Theoretical contributions to Monte Carlo methods, and applications to Statistics

Abstract : The first part of this thesis concerns the inference of un-normalized statistical models. We study two methods of inference based on sampling, known as Monte-Carlo MLE (Geyer, 1994), and Noise Contrastive Estimation (Gutmann and Hyvarinen, 2010). The latter method was supported by numerical evidence of improved stability, but no theoretical results had yet been proven. We prove that Noise Contrastive Estimation is more robust to the choice of the sampling distribution. We assess the gain of accuracy depending on the computational budget. The second part of this thesis concerns approximate sampling for high dimensional distributions. The performance of most samplers deteriorates fast when the dimension increases, but several methods have proven their effectiveness (e.g. Hamiltonian Monte Carlo, Langevin Monte Carlo). In the continuity of some recent works (Eberle et al., 2017; Cheng et al., 2018), we study some discretizations of the kinetic Langevin diffusion process and establish explicit rates of convergence towards the sampling distribution, that scales polynomially fast when the dimension increases. Our work improves and extends the results established by Cheng et al. for log-concave densities.
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Lionel Riou-Durand. Theoretical contributions to Monte Carlo methods, and applications to Statistics. Statistics [math.ST]. Université Paris-Saclay, 2019. English. ⟨NNT : 2019SACLG006⟩. ⟨tel-02266361⟩

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