Assessing the intensity of a epidemic, such as the COVID19 pandemic, during the epidemic outbreak, constitutes a significant technical challenge with high societal stakes. Elaborating on classical epidemiological models, this work aims to define a hierarchical Bayesian model that permits the robust estimation of the temporal evolution of the pandemic intensity despite highly corrupted daily new infection counts. It also outputs uncertainty assessment, in the form of credibility intervals robust to the priors choice, accounting for uncertainties on model parameters. The estimation is performed by carefully designed Monte Carlo samplers. The relevance of the proposed estimation procedure is illustrated on real COVID-19 pandemic data for several countries and periods, made available from the Johns Hopkins University repository.

Sampling from a distribution on the real d-space, whose density is nonsmooth and log-concave, is a computational issue that often arises in Machine Learning and Statistics. Langevin-based Hastings-Metropolis methods were proposed: they extend the Unadjusted Langevin Algorithm by using proximal methods to define a smoothed version of the density of interest. We consider the case when these extensions do not apply: the involved proximal operators do not have closed-form expressions and the density is defined on a subset of the real d-space. We derive new Gaussian proposal mechanisms in a Metropolis Adjusted Langevin Algorithm, which use first-order information about the density function. We numerically compare these strategies and discuss the benefits of a change of geometry. The gain in using partial updates of the parameter instead of global updates is also illustrated.

This paper provides a coherent framework for studying longitudinal manifold-valued data for which the dynamic changes over time. We introduce a Bayesian mixed-effects model that allows estimating both a group-representative piecewise-geodesic trajectory in the Riemannian space of shape and interindividual variability. We prove the existence of the maximum a posteriori estimate and its asymptotic consistency under reasonable assumptions. Due to the nonlinearity of the proposed model, we use a stochastic version of the expectation-maximization algorithm to estimate the model parameters. Our simulations show that our model is not noise-sensitive and succeeds in explaining various paths of progression.

We introduce a hierarchical model which allows to estimate a group-average piecewise-geodesic trajectory in the Riemannian space of measurements and individual variability. This model falls into the well defined mixed-effect models. The subject-specific trajectories are defined through spatial and temporal transformations of the group-average piecewise-geodesic path, component by component. Thus we can apply our model to a wide variety of situations. Due to the non-linearity of the model, we use the Stochastic Approximation Expectation-Maximization algorithm to estimate the model parameters. Experiments on synthetic data validate this choice. The model is then applied to the metastatic renal cancer chemotherapy monitoring: we run estimations on RECIST scores of treated patients and estimate the time they escape from the treatment. Experiments highlight the role of the different parameters on the response to treatment.