are multiple diagnostic screening test outcomes. [ABSTRACT FROM AUTHOR]

Together, graphical models and the Bayesian paradigm provide powerful new tools that promise to change the way that environmental science is done. The capacity to merge theory with mechanistic understanding and empirical evidence, to assimilate diverse sources of information and to accommodate complexity will transform the collection and interpretation of data. As we discuss here, we specifically expect a shift from a focus on simple experiments with inflexible design and selection among models that embrace parts of processes to a synthesis of integrated process models. With this potential come new challenges, including some that are specific and technical and others that are general and will involve reexamination of the role of inference and prediction. [Copyright &y& Elsevier]

Screening mammography is a widely used method for breast cancer detection. For each mammogram we propose a performance model based on order of outcomes. That is, we envision an initial assessment, a follow up assessment if the initial one is positive and, eventually, a determination of whether cancer was present or not. A model can be built at each stage reflecting effects due to patient characteristics, to the facility where mammogram was performed and to the radiologist reading the mammogram. Since assessment is not perfectly associated with outcome, familiar rates of agreement and disagreement are of interest. These rates can be investigated at various levels of risk factors of interest. The approach is illustrated with screening mammography data from the Group Health Cooperative in Seattle, WA. A Bayesian framework is adopted for inference and an analysis of the data set is presented. [ABSTRACT FROM AUTHOR]

Customary modeling for continuous point-referenced data assumes a Gaussian process that is often taken to be stationary. When such models are fitted within a Bayesian framework, the unknown parameters of the process are assumed to be random, so a random Gaussian process results. Here we propose a novel spatial Dirichlet process mixture model to produce a random spatial process that is neither Gaussian nor stationary. We first develop a spatial Dirichlet process model for spatial data and discuss its properties. Because of familiar limitations associated with direct use of Dirichlet process models, we introduce mixing by convolving this process with a pure error process. We then examine properties of models created through such Dirichlet process mixing. In the Bayesian framework, we implement posterior inference using Gibbs sampling. Spatial prediction raises interesting questions, but these can be handled. Finally, we illustrate the approach using simulated data, as well as a dataset involving precipitation measurements over the Languedoc-Roussillon region in southern France. [ABSTRACT FROM AUTHOR]

Abstract: There is by now a substantial literature on spatio-temporal modeling. However, to date, there exists essentially no literature which addresses the issue of process change from a certain time. In fact, if we look at change points for purely time series data, the customary form is to propose a model involving a mean or level shift. We see little attempting to capture a change in association structure. Part of the concern is how to specify flexible ways to bridge the association across the time point and still ensure that a proper joint distribution has been defined for all of the data. Introducing a spatial component evidently adds further complication. We want to allow for a change-point reflecting change in both temporal and spatial association. In this paper we propose a constructive, flexible model formulation through additive specifications. We also demonstrate how computational concerns benefit from the availability of temporal order. Finally, we illustrate with several simulated datasets to examine the capability of the model to detect different types of structural changes. [Copyright &y& Elsevier]

Spatial process models are now widely used for inference in many areas of application. In such contexts interest is often in the rate of change of a spatial surface at a given location in a given direction. Examples include temperature or rainfall gradients in meteorology, pollution gradients for environmental data, and surface roughness assessment for digital elevation models. Because the spatial surface is viewed as a random realization, all such rates of change are random as well. We formalize the notions of directional finite difference processes and directional derivative processes building upon the concept of mean square differentiability as developed by Stein and Banerjee and Gelfand. We obtain complete distribution theory results under the assumptions of a stationary Gaussian process model either for the data or for spatial random effects. We present inference under a Bayesian framework which, in this setting, presents several advantages. Finally, we illustrate our methodology with a simulated dataset and also with a real estate dataset consisting of selling prices of individual homes. [ABSTRACT FROM AUTHOR]

With survival data there is often interest not only in the survival time distribution but also in the residual survival time distribution. In fact, regression models to explain residual survival time might be desired. Building upon recent work of Kottas & Gelfand [ J. Amer. Statist. Assoc. 96 (2001) 1458], we formulate a semiparametric median residual life regression model induced by a semiparametric accelerated failure time regression model. We utilize a Bayesian approach which allows full and exact inference. Classical work essentially ignores covariates and is either based upon parametric assumptions or is limited to asymptotic inference in non-parametric settings. No regression modelling of median residual life appears to exist. The Bayesian modelling is developed through Dirichlet process mixing. The models are fitted using Gibbs sampling. Residual life inference is implemented extending the approach of Gelfand & Kottas [ J. Comput. Graph. Statist. 11 (2002) 289]. Finally, we present a fairly detailed analysis of a set of survival times with moderate censoring for patients with small cell lung cancer. [ABSTRACT FROM AUTHOR]

Dans un modèle multi-couches, il y a plusieurs façons de procéder à une marginalisation, et donc plus d'une vraisemblance marginale. Sachant que les log-vraisemblances sont souvent employées pour comparer des modèles entre eux, on se demande quelle vraisemblance choisir pour un modèle particulier. Adoptant une approche bayésienne, les auteurs jettent un éclairage nouveau sur la comparaison qualitative des diverses vraisemblances associées à un modèle donné. Ils lient ces résultats à des questions périphéri-ques concernant le nombre de paramètres, la pénalisation et la définition d'un critère de choix de modèle cohérent avec la vraisemblance. Dans le cadre particulier d'un modèle à deux couches, ils montrent que sous des conditions très générales, l'espérance de la vraisemblance de premier niveau est, a priori, plus petite que celle de la vraisemblance marginale, et ce peu importe le choix de la loi a priori sur les hyper-paramètres, le nombre ou la valeur des observations recueillies. A posteriori, l'inégalité est renversée et l'écart se creuse à mesure qu'augmentent la taille échantillonnale et le nombre de couches du modèle. [ABSTRACT FROM AUTHOR]

In many applications, the objective is to build regression models to explain a response variable over a region of interest under the assumption that the responses are spatially correlated. In nearly all of this work, the regression coefficients are assumed to be constant over the region. However, in some applications, coefficients are expected to vary at the local or subregional level. Here we focus on the local case. Although parametric modeling of the spatial surface for the coefficient is possible, here we argue that it is more natural and flexible to view the surface as a realization from a spatial process. We show how such modeling can be formalized in the context of Gaussian responses providing attractive interpretation in terms of both random effects and explaining residuals. We also offer extensions to generalized linear models and to spatio-temporal setting. We illustrate both static and dynamic modeling with a dataset that attempts to explain (log) selling price of single-family houses. [ABSTRACT FROM AUTHOR]

Model choice is a fundamental problem in data analysis. With interest in hierarchical models which typically arise as Bayesian specifications, we confine ourselves to Bayesian model choice criteria. If Y denotes the observed data and T(Y) is the criterion, our goal is to calibrate T(Y) in order to assess how large or small it is under a given model. In particular, if we have the distribution of T(Y), then we can compute any probabilities or determine any quantities of interest.Apart from very special cases, analytic development of such distributions is intractable. Standard analytic approximations may be inapplicable if usual random effects are introduced at the various modeling levels. Indeed, calculation of T(Y) itself is often difficult enough.We suggest a generic simulation-intensive approach for obtaining the distribution of T(Y) to arbitrary accuracy. We focus on various Bayes factors, e.g., the usual Bayes factor, the posterior Bayes factor and the pseudo-Bayes factor. We illustrate with a binomial regression example. [Copyright &y& Elsevier]

Provides a computational approach to obtain the entire posterior distribution for more general functionals. Discussion on dirichlet process mixing; Inference for functionals; Comparison of populations of survival data.

Median regression models become an attractive alternative to mean regression models when employing flexible families of distributions for the errors. Classical approaches are typically algorithmic with desirable properties emerging asymptotically. However, nonparametric error models may be most attractive in the case of smaller sample sizes where parametric specifications are difficult to justify. Hence, a Bayesian approach. enabling exact inference given the observed data. may he appealing. In this context there is little Bayesian work. We develop two fully Bayesian modeling approaches. employing mixture models, for the errors in a median regression model. The associated families of error distributions allow increased variability, skewness, and flexible tail behavior. The first family is semiparametric with extra variability captured nonparametrically through mixing and handled parametrically. The second family, a fully nonparametric one, includes all unimodal densities on the real line with median (and model) equal to zero. In conjunction with a parametric regression specification, two semiparantetric median regression models arise. Alter fitting such models by using Gibbs sampling, full posterior inference for general population is possible. The approach can also be applied when censored observations are present, leading to semiparametric censored median regression modeling. We illustrate with two examples. one involving censuring. [ABSTRACT FROM AUTHOR]

This article provides information on the Gibbs sampling method in statistics. During the course of the 1990s, the technology generally referred to as Markov chain Monte Carlo (MCMC) has revolutionized the way statistical models are fitted and, in the process, dramatically revised the scope of models which can be entertained. I reviewed its origins and its crossover into the mainstream statistical literature. I then attempt an assessment of the impact of Gibbs sampling on the research community, on both statisticians and subject area scientists. Finally, I offer some thoughts on where the technology is headed and what needs to be done as we move into the next millennium. Gibbs sampling is a simulation tool for obtaining samples from a nonnormalized joint density function. Such samples may be marginalized providing samples from the marginal distributions associated with the joint density. The Gibbs sampler was introduced as a MCMC tool in the context of image restoration by Geman and Geman in 1984 and Geifand and Smith in 1990 offered the Gibbs sampler as a very general approach for fitting statistical models, extending the applicability of the work of Geman and Geman and also broadening the substitution sampling ideas that Tanner and Wong proposed under the name of data augmentation.

Presents a study which analyzed generalized linear models, using parameter identifiability and posterior propriety. Discussion of the identifiability issue; Information on the propriety problem; Development of the embedding notion.

Model choice is a fundamental and much discussed activity in the analysis of datasets. Nonnested hierarchical models introducing random effects may not be handled by classical methods. Bayesian approaches using predictive distributions can be used though the formal solution, which includes Bayes factors as a special case, can be criticised. We propose a predictive criterion where the goal is good prediction of a replicate of the observed data but tempered by fidelity to the observed values. We obtain this criterion by minimising posterior loss for a given model and then, for models under consideration, selecting the one which minirnises this criterion. For a broad range of losses, the criterion emerges as a form partitioned into a goodness-of-fit term and a penalty term. We illustrate its performance with an application to a large dataset involving residential property transactions. [ABSTRACT FROM AUTHOR]

Les tenants de l'approche non paramétrique bayésienne, lorsqu'ils estiment une loi de probabilité inconnue à partir d'un échantillon aléatoire de sa population, représentent généralement l'information a priori au moyen d'un processus de Dirichlet. Jusqu'à présent, les travaux réalisés dans ce domaine ont surtout porté sur l'espérance a posteriori de ce processus, qui permet de faire des prévisions sur la valeur éventuelle d'une future observation, étant donné les données déjà recueillies. L'objet de cet article est de mettre en lumière d'autres caractéristiques de ce processus a posteriori et des processus a posteriori correspondant aux fonctionnelles de la loi des données, notamment des fonctionnelles linéaires. À titre d'illustration, on montre comment il est possible d'identifier et d'étudier les propriétés du processus a posteriori, lorsque les données sont issues d'une loi gamma et qu'a priori, le processus de Dirichlet s'appuie sur un mélange de lois normales. [ABSTRACT FROM AUTHOR]

Les modèles statistiques comportant des données contraintes ou manquantes produisent des vraisemblances contenant des intégrales. Dans le cas où les données sont à la fois contraintes et manquantes, le résultat est un rapport d'intégrales. Il est parfois impossible d'obtenir une expression exacte ou měme approximative de ce rapport pour des données à plusieurs variables. Dans le but d'obtenir des estimateurs du maximum de vraisemblance, nous proposons des approximations Monte Carlo pour ces intégrales, et par la suite maximisons la vraisemblance approximative qui en résulte. L'itération de cette stratégie accélère la maximisation, tandis que l'échantillonnage de Gibbs est utile pour la génération Monte Carlo. En conséquence, nous considérons une classe de modèles plus vaste que le cadre EM habituel, sans utiliser un algorithme de type EM. La mise en pratique de cette méthodologie est illustrée par deux exemples numériques. [ABSTRACT FROM AUTHOR]

Biological size relationships (i.e., regression of one size variable on another) are typically monotone but often break down at extreme values of the variables. Here we study the way in which a plant's biomass is apportioned to reproductive and other life activities, Working with a theory proposed by Weiner (1988) based an analogy between a biological plant and an industrial plant leads to a truncated regression model formulation. We consider a dataset involving 542 goldenrod plants that has been analyzed in a limited fashion by others. Important extensions that we provide include nonparametric modeling of the size relationship, introduction of covariate information, incorporation of heterogeneity across plant families, and inclusion of measurement error models for both response and explanatory variables. Our approach is through hierarchical models taking advantage of available prior information on the magnitudes of the site variables. Models are fitted using simulation methods enabling a full range of inference. An attractive model choice criterion demonstrates the need to accommodate all of the aforementioned aspects for the given dataset. Our flexible modeling approach can be adapted to investigate other biological size relationships. [ABSTRACT FROM AUTHOR]

Considers generalized linear models (GLMs) as an attractive and widely used class of models and limited in the range of density shapes that they can provide. Basics of Dirichlet process (DP) mixing; How model determination may be investigated; Extended generalized linear models; Model choice for Dirichlet process mixed model; Conclusions.

Expert opinion is often sought with regard to unknowns in a decision-making setting. For a univariate unknown, θ, our presumption is that such opinion is elicited as a partial probabilistic specification in the form of either probability assignments regarding the chance of θ falling in a fixed set of disjoint exhaustive intervals or selected quantiles for θ. Treating such specification as "data," our focus is on the development of suitable probability densities for these data given the true θ. In particular, we advocate a rich class of densities created by transformation of random mixtures of beta distributions. These densities become likelihoods when viewed as a function of θ given the data. We presume that a decision-maker (here a so-called supra Bayesian) presides over the opinion collection, offering his or her assessment as well. All of this opinion is synthesized using Bayes's theorem, resulting in the posterior distribution as the pooling mechanism. The models are applied to opinion collected regarding points per game for participants in the 1991 National Basketball Association championship basketball series. [ABSTRACT FROM AUTHOR]

Constrained parameter problems arise in a wide variety of applications, including bioassay, actuarial graduation, ordinal categorical data, response surfaces, reliability development testing, and variance component models. Truncated data problems arise naturally in survival and failure time studies. ordinal data models, and categorical data studies aimed at uncovering underlying continuous distributions. In many applications both parameter constraints and data truncation are present. The statistical literature on such problems is very extensive, reflecting both the problems' widespread occurrence in applications and the methodological challenges that they pose. However. it is striking that so little of this applied and theoretical literature involves a parametric Bayesian perspective. From a technical viewpoint, this perhaps is not difficult to understand. The fundamental toot for Bayesian calculations in typical realistic models is (multidimensional) numerical integration, which often is problematic in unconstrained contexts and can be well- nigh impossible for the kinds of constrained problems we consider. In this article we show that Bayesian calculations can be implemented routinely for constrained parameter and truncated data problems by means of the Gibbs sampler. Specific models discussed include constrained multinormal parameters. constrained linear model parameters, ordered parameters in experimental family models, data and order restricted parameters from exponential distributions, straight line regression with censoring and bivariate grouped data models. Analysis of data sets illustrating the first two of these settings is provided. [ABSTRACT FROM AUTHOR]

Stochastic substitution, the Gibbs sampler, and the sampling--importance--resampling algorithm can be viewed as three alternative sampling- (or Monte Carlo- ) based approaches to the calculation of numerical estimates of marginal probability distributions. The three approaches will be reviewed, compared, and contrasted in relation to various joint probability structures frequently encountered in applications. In particular, the relevance of the approaches to calculating Bayesian posterior densities for a variety of structured models will be discussed and illustrated. [ABSTRACT FROM AUTHOR]

Proposes a conditional bias correction method for developing expectation-maximum intervals that corrects the deficiencies in the naive empirical Bayes confidence intervals. Bias-corrected naive approach; Marginal posterior approach; Simulated coverage probabilities and interval lengths; Conclusions.

Extending earlier work by the authors [3, 4], this article examines in more detail the internal and external behavior of a jury. The internal analysis sheds light on the decision-making process from the first ballot position to the final outcome. Data in Kalven and Zeisel [6] have been employed to test the tenability of the models. The external analysis explores questions regarding the correctness of jury decisions and the effect of changes in jury size and in majority requirements for decisions. [ABSTRACT FROM AUTHOR]

Jury size, majorities required for acquittal or conviction and correctness of juror decisions in criminal trials are studied. The article draws heavily on ideas presented in a previous article [1] which updated a model suggested by Poisson. Briefly, a model is developed and then examined, employing data from Kalven and Zeisel [2] on the American legal system. Comparisons of the American experience with French data from the early 19th century are made through the use of the model. Recent U. S. Supreme Court decisions in criminal trials regarding jury size and the relaxation of unanimity for decisions in criminal trials make these studies pertinent. [ABSTRACT FROM AUTHOR]

Supports two distinct adaptive strategies that typically accelerate the convergence of a Markov chain Monte Carlo (MCMC) algorithm. Background information about MCMC algorithms; Brief review of aspects of MCMC simulation; Descriptions of the two acceleration strategies; Examples employing the two proposed acceleration.

Evidence of global warming induced from the increasing concentration of greenhouse gases in the atmosphere suggests more frequent warm days and heat waves. The concept of an extreme heat event (EHE), defined locally based on exceedance of a suitable local threshold, enables us to capture the notion of a period of persistent extremely high temperatures. Modeling for extreme heat events is customarily implemented using time series of temperatures collected at a set of locations. Since spatial dependence is anticipated in the occurrence of EHE's, a joint model for the time series, incorporating spatial dependence is needed. Recent work by Schliep et al. (J R Stat Soc Ser A Stat Soc 184(3):1070–1092, 2021) develops a space-time model based on a point-referenced collection of temperature time series that enables the prediction of both the incidence and characteristics of EHE's occurring at any location in a study region. The contribution here is to introduce a formal definition of the notion of the spatial extent of an extreme heat event and then to employ output from the Schliep et al. (J R Stat Soc Ser A Stat Soc 184(3):1070–1092, 2021) modeling work to illustrate the notion. For a specified region and a given day, the definition takes the form of a block average of indicator functions over the region. Our risk assessment examines extents for the Comunidad Autónoma de Aragón in northeastern Spain. We calculate daily, seasonal and decadal averages of the extents for two subregions in this comunidad. We generalize our definition to capture extents of persistence of extreme heat and make comparisons across decades to reveal evidence of increasing extent over time. [ABSTRACT FROM AUTHOR]

This article gives an exposition of models provided by Poisson to account for actual jury decisions in criminal and civil trials in France in the first half of the 19th century. The model parameters are jury size, juror error, and probability that the accused is guilty. Poisson's models are cast in a modern setting, and this provides a framework in which new investigations can be initiated. A recent decision by the U.S. Supreme Court on jury size for felony proceedings in state courts should provoke more analyses that can build on the basis provided. [ABSTRACT FROM AUTHOR]

In this article, we propose a doubly stochastic spatial point process model with both aggregation and repulsion. This model combines the ideas behind Strauss processes and log Gaussian Cox processes. The likelihood for this model is not expressible in closed form but it is easy to simulate realizations under the model. We therefore explain how to use approximate Bayesian computation (ABC) to carry out statistical inference for this model. We suggest a method for model validation based on posterior predictions and global envelopes. We illustrate the ABC procedure and model validation approach using both simulated point patterns and a real data example. [ABSTRACT FROM AUTHOR]

Presence/absence data and presence‐only data are the two customary sources for learning about species distributions over a region. We present an ambitious agenda with regard to the analysis of such data. We illuminate the fundamental modeling differences between the two types of data. Most simply, locations are considered to be fixed under presence/absence data; locations are random under presence‐only data. The definition of "probability of presence" is incompatible between the two. We are not comfortable with modeling strategies in the literature that ignore this incompatibility and that assume that presence/absence modeling can be induced from presence‐only specifications and, therefore, that fusion of presence‐only and presence/absence data sources is routine. While, in some cases, data collection may not support this, we propose that, since, in nature, presence/absence is seen at the point locations, presence/absence data should be modeled at point level. If so, we need to specify two surfaces. The first provides the probability of presence at any location in the region. The second provides a realization from this surface in the form of a binary map yielding the results of Bernoulli trials across all locations; this surface is only partially observed. On the other hand, presence‐only data should be modeled as a (partially observed) point pattern, arising from a random number of individuals seen at random locations, driven by specification of an intensity function. There is no notion of Bernoulli trials; events are associated with areas. We further suggest that, with just presence/absence data, preferential sampling of locations may arise. Accounting for this, using a shared process perspective, can improve our estimated presence/absence surface as well as prediction of presence. We further propose that preferential sampling can enable a probabilistically coherent fusion of the two data types. We illustrate with two real data sets, one presence/absence, one presence‐only, for invasive species presence in New England in the United States. We demonstrate that potential bias in sampling locations can affect inference with regard to presence/absence and show that inference can be improved with preferential sampling ideas. We also provide a probabilistically coherent fusion of the two data sets again with the goal of improving inference for presence/absence. The importance of our work is to encourage more careful modeling when studying species distributions. Ignoring incompatibility between data types and adopting nongenerative modeling specifications results in invalid inference; the quantitative ecological community should benefit from this recognition. [ABSTRACT FROM AUTHOR]

Understanding unexposed/baseline behavior of marine mammals is required to assess the effects of increasing levels of anthropogenic noise exposure in the marine environment. However, quantifying variation in the baseline behavior of whales is challenging due to the fact that they spend much of their time at depth, and therefore, their diving behavior is not directly observable. Data collection employs tags as measurement devices to record vertical movement. We focus here on satellite tags, which have the advantage of collection over a time window of weeks. The type of data we analyze here suffers the disadvantage of being in the form of depths attached to an arbitrarily created set of depth bins and being sparse in time. We provide a multi-stage generative model for deep dives using a continuous-time discrete-space Markov chain. Then, we build a likelihood, incorporating dive-specific random effects, in order to fit this model to a set of satellite tag records, each consisting of a temporally misaligned collection of deep dives with sparse binned depths for each dive. Through simulation, we demonstrate the ability to recover true model parameters. With real satellite tag records, we validate the model out of sample and also provide inference regarding stage behavior, inter-tag record behavior, dive duration, and maximum dive depth. Supplementary materials accompanying this paper appear online. [ABSTRACT FROM AUTHOR]