Efficient Stochastic Approaches for Multidimensional Integrals in Bayesian Statistics A fundamental problem in Bayesian statistics is the accurate evaluation of multidimensional integrals. A comprehensive experimental study of quasiMonte Carlo algorithms based on Sobol sequence combined with Matousek linear scrambling and a comparison with adaptive Monte Carlo approach and a lattice rule based on generalized Fibonacci numbers has been presented. The numerical tests show that the stochastic algorithms under consideration are efficient for multidimensional integration and especially for computing high dimensional integrals. It is a crucial element since this may be important to be estimated in order to achieve a more accurate and reliable interpretation of the results in Bayesian statistics which is foundational in applications such as machine learning.

New Weak Error bounds and expansions for Optimal Quantization We propose new weak error bounds and expansion in dimension one for optimal quantizationbased cubature formula for different classes of functions, such that piecewise affine functions, Lipschitz convex functions or differentiable function with piecewisedefined locally Lipschitz or xcexb1Holder derivatives. This new results rest on the local behaviors of optimal quantizers, the L rL s distribution mismatch problem and Zadoru0027s Theorem. This new expansion supports the definition of a RichardsonRomberg extrapolation yielding a better rate of convergence for the cubature formula. An extension of this expansion is then proposed in higher dimension for the first time. We then propose a novel variance reduction method for Monte Carlo estimators, based on one dimensional optimal quantizers.