Abstract: We consider the problem of performing statistical inference on the states of a dynamical system whose infinitesimal dynamics are governed by a non-linear PDE. These states depend on an unknown initial condition which we model, a priori, as a Gaussian random field, resulting in `a priori’ random trajectories. Given discrete measurements of the process corrupted by additive noise, one then wishes to update the `prior trajectory’ to the best `posterior’ inference on the states of the dynamical system. This can be regarded as a problem of Bayesian inference in an infinite-dimensional regression context where the regression function is a solution to the PDE governing the dynamics and where the posterior distribution arises from a Gaussian process prior on the initial condition. We will highlight how the statistical properties of this methodology naturally depend on the `deterministic dynamics’. The focus will be on dissipative systems for which, from a PDE perspective, inference on the initial state is expected to be achieved at poor rates only; we will explain, however, why this is in fact good news for the statistician. We will show that the `posterior’ trajectories concentrate at essentially non-parametric minimax rate around the true trajectory and are asymptotically Gaussian processes (in a Wasserstein sense with respect to a strong topology) centered at the true trajectory with optimal Fisher-information covariance. We will then explain how this allows one to construct (asymptotically) valid uniform-in-time-and-space confidence bands for the true trajectory at any given confidence level.

Abstract: Consider stochastic heat and wave equation, driven by a space-time white noise or spatial fractional noise and temporal white noise. We consider solution in a mild sense, imposing some conditions on the dimension of the space. It is known, according to the work initiated by Nualart in 2020 and 2021, that for those processes, their spatial average on a ball with radius R satisfies a CLT when R grows to infinity. Thanks to a Stein's method initiated by Pimentel in 2021, and by taking back the ideas of Nourdin and Peccati, we will show that the couple of random variables composed by the spatial average and an evaluation of the solution, converges in distribution, under normalization, toward the independent couple between a Gaussian variable and the evaluation. We interpret this result as asymptotic independence : to know the precise value of a point does not change the knowledge we have about the spatial average. We will conclude by raising some questions we could answer thanks to our approach. This will be a blackboard talk!

Abstract: Predicting the three-dimensional folding structure of a protein from its known one-dimensional amino acid structure is among the most important yet hardest scientific challenges. In this presentation, two new statistical approaches are proposed to tackle problems related to protein data which can be viewed as data on the torus. First, a new flexible distribution for fitting data on the three-dimensional torus is proposed, which we call a trivariate wrapped Cauchy copula. This distribution has a lot of desirable properties, such as a tractable form of density, simple data generating mechanism and known marginal and conditional distributions. Second, novel semi-parametric tests for symmetry on the torus are presented and studied. These tests are valid not only under a given parametric hypothesis but also under a very broad class of symmetric distributions, for both known and unknown symmetry centers. The asymptotic properties under the null and alternative hypotheses are established. Using Stein's method, bounds for the rate of convergence of these test statistics are derived, and their finite sample behavior is investigated by means of Monte Carlo simulations.

This is joint work with Andreas Anastasiou, Shogo Kato, Christophe Ley and Kanti Mardia.

Abstract: We develop two new classes of tests for the Weibull distribution based on Stein’s method. The proposed tests are applied in the full sample case as well as in the presence of random right censoring. We investigate the finite sample performance of the new tests using a comprehensive Monte Carlo study. In both the absence and presence of censoring, it is found that the newly proposed classes of tests outperform competing tests against the majority of the distributions considered. In the cases where censoring is present we consider various censoring distributions. We present another result of independent interest; a test initially proposed for use with full samples is amended to allow for testing for the Weibull distribution in the presence of censoring.

Abstract: In this talk we will review some recent results around Gaussian correlation inequalities with a special focus on the Gaussian product conjecture. We will make a proof in the case of the dimension three using simple optimization arguments and provide some insights for the dimension four which is still an open case at this moment.