Sophie Donnet

Combining Expert Opinions in Prior Elicitation

We consider the problem of combining opinions from different experts in an explicitly model-based way to construct a valid subjective prior in a Bayesian statistical approach. We propose a generic approach by considering a hierarchical model accounting for various sources of variation as well as accounting for potential dependence between experts. We apply this approach to two problems. The first problem deals with a food risk assessment problem involving modelling dose-response for Listeria monocytogenes contamination of mice. Two hierarchical levels of variation are considered (between and within experts) with a complex mathematical situation due to the use of an indirect probit regression. The second concerns the time taken by PhD students to submit their thesis in a particular school. It illustrates a complex situation where three hierarchical levels of variation are modelled but with a simpler underlying probability distribution (log-Normal).

Posterior concentration rates for empirical Bayes procedures with applications to Dirichlet process mixtures

In this paper we provide general conditions to check on the model and the prior to derive posterior concentration rates for data-dependent priors (or empirical Bayes approaches). We aim at providing conditions that are close to the conditions provided in the seminal paper by Ghosal & van der Vaart (2007). We then apply the general theorem to two different settings: the estimation of a density using Dirichlet process mixtures of Gaussian random variables with base measure depending on some empirical quantities and the estimation of the intensity of a counting process under the Aalen model. A simulation study for inhomogeneous Poisson processes also illustrates our results. In the former case we also derive some results on the estimation of the mixing density and on the deconvolution problem. In the latter, we provide a general theorem on posterior concentration rates for counting processes with Aalen multiplicative intensity with priors not depending on the data.

Posterior concentration rates for counting processes with Aalen multiplicative intensities

We provide general conditions to derive posterior concentration rates for Aalen counting processes. The conditions are designed to resemble those proposed in the literature for the problem of density estimation, so that existing results on density estimation can be adapted to the present setting. We apply the general theorem to some prior models including Dirichlet process mixtures of uniform densities to estimate monotone non-increasing intensities and log-splines.

Bayesian Inference for Partially Observed Branching Processes

Poisson processes are used in various application fields applications (public health biology, reliability and so on). In their homogeneous version, the intensity process is a deterministic constant. In their inhomogeneous version, it depends on time. To allow for an endogenous evolution of the intensity process we consider multiplicative intensity processes. Inference methods have been developed when the trajectories are fully observed. We deal with the case of a partially observed process. As a motivating example, consider the analysis of an electrical network through time. This network is composed of cables and accessories (joints). When a cable fails, the cable is replaced by a new cable connected to the network by two new accessories. When an accessory fails, the same kind of reparation is done leading to the addition of only one accessory. The failure rate depends on the stochastically evolving number of accessories. We only observe the times events; the initial number of accessories and the cause of the incident (cable or accessory) are only partially observed. The aim is to estimate the different failure rates or to make predictions. The inference is strongly influenced by the initial number of accessories, which is typically an unknown quantity. We deduce a sensible prior on the initial number of accessories using the probabilistic properties of the process . We illustrate the performances of our methodology on a large simulation study.