We have developed models and theory for networks, deep neural networks and different stochastic processes. We have developped Bernstein von Mises properties for semiparametric models; we have studied and developped Bayesian methods for manifold learning These works have lead to 21 publications and 10 preprints and 7 PhD theses. All these results are available on arxiv.
1. Complex modelling and inference. We studied 4 families of complex models: Point processes, Networks, Diffusion models and Deep Neural networks. We also investigated Bayesian manifold learning.
Point processes: We developed theory and methodology for Bayesian nonparametric methods for multivariate nonlinear Hawkes processes and estimation of the graph of interactions: published in Bernoulli and under minor revision at JMLR. We developed and studied Bayesian nonparametric methods for spatial point processes with covariates, published in Bernoulli. We also used discrete times Hawkes processes to model the early stage of the Covid epidemic, published in PLoSone and for conflict data, submitted on arxiv.
Networks. We developed new generative models for networks, we studied the asymptotic behaviour of a wide class of models called the graphex models and the asymptotic behaviour of a Bayesian approach. These have been published in Bernoulli, Advances in applied probability, EJS and ECML PKDD.
diffusion models: We studied Bayesian nonparametric methods for estimating the drift vector field in a multi-dimensional, published in the Annals of Statistics.
Deep neural networks: We studied the impact of initialisation in the large width large depth regimes, leading to 3 publications in AISTAT, Neurips workshop, TMLR and 1 preprint. We also studied the interplay between penalties and priors in the context of DNN, submitted on arxiv. We compared Gaussian processes and deep learning approaches for Gaussian regression, showing the superiority of the latter in some situations, published in Neurips. We studied convergence rates of score based diffusion generative models based on DNN under the manifold hypothesis, submitted on arxiv..
Bayesian manifold learning. We developed and studied a Bayesian procedure for estimating the density living near a manifold. We also studied regression under the manifold hypothesis, under both extrinsic an intrinsic Gaussian process priors. We obtained minimax rates of convergence for Bayesian nonparametric regression using based graph Laplacians in large dimension,. These works have been published respectively in Annals of Statistics, Neurips and (minor revision) JMLR.
2.Semiparametric inference
(i) Non standard losses: We studied concentration of the posterior distribution under local losses; published in JASA and Bernoulli. We also studied the multivariate deconvolution problem in max-sliced Wasserstein, from a frequentist a Bayesian perspective, published in the Annals of Statistics.
(ii) Bernstein von Mises theorems (BvM): we proposed two different Bayesian quasi posterior methods to ensure that the resulting distributions behave well for multiple targets, published in EJS and in JRSS B. Finally we derived a BvM for a class on nonparametric mixture models, submitted on arxiv.
(iii) Empirical Bayes. We characterised the behaviour of the Maximum marginal likelihood estimator in various irregular scenarios
3. Approximate Bayesian approaches:
(i) We derived robust Bayesian methods using Generalized Bayesian inference, published in AISTAT. We also generalised the notion of minimax convergence rates under random sample sizes,submitted on arxiv.
(ii) We studied 3 families of approximate Bayesian computations: coresets, ABC and variational algorithms, published in Neurips, arxiv and JMLR (under minor revision) respectively.
(ii) We studied efficient algorithms to compute the marginal likelihoods in mixture models investigating also distributed inference, published in a thesis.
