The seasonal forecast of electricity demand: A hierarchical Bayesian model with climatological weather generator

In this paper we focus on the one year ahead prediction of the electricity peak-demand daily trajectory during the winter season in Central England and Wales. We define a Bayesian hierarchical model for predicting the winter trajectories and present results based on the past observed weather. Thanks to the flexibility of the Bayesian approach, we are able to produce the marginal posterior distributions of all the predictands of interest. This is a fundamental progress with respect to the classical methods. The results are encouraging in both skill and representation of uncertainty. Further extensions are straightforward at least in principle. The main two of those consist in conditioning the weather generator model with respect to additional information like the knowledge of the first part of the winter and/or the seasonal weather forecast. Copyright (C) 2006 John Wiley & Sons, Ltd.

The seasonal forecast of electricity demand: A hierarchical Bayesian model with climatological weather generator

In this paper we focus on the one year ahead prediction of the electricity peak-demand daily trajectory during the winter season in Central England and Wales. We define a Bayesian hierarchical model for predicting the winter trajectories and present results based on the past observed weather. Thanks to the flexibility of the Bayesian approach, we are able to produce the marginal posterior distributions of all the predictands of interest. This is a fundamental progress with respect to the classical methods. The results are encouraging in both skill and representation of uncertainty. Further extensions are straightforward at least in principle. The main two of those consist in conditioning the weather generator model with respect to additional information like the knowledge of the first part of the winter and/or the seasonal weather forecast. Copyright (C) 2006 John Wiley & Sons, Ltd.

Global trends and factors associated with the illegal killing of elephants: a hierarchical bayesian analysis of carcass encounter data

Elephant poaching and the ivory trade remain high on the agenda at meetings of the Convention on International Trade in Endangered Species of Wild Fauna and Flora (CITES). Well-informed debates require robust estimates of trends, the spatial distribution of poaching, and drivers of poaching. We present an analysis of trends and drivers of an indicator of elephant poaching of all elephant species. The site-based monitoring system known as Monitoring the Illegal Killing of Elephants (MIKE), set up by the 10th Conference of the Parties of CITES in 1997, produces carcass encounter data reported mainly by anti-poaching patrols. Data analyzed were site by year totals of 6,337 carcasses from 66 sites in Africa and Asia from 2002–2009. Analysis of these observational data is a serious challenge to traditional statistical methods because of the opportunistic and non-random nature of patrols, and the heterogeneity across sites. Adopting a Bayesian hierarchical modeling approach, we used the proportion of carcasses that were illegally killed (PIKE) as a poaching index, to estimate the trend and the effects of site- and country-level factors associated with poaching. Important drivers of illegal killing that emerged at country level were poor governance and low levels of human development, and at site level, forest cover and area of the site in regions where human population density is low. After a drop from 2002, PIKE remained fairly constant from 2003 until 2006, after which it increased until 2008. The results for 2009 indicate a decline. Sites with PIKE ranging from the lowest to the highest were identified. The results of the analysis provide a sound information base for scientific evidence-based decision making in the CITES process.

Regression Models with Heteroscedasticity using Bayesian Approach

In this paper, we compare the performance of two statistical approaches for the analysis of data obtained from the social research area. In the first approach, we use normal models with joint regression modelling for the mean and for the variance heterogeneity. In the second approach, we use hierarchical models. In the first case, individual and social variables are included in the regression modelling for the mean and for the variance, as explanatory variables, while in the second case, the variance at level 1 of the hierarchical model depends on the individuals (age of the individuals), and in the level 2 of the hierarchical model, the variance is assumed to change according to socioeconomic stratum. Applying these methodologies, we analyze a Colombian tallness data set to find differences that can be explained by socioeconomic conditions. We also present some theoretical and empirical results concerning the two models. From this comparative study, we conclude that it is better to jointly modelling the mean and variance heterogeneity in all cases. We also observe that the convergence of the Gibbs sampling chain used in the Markov Chain Monte Carlo method for the jointly modeling the mean and variance heterogeneity is quickly achieved.

Assessing the impacts of grazing levels on bird density in woodland habitat: a Bayesian approach using expert opinion

Many studies on birds focus on the collection of data through an experimental design, suitable for investigation in a classical analysis of variance (ANOVA) framework. Although many findings are confirmed by one or more experts, expert information is rarely used in conjunction with the survey data to enhance the explanatory and predictive power of the model. We explore this neglected aspect of ecological modelling through a study on Australian woodland birds, focusing on the potential impact of different intensities of commercial cattle grazing on bird density in woodland habitat. We examine a number of Bayesian hierarchical random effects models, which cater for overdispersion and a high frequency of zeros in the data using WinBUGS and explore the variation between and within different grazing regimes and species. The impact and value of expert information is investigated through the inclusion of priors that reflect the experience of 20 experts in the field of bird responses to disturbance. Results indicate that expert information moderates the survey data, especially in situations where there are little or no data. When experts agreed, credible intervals for predictions were tightened considerably. When experts failed to agree, results were similar to those evaluated in the absence of expert information. Overall, we found that without expert opinion our knowledge was quite weak. The fact that the survey data is quite consistent, in general, with expert opinion shows that we do know something about birds and grazing and we could learn a lot faster if we used this approach more in ecology, where data are scarce. Copyright (c) 2005 John Wiley & Sons, Ltd.

Bayesian ratemaking procedure of crop insurance contracts with skewed distribution

Over the years, crop insurance programs became the focus of agricultural policy in the USA, Spain, Mexico, and more recently in Brazil. Given the increasing interest in insurance, accurate calculation of the premium rate is of great importance. We address the crop-yield distribution issue and its implications in pricing an insurance contract considering the dynamic structure of the data and incorporating the spatial correlation in the Hierarchical Bayesian framework. Results show that empirical (insurers) rates are higher in low risk areas and lower in high risk areas. Such methodological improvement is primarily important in situations of limited data.

Modelling HIV/AIDS length of stay in Portugal

Tese apresentada como requisito parcial para obtenção do grau de Doutor em Estatística e Gestão de Informação pelo Instituto Superior de Estatística e Gestão de Informação da Universidade Nova de Lisboa

Métodos estatísticos aplicados à modelação de dados oncológicos

RESUMO: A estrutura demográfica portuguesa é marcada por baixas taxas de natalidade e mortalidade, onde a população idosa representa uma fatia cada vez mais representativa, fruto de uma maior longevidade. A incidência do cancro, na sua generalidade, é maior precisamente nessa classe etária. A par de outras doenças igualmente lesivas (e.g. cardiovasculares, degenerativas) cuja incidência aumenta com a idade, o cancro merece relevo. Estudos epidemiológicos apresentam o cancro como líder mundial na mortalidade. Em países desenvolvidos, o seu peso representa 25% do número total de óbitos, percentagem essa que mais que duplica noutros países. A obesidade, a baixa ingestão de frutas e vegetais, o sedentarismo, o consumo de tabaco e a ingestão de álcool, configuram-se como cinco dos fatores de risco presentes em 30% das mortes diagnosticadas por cancro. A nível mundial e, em particular no Sul de Portugal, os cancros do estômago, recto e cólon apresentam elevadas taxas de incidência e de mortalidade. Do ponto de vista estritamente económico, o cancro é a doença que mais recursos consome enquanto que do ponto de vista físico e psicológico é uma doença que não limita o seu raio de ação ao doente. O cancro é, portanto, uma doença sempre atual e cada vez mais presente, pois reflete os hábitos e o ambiente de uma sociedade, não obstante as características intrínsecas a cada indivíduo. A adoção de metodologia estatística aplicada à modelação de dados oncológicos é, sobretudo, valiosa e pertinente quando a informação é oriunda de Registos de Cancro de Base Populacional (RCBP). A pertinência é justificada pelo fato destes registos permitirem aferir numa população específica, o risco desta sofrer e/ou vir a sofrer de uma dada neoplasia. O peso que as neoplasias do estômago, cólon e recto assumem foi um dos elementos que motivou o presente estudo que tem por objetivo analisar tendências, projeções, sobrevivências relativas e a distribuição espacial destas neoplasias. Foram considerados neste estudo todos os casos diagnosticados no período 1998-2006, pelo RCBP da região sul de Portugal (ROR-Sul). O estudo descritivo inicial das taxas de incidência e da tendência em cada uma das referidas neoplasias teve como base uma única variável temporal - o ano de diagnóstico - também designada por período. Todavia, uma metodologia que contemple apenas uma única variável temporal é limitativa. No cancro, para além do período, a idade à data do diagnóstico e a coorte de nascimento, são variáveis temporais que poderão prestar um contributo adicional na caracterização das taxas de incidência. A relevância assumida por estas variáveis temporais justificou a sua inclusão numaclasse de modelos designada por modelos Idade-Período-Coorte (Age-Period-Cohort models - APC), utilizada na modelação das taxas de incidência para as neoplasias em estudo. Os referidos modelos permitem ultrapassar o problema de relações não lineares e/ou de mudanças súbitas na tendência linear das taxas. Nos modelos APC foram consideradas a abordagem clássica e a abordagem com recurso a funções suavizadoras. A modelação das taxas foi estratificada por sexo. Foram ainda estudados os respectivos submodelos (apenas com uma ou duas variáveis temporais). Conhecido o comportamento das taxas de incidência, uma questão subsequente prende-se com a sua projeção em períodos futuros. Porém, o efeito de mudanças estruturais na população, ao qual Portugal não é alheio, altera substancialmente o número esperado de casos futuros com cancro. Estimativas da incidência de cancro a nível mundial obtidas a partir de projeções demográficas apontam para um aumento de 25% dos casos de cancro nas próximas duas décadas. Embora a projeção da incidência esteja associada a alguma incerteza, as projeções auxiliam no planeamento de políticas de saúde para a afetação de recursos e permitem a avaliação de cenários e de intervenções que tenham como objetivo a redução do impacto do cancro. O desconhecimento de projeções da taxa de incidência destas neoplasias na área abrangida pelo ROR-Sul, levou à utilização de modelos de projeção que diferem entre si quanto à sua estrutura, linearidade (ou não) dos seus coeficientes e comportamento das taxas na série histórica de dados (e.g. crescente, decrescente ou estável). Os referidos modelos pautaram-se por duas abordagens: (i)modelos lineares no que concerne ao tempo e (ii) extrapolação de efeitos temporais identificados pelos modelos APC para períodos futuros. Foi feita a projeção das taxas de incidência para os anos de 2007 a 2010 tendo em conta o género, idade e neoplasia. É ainda apresentada uma estimativa do impacto económico destas neoplasias no período de projeção. Uma questão pertinente e habitual no contexto clínico e a que o presente estudo pretende dar resposta, reside em saber qual a contribuição da neoplasia em si para a sobrevivência do doente. Nesse sentido, a mortalidade por causa específica é habitualmente utilizada para estimar a mortalidade atribuível apenas ao cancro em estudo. Porém, existem muitas situações em que a causa de morte é desconhecida e, mesmo que esta informação esteja disponível através dos certificados de óbito, não é fácil distinguir os casos em que a principal causa de morte é devida ao cancro. A sobrevivência relativa surge como uma medida objetiva que não necessita do conhecimento da causa específica da morte para o seu cálculo e dar-nos-á uma estimativa da probabilidade de sobrevivência caso o cancro em análise, num cenário hipotético, seja a única causa de morte. Desconhecida a principal causa de morte nos casos diagnosticados com cancro no registo ROR-Sul, foi determinada a sobrevivência relativa para cada uma das neoplasias em estudo, para um período de follow-up de 5 anos, tendo em conta o sexo, a idade e cada uma das regiões que constituem o registo. Foi adotada uma análise por período e as abordagens convencional e por modelos. No epílogo deste estudo, é analisada a influência da variabilidade espaço-temporal nas taxas de incidência. O longo período de latência das doenças oncológicas, a dificuldade em identificar mudanças súbitas no comportamento das taxas, populações com dimensão e riscos reduzidos, são alguns dos elementos que dificultam a análise da variação temporal das taxas. Nalguns casos, estas variações podem ser reflexo de flutuações aleatórias. O efeito da componente temporal aferida pelos modelos APC dá-nos um retrato incompleto da incidência do cancro. A etiologia desta doença, quando conhecida, está associada com alguma frequência a fatores de risco tais como condições socioeconómicas, hábitos alimentares e estilo de vida, atividade profissional, localização geográfica e componente genética. O “contributo”, dos fatores de risco é, por vezes, determinante e não deve ser ignorado. Surge, assim, a necessidade em complementar o estudo temporal das taxas com uma abordagem de cariz espacial. Assim, procurar-se-á aferir se as variações nas taxas de incidência observadas entre os concelhos inseridos na área do registo ROR-Sul poderiam ser explicadas quer pela variabilidade temporal e geográfica quer por fatores socioeconómicos ou, ainda, pelos desiguais estilos de vida. Foram utilizados os Modelos Bayesianos Hierárquicos Espaço-Temporais com o objetivo de identificar tendências espaço-temporais nas taxas de incidência bem como quantificar alguns fatores de risco ajustados à influência simultânea da região e do tempo. Os resultados obtidos pela implementação de todas estas metodologias considera-se ser uma mais valia para o conhecimento destas neoplasias em Portugal.------------ABSTRACT: mortality rates, with the elderly being an increasingly representative sector of the population, mainly due to greater longevity. The incidence of cancer, in general, is greater precisely in that age group. Alongside with other equally damaging diseases (e.g. cardiovascular,degenerative), whose incidence rates increases with age, cancer is of special note. In epidemiological studies, cancer is the global leader in mortality. In developed countries its weight represents 25% of the total number of deaths, with this percentage being doubled in other countries. Obesity, a reduce consumption of fruit and vegetables, physical inactivity, smoking and alcohol consumption, are the five risk factors present in 30% of deaths due to cancer. Globally, and in particular in the South of Portugal, the stomach, rectum and colon cancer have high incidence and mortality rates. From a strictly economic perspective, cancer is the disease that consumes more resources, while from a physical and psychological point of view, it is a disease that is not limited to the patient. Cancer is therefore na up to date disease and one of increased importance, since it reflects the habits and the environment of a society, regardless the intrinsic characteristics of each individual. The adoption of statistical methodology applied to cancer data modelling is especially valuable and relevant when the information comes from population-based cancer registries (PBCR). In such cases, these registries allow for the assessment of the risk and the suffering associated to a given neoplasm in a specific population. The weight that stomach, colon and rectum cancers assume in Portugal was one of the motivations of the present study, that focus on analyzing trends, projections, relative survival and spatial distribution of these neoplasms. The data considered in this study, are all cases diagnosed between 1998 and 2006, by the PBCR of Portugal, ROR-Sul.Only year of diagnosis, also called period, was the only time variable considered in the initial descriptive analysis of the incidence rates and trends for each of the three neoplasms considered. However, a methodology that only considers one single time variable will probably fall short on the conclusions that could be drawn from the data under study. In cancer, apart from the variable period, the age at diagnosis and the birth cohort are also temporal variables and may provide an additional contribution to the characterization of the incidence. The relevance assumed by these temporal variables justified its inclusion in a class of models called Age-Period-Cohort models (APC). This class of models was used for the analysis of the incidence rates of the three cancers under study. APC models allow to model nonlinearity and/or sudden changes in linear relationships of rate trends. Two approaches of APC models were considered: the classical and the one using smoothing functions. The models were stratified by gender and, when justified, further studies explored other sub-models where only one or two temporal variables were considered. After the analysis of the incidence rates, a subsequent goal is related to their projections in future periods. Although the effect of structural changes in the population, of which Portugal is not oblivious, may substantially change the expected number of future cancer cases, the results of these projections could help planning health policies with the proper allocation of resources, allowing for the evaluation of scenarios and interventions that aim to reduce the impact of cancer in a population. Worth noting that cancer incidence worldwide obtained from demographic projections point out to an increase of 25% of cancer cases in the next two decades. The lack of projections of incidence rates of the three cancers under study in the area covered by ROR-Sul, led us to use a variety of forecasting models that differ in the nature and structure. For example, linearity or nonlinearity in their coefficients and the trend of the incidence rates in historical data series (e.g. increasing, decreasing or stable).The models followed two approaches: (i) linear models regarding time and (ii) extrapolation of temporal effects identified by the APC models for future periods. The study provide incidence rates projections and the numbers of newly diagnosed cases for the year, 2007 to 2010, taking into account gender, age and the type of cancer. In addition, an estimate of the economic impact of these neoplasms is presented for the projection period considered. This research also try to address a relevant and common clinical question in these type of studies, regarding the contribution of the type of cancer to the patient survival. In such studies, the primary cause of death is commonly used to estimate the mortality specifically due to the cancer. However, there are many situations in which the cause of death is unknown, or, even if this information is available through the death certificates, it is not easy to distinguish the cases where the primary cause of death is the cancer. With this in mind, the relative survival is an alternative measure that does not need the knowledge of the specific cause of death to be calculated. This estimate will represent the survival probability in the hypothetical scenario of a certain cancer be the only cause of death. For the patients with unknown cause of death that were diagnosed with cancer in the ROR-Sul, the relative survival was calculated for each of the cancers under study, for a follow-up period of 5 years, considering gender, age and each one of the regions that are part the registry. A period analysis was undertaken, considering both the conventional and the model approaches. In final part of this study, we analyzed the influence of space-time variability in the incidence rates. The long latency period of oncologic diseases, the difficulty in identifying subtle changes in the rates behavior, populations of reduced size and low risk are some of the elements that can be a challenge in the analysis of temporal variations in rates, that, in some cases, can reflect simple random fluctuations. The effect of the temporal component measured by the APC models gives an incomplete picture of the cancer incidence. The etiology of this disease, when known, is frequently associated to risk factors such as socioeconomic conditions, eating habits and lifestyle, occupation, geographic location and genetic component. The "contribution"of such risk factors is sometimes decisive in the evolution of the disease and should not be ignored. Therefore, there was the need to consider an additional approach in this study, one of spatial nature, addressing the fact that changes in incidence rates observed in the ROR-Sul area, could be explained either by temporal and geographical variability or by unequal socio-economic or lifestyle factors. Thus, Bayesian hierarchical space-time models were used with the purpose of identifying space-time trends in incidence rates together with the the analysis of the effect of the risk factors considered in the study. The results obtained and the implementation of all these methodologies are considered to be an added value to the knowledge of these neoplasms in Portugal.

Modelling hydrodynamics in the Rio Parana, Argentina : an evaluation and inter-comparison of reduced-complexity and physics based models applied to a large sand-bed river

Depth-averaged velocities and unit discharges within a 30 km reach of one of the world's largest rivers, the Rio Parana, Argentina, were simulated using three hydrodynamic models with different process representations: a reduced complexity (RC) model that neglects most of the physics governing fluid flow, a two-dimensional model based on the shallow water equations, and a three-dimensional model based on the Reynolds-averaged Navier-Stokes equations. Row characteristics simulated using all three models were compared with data obtained by acoustic Doppler current profiler surveys at four cross sections within the study reach. This analysis demonstrates that, surprisingly, the performance of the RC model is generally equal to, and in some instances better than, that of the physics based models in terms of the statistical agreement between simulated and measured flow properties. In addition, in contrast to previous applications of RC models, the present study demonstrates that the RC model can successfully predict measured flow velocities. The strong performance of the RC model reflects, in part, the simplicity of the depth-averaged mean flow patterns within the study reach and the dominant role of channel-scale topographic features in controlling the flow dynamics. Moreover, the very low water surface slopes that typify large sand-bed rivers enable flow depths to be estimated reliably in the RC model using a simple fixed-lid planar water surface approximation. This approach overcomes a major problem encountered in the application of RC models in environments characterised by shallow flows and steep bed gradients. The RC model is four orders of magnitude faster than the physics based models when performing steady-state hydrodynamic calculations. However, the iterative nature of the RC model calculations implies a reduction in computational efficiency relative to some other RC models. A further implication of this is that, if used to simulate channel morphodynamics, the present RC model may offer only a marginal advantage in terms of computational efficiency over approaches based on the shallow water equations. These observations illustrate the trade off between model realism and efficiency that is a key consideration in RC modelling. Moreover, this outcome highlights a need to rethink the use of RC morphodynamic models in fluvial geomorphology and to move away from existing grid-based approaches, such as the popular cellular automata (CA) models, that remain essentially reductionist in nature. In the case of the world's largest sand-bed rivers, this might be achieved by implementing the RC model outlined here as one element within a hierarchical modelling framework that would enable computationally efficient simulation of the morphodynamics of large rivers over millennial time scales. (C) 2012 Elsevier B.V. All rights reserved.

Fire management, climate change and their interacting effects on birds in complex Mediterranean landscapes: dynamic distribution modelling of an early-successional species - the near-threatened Dartford Warbler (Sylvia undata)

The current challenge in a context of major environmental changes is to anticipate the responses of species to future landscape and climate scenarios. In the Mediterranean basin, climate change is one the most powerful driving forces of fire dynamics, with fire frequency and impact having markedly increased in recent years. Species distribution modelling plays a fundamental role in this challenge, but better integration of available ecological knowledge is needed to adequately guide conservation efforts. Here, we quantified changes in habitat suitability of an early-succession bird in Catalonia, the Dartford Warbler (Sylvia undata) ― globally evaluated as Near Threatened in the IUCN Red List. We assessed potential changes in species distributions between 2000 and 2050 under different fire management and climate change scenarios and described landscape dynamics using a spatially-explicit fire-succession model that simulates fire impacts in the landscape and post-fire regeneration (MEDFIRE model). Dartford Warbler occurrence data were acquired at two different spatial scales from: 1) the Atlas of European Breeding Birds (EBCC) and 2) Catalan Breeding Bird Atlas (CBBA). Habitat suitability was modelled using five widely-used modelling techniques in an ensemble forecasting framework. Our results indicated considerable habitat suitability losses (ranging between 47% and 57% in baseline scenarios), which were modulated to a large extent by fire regime changes derived from fire management policies and climate changes. Such result highlighted the need for taking the spatial interaction between climate changes, fire-mediated landscape dynamics and fire management policies into account for coherently anticipating habitat suitability changes of early succession bird species. We conclude that fire management programs need to be integrated into conservation plans to effectively preserve sparsely forested and early succession habitats and their associated species in the face of global environmental change.

Bayesian ranking and selection of fishing boat efficiencies

The steadily accumulating literature on technical efficiency in fisheries attests to the importance of efficiency as an indicator of fleet condition and as an object of management concern. In this paper, we extend previous work by presenting a Bayesian hierarchical approach that yields both efficiency estimates and, as a byproduct of the estimation algorithm, probabilistic rankings of the relative technical efficiencies of fishing boats. The estimation algorithm is based on recent advances in Markov Chain Monte Carlo (MCMC) methods—Gibbs sampling, in particular—which have not been widely used in fisheries economics. We apply the method to a sample of 10,865 boat trips in the US Pacific hake (or whiting) fishery during 1987–2003. We uncover systematic differences between efficiency rankings based on sample mean efficiency estimates and those that exploit the full posterior distributions of boat efficiencies to estimate the probability that a given boat has the highest true mean efficiency.

Bayesian ranking and slection of fishing boat efficiencies

The steadily accumulating literature on technical efficiency in fisheries attests to the importance of efficiency as an indicator of fleet condition and as an object of management concern. In this paper, we extend previous work by presenting a Bayesian hierarchical approach that yields both efficiency estimates and, as a byproduct of the estimation algorithm, probabilistic rankings of the relative technical efficiencies of fishing boats. The estimation algorithm is based on recent advances in Markov Chain Monte Carlo (MCMC) methods— Gibbs sampling, in particular—which have not been widely used in fisheries economics. We apply the method to a sample of 10,865 boat trips in the US Pacific hake (or whiting) fishery during 1987–2003. We uncover systematic differences between efficiency rankings based on sample mean efficiency estimates and those that exploit the full posterior distributions of boat efficiencies to estimate the probability that a given boat has the highest true mean efficiency.

Dissecting the illegal ivory trade: an analysis of ivory seizures data

Reliable evidence of trends in the illegal ivory trade is important for informing decision making for elephants but it is difficult to obtain due to the covert nature of the trade. The Elephant Trade Information System, a global database of reported seizures of illegal ivory, holds the only extensive information on illicit trade available. However inherent biases in seizure data make it difficult to infer trends; countries differ in their ability to make and report seizures and these differences cannot be directly measured. We developed a new modelling framework to provide quantitative evidence on trends in the illegal ivory trade from seizures data. The framework used Bayesian hierarchical latent variable models to reduce bias in seizures data by identifying proxy variables that describe the variability in seizure and reporting rates between countries and over time. Models produced bias-adjusted smoothed estimates of relative trends in illegal ivory activity for raw and worked ivory in three weight classes. Activity is represented by two indicators describing the number of illegal ivory transactions--Transactions Index--and the total weight of illegal ivory transactions--Weights Index--at global, regional or national levels. Globally, activity was found to be rapidly increasing and at its highest level for 16 years, more than doubling from 2007 to 2011 and tripling from 1998 to 2011. Over 70% of the Transactions Index is from shipments of worked ivory weighing less than 10 kg and the rapid increase since 2007 is mainly due to increased consumption in China. Over 70% of the Weights Index is from shipments of raw ivory weighing at least 100 kg mainly moving from Central and East Africa to Southeast and East Asia. The results tie together recent findings on trends in poaching rates, declining populations and consumption and provide detailed evidence to inform international decision making on elephants.

Elicitation of multivariate prior distributions: A nonparametric Bayesian approach

In the context of Bayesian statistical analysis, elicitation is the process of formulating a prior density f(.) about one or more uncertain quantities to represent a person's knowledge and beliefs. Several different methods of eliciting prior distributions for one unknown parameter have been proposed. However, there are relatively few methods for specifying a multivariate prior distribution and most are just applicable to specific classes of problems and/or based on restrictive conditions, such as independence of variables. Besides, many of these procedures require the elicitation of variances and correlations, and sometimes elicitation of hyperparameters which are difficult for experts to specify in practice. Garthwaite et al. (2005) discuss the different methods proposed in the literature and the difficulties of eliciting multivariate prior distributions. We describe a flexible method of eliciting multivariate prior distributions applicable to a wide class of practical problems. Our approach does not assume a parametric form for the unknown prior density f(.), instead we use nonparametric Bayesian inference, modelling f(.) by a Gaussian process prior distribution. The expert is then asked to specify certain summaries of his/her distribution, such as the mean, mode, marginal quantiles and a small number of joint probabilities. The analyst receives that information, treating it as a data set D with which to update his/her prior beliefs to obtain the posterior distribution for f(.). Theoretical properties of joint and marginal priors are derived and numerical illustrations to demonstrate our approach are given. (C) 2010 Elsevier B.V. All rights reserved.