-
Marc Hoffmann
Statistical inference in growth-fragmentation models
We will review some results about the statistical inference of the branching rate of certain piecewise deterministic Markov models. Whereas their abstract statistical structure is relatively well-known from a parametric point of view, some recent applications (arising for instance from cell division models in biology) have renewed the interest of such statistical models, in particular from a non-parametric and testing point of view. In that context, new difficulties emerge, in particular from the perspective of implementing procedures. We will present some generic inference results (including real-data study on Escherichia Coli) and explain how fragile the information is with respect to the observation scheme (namely observing data in a stationary regime, at branching times or simply the whole genealogy over a given fixed time), a point that is sometimes overlooked by practitioners.
Download the slides
-
Ivan Vujacic
Another look at estimating systems of ordinary differential equations via regularization
We consider estimation of parameters in systems of ordinary differential equations (ODEs). The problem is approached from the viewpoint of M-estimation. In general the true solution of the system is unavailable, therefore any M-criterion function is necessarily defined via an approximation to the true solution. We define an approximation by viewing the system of ODEs as an operator equation and exploiting the connection with the regularization theory.
Combining the introduced regularized solution with M-criterion function we lay out a general framework for estimating parameters in ODEs which can handle partially observed systems. Well-known methods like generalized profiling and smooth and match estimators fit into the proposed framework. If M-criterion function is loglikelihood choosing suitable regularized solution yields estimator which is consistent and asymptotically normal.
Download the slides
-
Moritz Schauer
Bayesian inference for discretely observed diffusion processes
We consider the problem of Bayesian inference for discretely observed multidimensional
diffusion processes using data augmentation a Markov chain Monte Carlo algorithm to
sample simultaneously from the joint posterior distribution of the unobserved bridge
segments between the observations and the parameters. A main computational difficulty
in this approach is the generation of "good" proposals for the bridge segments to be used
in the imputation step of the algorithm. A second difficulty is to handle unknown parameters
appearing in the diffusion coefficient. A direct application of the data-augmentation technique
results in a Markov chain which is reducible if the diffusion coefficient is unknown.
To address the first difficulty we developed a Monte Carlo method for simulating a multi-dimensional
diffusion process conditioned on hitting a fixed point at a fixed future time. Proposals for such
diffusion bridges are obtained by superimposing an additional guiding term to the drift of the process
under consideration.
The second problem can be solved using a transformation of the state space of the Markov chain.
The specific form of proposal bridges used for the imputation of the missing data naturally provides
a mapping that decouples the dependence between the diffusion coefficient and the missing data.
Download the slides
