We first introduce the Metropolis-Hastings algorithm. We then consider the Random Walk Metropolis algorithm on $R^n$ with Gaussian proposals, and when the target probability measure is the $n$-fold product of a one dimensional law. It is well-known that, in the limit $n$ tends to infinity, starting at equilibrium and for an appropriate scaling of the variance and of the timescale as a function of the dimension $n$, a diffusive limit is obtained for each component of the Markov chain. We generalize this result when the initial distribution is not the target probability measure. The obtained diffusive limit is the solution to a stochastic differential equation nonlinear in the sense of McKean. We prove convergence to equilibrium for this equation. We discuss practical counterparts in order to optimize the variance of the proposal distribution to accelerate convergence to equilibrium. Our analysis confirms the interest of the constant acceptance rate strategy (with acceptance rate between 1/4 and 1/3).[-]

Consider a problem of Markovian trajectories of particles for which you are trying to estimate the probability of a event.
Under the assumption that you can represent this event as the last event of a nested sequence of events, it is possible to design a splitting algorithm to estimate the probability of the last event in an efficient way. Moreover you can obtain a sequence of trajectories which realize this particular event, giving access to statistical representation of quantities conditionally to realize the event.
In this talk I will present the "Adaptive Multilevel Splitting" algorithm and its application to various toy models. I will explain why it creates an unbiased estimator of a probability, and I will give results obtained from numerical simulations.[-]

This course will give a gentle introduction to SMC (Sequential Monte Carlo algorithms):
• motivation: state-space (hidden Markov) models, sequential analysis of such models; non-sequential problems that may be tackled using SMC.
• Formalism: Markov kernels, Feynman-Kac distributions.
• Monte Carlo tricks: importance sampling and resampling
• standard particle filters: bootstrap, guided, auxiliary
• maximum likelihood estimation of state-stace models
• Bayesian estimation of these models: PMCMC, SMC$^2$.[-]

Learning neural networks using only a small amount of data is an important ongoing research topic with tremendous potential for applications. We introduce a regularizer for the variational modeling of inverse problems in imaging based on normalizing flows, called patchNR. It involves a normalizing flow learned on patches of very few images. The subsequent reconstruction method is completely unsupervised and the same regularizer can be used for different forward operators acting on the same class of images.
By investigating the distribution of patches versus those of the whole image class, we prove that our variational model is indeed a MAP approach. Numerical examples for low-dose CT, limited-angle CT and superresolution of material images demonstrate that our method provides high quality results among unsupervised methods, but requires only very few data. Further, the appoach also works if only the low resolution image is available.
In the second part of the talk I will generalize normalizing flows to stochastic normalizing flows to improve their expressivity.Normalizing flows, diffusion normalizing flows and variational autoencoders are powerful generative models. A unified framework to handle these approaches appear to be Markov chains. We consider stochastic normalizing flows as a pair of Markov chains fulfilling some properties and show how many state-of-the-art models for data generation fit into this framework. Indeed including stochastic layers improves the expressivity of the network and allows for generating multimodal distributions from unimodal ones. The Markov chains point of view enables us to couple both deterministic layers as invertible neural networks and stochastic layers as Metropolis-Hasting layers, Langevin layers, variational autoencoders and diffusion normalizing flows in a mathematically sound way. Our framework establishes a useful mathematical tool to combine the various approaches.
Joint work with F. Altekrüger, A. Denker, P. Hagemann, J. Hertrich, P. Maass[-]