Jérôme A. Dupuis

Bayesian variable selection in qualitative models by Kullback-Leibler projections

Abstract The variable selection method proposed in the paper is based on the evaluation of the Kullback–Leibler distance between the full (or encompassing) model and its submodels. The Bayesian implementation of the method does not require a separate prior modeling on the submodels since the corresponding parameters for the submodels are defined as the Kullback–Leibler projections of the full model parameters. The result of the selection procedure is the submodel with the smallest number of covariates which is at an acceptable distance of the full model. We introduce the notion of explanatory power of a model and scale the maximal acceptable distance in terms of the explanatory power of the full model. Moreover, an additivity property between embedded submodels shows that our selection procedure is equivalent to select the submodel with the smallest number of covariates which has a sufficient explanatory power. We illustrate the performances of this method on a breast cancer dataset

Prior distributions for stratified capture-recapture models

We consider the Arnason-Schwarz model, usually used to estimate survival and movement probabilities from capture-recapture data. A missing data structure of this model is constructed which allows a clear separation of information relative to capture and relative to movement. Extensions of the Arnason-Schwarz model are considered. For example, we consider a model that takes into account both the individual migration history and the individual reproduction history. Biological assumptions of these extensions are summarized via a directed graph. Owing to missing data, the posterior distribution of parameters is numerically intractable. To overcome those computational difficulties we advocate a Gibbs sampling algorithm that takes advantage of the missing data structure inherent in capture-recapture models. Prior information on survival, capture and movement probabilities typically consists of a prior mean and of a prior 95% credible confidence interval. Dirichlet distributions are used to incorporate some prior information on capture, survival probabilities, and movement probabilities. Finally, the influence of the prior on the Bayesian estimates of movement probabilities is examined.

Estimating the occupancy rate of spatially rare or hard to detect species: a conditional approach.

We consider the problem of estimating the occupancy rate of a target species in a region divided in spatial units (called quadrats); this quantity being defined as the proportion of quadrats occupied by this species. We mainly focus on spatially rare or hard to detect species that are typically detected in very few quadrats, and for which estimating the occupancy rate (with an acceptable precision) is problematic. We develop a conditional approach for estimating the quantity of interest; we condition on the presence of the target species in the region of study. We show that conditioning makes identifiable the occurrence and detectability parameters, regardless of the number of visits made in the sampled quadrats. Compared with an unconditional approach, it proves to be complementary, in that this allows us to deal with biological questions that cannot be addressed by the former. Two Bayesian analyses of the data are performed: one is noninformative, and the other takes advantage of the fact that some prior information on detectability is available. It emerges that taking such a prior into account significantly improves the precision of the estimate when the target species has been detected in few quadrats and is known to be easily detectable.

Estimating Upper Bounds for Occupancy and Number of Manatees in Areas Potentially Affected by Oil from the Deepwater Horizon Oil Spill

The explosion of the Deepwater Horizon drilling platform created the largest marine oil spill in U.S. history. As part of the Natural Resource Damage Assessment process, we applied an innovative modeling approach to obtain upper estimates for occupancy and for number of manatees in areas potentially affected by the oil spill. Our data consisted of aerial survey counts in waters of the Florida Panhandle, Alabama and Mississippi. Our method, which uses a Bayesian approach, allows for the propagation of uncertainty associated with estimates from empirical data and from the published literature. We illustrate that it is possible to derive estimates of occupancy rate and upper estimates of the number of manatees present at the time of sampling, even when no manatees were observed in our sampled plots during surveys. We estimated that fewer than 2.4% of potentially affected manatee habitat in our Florida study area may have been occupied by manatees. The upper estimate for the number of manatees present in potentially impacted areas (within our study area) was estimated with our model to be 74 (95%CI 46 to 107). This upper estimate for the number of manatees was conditioned on the upper 95%CI value of the occupancy rate. In other words, based on our estimates, it is highly probable that there were 107 or fewer manatees in our study area during the time of our surveys. Because our analyses apply to habitats considered likely manatee habitats, our inference is restricted to these sites and to the time frame of our surveys. Given that manatees may be hard to see during aerial surveys, it was important to account for imperfect detection. The approach that we described can be useful for determining the best allocation of resources for monitoring and conservation.

Bayesian test of homogeneity for Markov chains

The test we develop expresses the null hypothesis in terms of proximity of the distribution of a Markov chain (yt) to the subspace of homogeneous Markov chains. The distance we use is the Kullback distance which turns out to be conceptually appropriate. Departure from the point null hypothesis allows us to formulate the question of interest in meaningful terms, but implementing this approach comes up against a scaling problem. In this paper, we propose a new approach in order to solve this scaling problem by formulating the proximity to homogeneity as a percentage of the maximum distance to .

Impact of climatic variations on bird species occupancy rate in a southern European forest

Species that are affected by climatic variations can undergo modification in range and/or abundance. Knowing how individuals or species occupy their habitat is essential to understand how species use their environment, and detecting variations that might affect this use can be determinant in species management. Hierarchical modeling is regularly used to assess for occupancy rate (i.e. proportion of patches occupied in a region), particularly when it is required to consider detectability-related issues. The present study is the first application of the conditional model presented in Dupuis et al. (Biometrics 2010), which is applied in the case of a heterogeneous area that might be divided into homogeneous sub-areas. Their approach is used to study the impact of three consecutive particularly cold winters on a selected set of bird species in a forest of southern France in the context of available prior information on birds detectability. We examined a limited range of factors that might influence the response of some bird species to climate. We considered the case of sedentary, partially migrating and migrating species. We also assessed if the biogeographical origins of the different species affect their occupancy rates. Globally, changes in occupancy rates between 1985 and 1987 indicates for the first time a continentalization of the regional forest fauna, reflected by the expansion of Palearctic and Turkestano-European faunistic type species, with depletion or extinction of European, Turkestano-Mediterranean and Mediterranean sedentary species. We have also shown the importance of prior information.

Estimating Species Richness from Quadrat Sampling Data: A General Approach

We consider the problem of estimating the number of species (denoted by S) of a biological community located in a region divided into n quadrats. To address this question, different hierarchical parametric approaches have been recently developed. Despite a detailed modeling of the underlying biological processes, they all have some limitations. Indeed, some assume that n is theoretically infinite; as a result, n and the sampling fraction are not a part of such models. Others require some prior information on S to be efficiently implemented. Our approach is more general in that it applies without limitation on the size of n, and it can be used in the presence, as well as in the absence, of prior information on S. Moreover, it can be viewed as an extension of the approach of Dorazio and Royle (2005, Journal of the American Statistical Association 100, 389-398) in that n is a part of the model and a prior distribution is placed on S. Despite serious computational difficulties, we have perfected an efficient Markov chain Monte Carlo algorithm, which allows us to obtain the Bayesian estimate of S. We illustrate our approach by estimating the number of species of a bird community located in a forest.

Response to Carl Schwarz

We think that the state-space formulation (suggested by Schwarz) is not appropriate to describe the missing data phenomenon inherent in multi-strata capturerecapture data. Contrary to the state-space models the observation process (that corresponds to the capture process in our paper) has no observation error. When animal i has been captured at time t, its position (and thus the observation) is known without error; but there is no observation error when animal i has not been captured at time t. Its position is simply not available from the data, and it is simply missing. Finally, we think that the ArnasonSchwarz model is typically a missing data model, in the same way as the mixture models or the hidden Markov chains models.