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In various industrial contexts, estimating the distribution of unobservedrandom vectors Xi from some noisy indirect observations H(Xi) + Ui is required.If the relation between Xi and the quantity H(Xi), measured with the error Ui,is implemented by a CPU-consuming computer model H, a major practicaldifficulty is to perform the statistical inference with a relatively smallnumber of runs of H. Following Fu et al. (2014), a Bayesian statisticalframework is considered to make use of possible prior knowledge on theparameters of the distribution of the Xi, which is assumed Gaussian. Moreover,a Markov Chain Monte Carlo (MCMC) algorithm is carried out to estimate theirposterior distribution by replacing H by a kriging metamodel build from alimited number of simulated experiments. Two heuristics, involving twodifferent criteria to be optimized, are proposed to sequentially design thesecomputer experiments in the limits of a given computational budget. The firstcriterion is a Weighted Integrated Mean Square Error (WIMSE). The second one,called Expected Conditional Divergence (ECD), developed in the spirit of theStepwise Uncertainty Reduction (SUR) criterion, is based on the discrepancybetween two consecutive approximations of the target posterior distribution.Several numerical comparisons conducted over a toy example then a motivatingreal case-study show that such adaptive designs can significantly outperformthe classical choice of a maximin Latin Hypercube Design (LHD) of experiments.Dealing with a major concern in hydraulic engineering, a particular emphasis isplaced upon the prior elicitation of the case-study, highlighting the overallfeasibility of the methodology. Faster convergences and manageabilityconsiderations lead to recommend the use of the ECD criterion in practicalapplications.

An adaptive kriging method for solving nonlinear inverse statistical problems