Probabilistic Inference from Arbitrary Uncertainty using Mixtures of Factorized Generalized Gaussians

This paper presents a general and efficient framework for probabilistic inference and learning from arbitrary uncertain information. It exploits the calculation properties of finite mixture models, conjugate families and factorization. Both the joint probability density of the variables and the likelihood function of the (objective or subjective) observation are approximated by a special mixture model, in such a way that any desired conditional distribution can be directly obtained without numerical integration. We have developed an extended version of the expectation maximization (EM) algorithm to estimate the parameters of mixture models from uncertain training examples (indirect observations). As a consequence, any piece of exact or uncertain information about both input and output values is consistently handled in the inference and learning stages. This ability, extremely useful in certain situations, is not found in most alternative methods. The proposed framework is formally justified from standard probabilistic principles and illustrative examples are provided in the fields of nonparametric pattern classification, nonlinear regression and pattern completion. Finally, experiments on a real application and comparative results over standard databases provide empirical evidence of the utility of the method in a wide range of applications

Uncertainty and sensitivity analysis of functional risk curves based on Gaussian processes

A functional risk curve gives the probability of an undesirable event as a function of the value of a critical parameter of a considered physical system. In several applicative situations, this curve is built using phenomenological numerical models which simulate complex physical phenomena. To avoid cpu-time expensive numerical models, we propose to use Gaussian process regression to build functional risk curves. An algorithm is given to provide confidence bounds due to this approximation. Two methods of global sensitivity analysis of the models' random input parameters on the functional risk curve are also studied. In particular, the PLI sensitivity indices allow to understand the effect of misjudgment on the input parameters' probability density functions

Dimension reduction for Gaussian process emulation: an application to the influence of bathymetry on tsunami heights

High accuracy complex computer models, or simulators, require large resources in time and memory to produce realistic results. Statistical emulators are computationally cheap approximations of such simulators. They can be built to replace simulators for various purposes, such as the propagation of uncertainties from inputs to outputs or the calibration of some internal parameters against observations. However, when the input space is of high dimension, the construction of an emulator can become prohibitively expensive. In this paper, we introduce a joint framework merging emulation with dimension reduction in order to overcome this hurdle. The gradient-based kernel dimension reduction technique is chosen due to its ability to drastically decrease dimensionality with little loss in information. The Gaussian process emulation technique is combined with this dimension reduction approach. Our proposed approach provides an answer to the dimension reduction issue in emulation for a wide range of simulation problems that cannot be tackled using existing methods. The efficiency and accuracy of the proposed framework is demonstrated theoretically, and compared with other methods on an elliptic partial differential equation (PDE) problem. We finally present a realistic application to tsunami modeling. The uncertainties in the bathymetry (seafloor elevation) are modeled as high-dimensional realizations of a spatial process using a geostatistical approach. Our dimension-reduced emulation enables us to compute the impact of these uncertainties on resulting possible tsunami wave heights near-shore and on-shore. We observe a significant increase in the spread of uncertainties in the tsunami heights due to the contribution of the bathymetry uncertainties. These results highlight the need to include the effect of uncertainties in the bathymetry in tsunami early warnings and risk assessments.Comment: 26 pages, 8 figures, 2 table