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We consider a generalization of the Bauer maximum principle. We work with tensorial products of convex measures sets, that are non necessarily compact but generated by their extreme points. We show that the maximum of a quasi-convex lower semicontinuous function on this product space is reached on the tensorial product of finite mixtures of extreme points. Our work is an extension of the Bauer maximum principle in three different aspects. First, we only assume that the objective functional is quasi-convex. Secondly, the optimization is performed over a space built as a product of measures sets. Finally, the usual compactness assumption is replaced with the existence of an integral representation on the extreme points. We focus on product of two different types of measures sets, called the moment class and the unimodal moment class. The elements of these classes are probability measures (respectively unimodal probability measures) satisfying generalized moment constraints. We show that an integral representation on the extreme points stands for such spaces and that it extends to their tensorial product. We give several applications of the Theorem, going from robust Bayesian analysis to the optimization of a quantile of a computer code output.

When computer codes are used for modeling complex physical systems, their unknown parameters are tuned by calibration techniques. A discrepancy function may be added to the computer code in order to capture its discrepancy with the real physical process. By considering the validation question of a computer code as a Bayesian selection model problem, Damblin et al. (2016) have highlighted a possible confounding effect in certain configurations between the code discrepancy and a linear computer code by using a Bayesian testing procedure based on the intrinsic Bayes factor. In this paper, we investigate the issue of code error identifiability by applying another Bayesian model selection technique which has been recently developed by Kamary et al. (2014). By embedding the competing models within an encompassing mixture model, Kamary et al. (2014)'s method allows each observation to belong to a different mixing component, providing a more flexible inference, while remaining competitive in terms of computational cost with the intrinsic Bayesian approach. By using the technique of sharing parameters mentioned in Kamary et al. (2014), an improper non-informative prior can be used for some computer code parameters and we demonstrate that the resulting posterior distribution is proper. We then check the sensitivity of our posterior estimates to the choice of the parameter prior distributions. We illustrate that the value of the correlation length of the discrepancy Gaussian process prior impacts the Bayesian inference of the mixture model parameters and that the model discrepancy can be identified by applying the Kamary et al. (2014) method when the correlation length is not too small. Eventually, the proposed method is applied on a hydraulic code in an industrial context.

We study an industrial computer code related to nuclear safety. A major topic of interest is to assess the uncertainties tainting the results of a computer simulation. In this work we gain robustness on the quantification of a risk measurement by accounting for all sources of uncertainties tainting the inputs of a computer code. To that extent, we evaluate the maximum quantile over a class of distributions defined only by constraints on their moments. Two options are available when dealing with such complex optimization problems: one can either optimize under constraints; or preferably, one should reformulate the objective function. We identify a well suited parameterization to compute the optimal quantile based on the theory of canonical moments. It allows an effective, free of constraints, optimization., Comment: arXiv admin note: substantial text overlap with arXiv:1811.12788

We gain robustness on the quantification of a risk measurement by accounting for all sources of uncertainties tainting the inputs of a computer code. We evaluate the maximum quantile over a class of distributions defined only by constraints on their moments. The methodology is based on the theory of canonical moments that appears to be a well-suited framework for practical optimization., Comment: 21 pages, 9 figures

In this article, we present a recently released R package for Bayesian calibration. Many industrial fields are facing unfeasible or costly field experiments. These experiments are replaced with numerical/computer experiments which are realized by running a numerical code. Bayesian calibration intends to estimate, through a posterior distribution, input parameters of the code in order to make the code outputs close to the available experimental data. The code can be time consuming while the Bayesian calibration implies a lot of code calls which makes studies too burdensome. A discrepancy might also appear between the numerical code and the physical system when facing incompatibility between experimental data and numerical code outputs. The package CaliCo deals with these issues through four statistical models which deal with a time consuming code or not and with discrepancy or not. A guideline for users is provided in order to illustrate the main functions and their arguments. Eventually, a toy example is detailed using CaliCo. This example (based on a real physical system) is in five dimensions and uses simulated data.

Field experiments are often difficult and expensive to make. To bypass these issues, industrial companies have developed computational codes. These codes intend to be representative of the physical system, but come with a certain amount of problems. The code intends to be as close as possible to the physical system. It turns out that, despite continuous code development, the difference between the code outputs and experiments can remain significant. Two kinds of uncertainties are observed. The first one comes from the difference between the physical phenomenon and the values recorded experimentally. The second concerns the gap between the code and the physical system. To reduce this difference, often named model bias, discrepancy, or model error, computer codes are generally complexified in order to make them more realistic. These improvements lead to time consuming codes. Moreover, a code often depends on parameters to be set by the user to make the code as close as possible to field data. This estimation task is called calibration. This paper proposes a review of Bayesian calibration methods and is based on an application case which makes it possible to discuss the various methodological choices and to illustrate their divergences. This example is based on a code used to predict the power of a photovoltaic plant.

A major issue of extreme value analysis is the determination of the shape parameter $\xi$ common to Generalized Extreme Value (GEV) and Generalized Pareto (GP) distributions, which drives the tail behavior, and is of major impact on the estimation of return levels and periods. Many practitioners make the choice of a Bayesian framework to conduct this assessment for accounting of parametric uncertainties, which are typically high in such analyses characterized by a low number of observations. Nonetheless, such approaches can provide large credibility domains for $\xi$, including negative and positive values, which does not allow to conclude on the nature of the tail. Considering the block maxima framework, a generic approach of the determination of the value and sign of $\xi$ arises from model selection between the Fr\'echet, Gumbel and Weibull possible domains of attraction conditionally to observations. Opposite to the common choice of the GEV as an appropriate model for {\it sampling} extreme values, this model selection must be conducted with great care. The elicitation of proper, informative and easy-to use priors is conducted based on the following principle: for all parameter dimensions they act as posteriors of noninformative priors and virtual samples. Statistics of these virtual samples can be assessed from prior predictive information, and a compatibility rule can be carried out to complete the calibration, even though they are only semi-conjugated. Besides, the model selection is conducted using a mixture encompassing framework, which allows to tackle the computation of Bayes factors. Motivating by a real case-study involving the elicitation of expert knowledge on meteorological magnitudes, the overall methodology is illustrated by toy examples too., Comment: Manuscript in preparation

A new approach for Bayesian model averaging (BMA) and selection is proposed, based on the mixture model approach for hypothesis testing in Kaniav et al., 2014. Inheriting from the good properties of this approach, it extends BMA to cases where improper priors are chosen for parameters that are common to all candidate models. From an algorithmic point of view, our approach consists in sampling from the posterior distribution of the single-datum mixture of all candidate models, weighted by their prior probabilities. We show that this posterior distribution is equal to the 'Bayesian-model averaged' posterior distribution over all candidate models, weighted by their posterior probability. From this BMA posterior sample, a simple Monte-Carlo estimate of each model's posterior probability is derived, as well as importance sampling estimates for expectations under each model's posterior distribution., Comment: 20 pages, 5 figures, submission in preparation

Making good predictions of a physical system using a computer code requires the inputs to be carefully specified. Some of these inputs called control variables have to reproduce physical conditions whereas other inputs, called parameters, are specific to the computer code and most often uncertain. The goal of statistical calibration consists in estimating these parameters with the help of a statistical model which links the code outputs with the field measurements. In a Bayesian setting, the posterior distribution of these parameters is normally sampled using MCMC methods. However, they are impractical when the code runs are high time-consuming. A way to circumvent this issue consists of replacing the computer code with a Gaussian process emulator, then sampling a cheap-to-evaluate posterior distribution based on it. Doing so, calibration is subject to an error which strongly depends on the numerical design of experiments used to fit the emulator. We aim at reducing this error by building a proper sequential design by means of the Expected Improvement criterion. Numerical illustrations in several dimensions assess the efficiency of such sequential strategies.

In [Lavielle and Ludena 07], a random thresholding metho d is intro duced to select the significant, or non null, mean terms among a collection of independent random variables, and applied to the problem of recovering the significant coefficients in non ordered model selection. We intro duce a simple modification which removes the dep endency of the proposed estimator on a window parameter while maintaining its asymptotic properties. A simulation study suggests that both procedures compare favorably to standard thresholding approaches, such as multiple testing or model-based clustering, in terms of the binary classification risk. An application of the method to the problem of activation detection on functional magnetic resonance imaging (fMRI) data is discussed., Comment: 22 pages, 7 figures. Submitted to Statistics and its Interface (SII)

Maximum likelihood estimation (MLE) and heuristic predictive estimation (HPE) are two widely used approaches in industrial uncertainty analysis. We review them from the point of view of decision theory, using Bayesian inference as a gold standard for comparison. The main drawback of MLE is that it may fail to properly account for the uncertainty on the physical process generating the data, especially when only a small amount of data are available. HPE offers an improvement in that it takes this uncertainty into account. However, we show that this approach is actually equivalent to Bayes estimation for a particular cost function that is not explicitly chosen by the decision maker. This may produce results that are suboptimal from a decisional perspective. These results plead for a systematic use of Bayes estimators based on carefully defined cost functions., Comment: 28 pages 5 figures, submitted to "Reliability Engineering & System Safety"

Spatial Independent Components Analysis (ICA) is increasingly used in the context of functional Magnetic Resonance Imaging (fMRI) to study cognition and brain pathologies. Salient features present in some of the extracted Independent Components (ICs) can be interpreted as brain networks, but the segmentation of the corresponding regions from ICs is still ill-controlled. Here we propose a new ICA-based procedure for extraction of sparse features from fMRI datasets. Specifically, we introduce a new thresholding procedure that controls the deviation from isotropy in the ICA mixing model. Unlike current heuristics, our procedure guarantees an exact, possibly conservative, level of specificity in feature detection. We evaluate the sensitivity and specificity of the method on synthetic and fMRI data and show that it outperforms state-of-the-art approaches.

This thesis is dedicated to the statistical analysis of multi-sub ject fMRI data, with the purpose of identifying bain structures involved in certain cognitive or sensori-motor tasks, in a reproducible way across sub jects. To overcome certain limitations of standard voxel-based testing methods, as implemented in the Statistical Parametric Mapping (SPM) software, we introduce a Bayesian model selection approach to this problem, meaning that the most probable model of cerebral activity given the data is selected from a pre-defined collection of possible models. Based on a parcellation of the brain volume into functionally homogeneous regions, each model corresponds to a partition of the regions into those involved in the task under study and those inactive. This allows to incorporate prior information, and avoids the dependence of the SPM-like approach on an arbitrary threshold, called the cluster- forming threshold, to define active regions. By controlling a Bayesian risk, our approach balances false positive and false negative risk control. Furthermore, it is based on a generative model that accounts for the spatial uncertainty on the localization of individual effects, due to spatial normalization errors. On both simulated and real fMRI datasets, we show that this new paradigm corrects several biases of the SPM-like approach, which either swells or misses the different active regions, depending on the choice of a cluster-forming threshold., Comment: PhD Thesis, 208 pages, Applied Statistics and Neuroimaging, University of Orsay, France