Alberto Pasanisi

Estimation of a quantity of interest in uncertainty analysis: Some help from Bayesian decision theory

In the context of risk analysis under uncertainty, we focus here on the problem of estimating a so-called quantity of interest of an uncertainty analysis problem, i.e. a given feature of the probability distribution function (pdf) of the output of a deterministic model with uncertain inputs. We will stay here in a fully probabilistic setting. A common problem is how to account for epistemic uncertainty tainting the parameter of the probability distribution of the inputs. In the standard practice, this uncertainty is often neglected (plug-in approach). When a specific uncertainty assessment is made, under the basis of the available information (expertise and/or data), a common solution consists in marginalizing the joint distribution of both observable inputs and parameters of the probabilistic model (i.e. computing the predictive pdf of the inputs), then propagating it through the deterministic model. We will reinterpret this approach in the light of Bayesian decision theory, and will put into evidence that this practice leads the analyst to adopt implicitly a specific loss function which may be inappropriate for the problem under investigation, and suboptimal from a decisional perspective. These concepts are illustrated on a simple numerical example, concerning a case of flood risk assessment.

A Critical Discussion and Practical Recommendations on Some Issues Relevant to the Nonprobabilistic Treatment of Uncertainty in Engineering Risk Assessment

Models for the assessment of the risk of complex engineering systems are affected by uncertainties due to the randomness of several phenomena involved and the incomplete knowledge about some of the characteristics of the system. The objective of this article is to provide operative guidelines to handle some conceptual and technical issues related to the treatment of uncertainty in risk assessment for engineering practice. In particular, the following issues are addressed: (1) quantitative modeling and representation of uncertainty coherently with the information available on the system of interest; (2) propagation of the uncertainty from the input(s) to the output(s) of the system model; (3) (Bayesian) updating as new information on the system becomes available; and (4) modeling and representation of dependences among the input variables and parameters of the system model. Different approaches and methods are recommended for efficiently tackling each of issues (1)-(4) above; the tools considered are derived from both classical probability theory as well as alternative, nonfully probabilistic uncertainty representation frameworks (e.g., possibility theory). The recommendations drawn are supported by the results obtained in illustrative applications of literature.

Adaptive Numerical Designs for the Calibration of Computer Codes

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, 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 reducing their uncertainty 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 typically sampled using Markov Chain Monte Carlo methods. However, they are impractical when the code runs are highly time-consuming. A way to circumvent this issue consists of replacing the computer code with a Gaussian process emulator, then sampling a surrogate 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. Under the assumption that there is no code discre...

Bayesian Model Selection for the Validation of Computer Codes

Complex physical systems are increasingly modeled by computer codes which aim at predicting the reality as accurately as possible. During the last decade, code validation has benefited from a large interest within the scientific community because of the requirement to assess the uncertainty affecting the code outputs. Inspiring frompast contributions to this task, a testing procedure is proposed in this paper to decide either a pure code prediction or a discrepancy-corrected one should be used to provide the best approximation of the physical system. In a particular case where the computer code depends on uncertain parameters, this problem of model selection can be carried out in a Bayesian setting. It requires the specification of proper prior distributions that are well known as having a strong impact on the results. Another way consists in specifying non-informative priors. However, they are sometimes improper, which is a major barrier for computing the Bayes factor. A way to overcome this issue is to use the so-called intrinsic Bayes factor (IBF) in order to replace the ill-defined Bayes factor when improper priors are used. For computer codes which depend linearly on their parameters, the computation of the IBF is made easier, thanks to some explicit marginalization. In the paper, we present a special case where the IBF is equal to the standard Bayes factor when the right-Haar prior is specified on the code parameters and the scale of the code discrepancy. On simulated data, the IBF has been computed for several prior distributions. A confounding effect between the code discrepancy and the linear code is pointed out. Finally, the IBF is computed for an industrial computer code used for monitoring power plant production.

Empirical comparison of two methods for the Bayesian update of the parameters of probability distributions in a two-level hybrid probabilistic-possibilistic uncertainty framework for risk assessment

In this paper, we address the issue of updating in a Bayesian framework, the possibilistic representation of the epistemically-uncertain parameters of (aleatory) probability distributions, as new information (e.g., data) becomes available. Two approaches are considered: the first is based on a purely possibilistic counterpart of the classical, well-grounded probabilistic Bayes’ theorem; the second relies on the hybrid combination of (i) Fuzzy Interval Analysis (FIA) to process the uncertainty described by possibility distributions and (ii) repeated Bayesian updating of the uncertainty represented by probability distributions. The feasibility of the two methods is shown on a literature case study involving the risk-based design of a flood protection dike.

Bayesian model selection for the validation of computer codes

Complex physical systems are increasingly modeled by computer codes which aim at predicting the reality as accurately as possible. During the last decade, code validation has benefited from a large interest within the scientific community because of the requirement to assess the uncertainty affecting the code outputs. Inspiring frompast contributions to this task, a testing procedure is proposed in this paper to decide either a pure code prediction or a discrepancy-corrected one should be used to provide the best approximation of the physical system. In a particular case where the computer code depends on uncertain parameters, this problem of model selection can be carried out in a Bayesian setting. It requires the specification of proper prior distributions that are well known as having a strong impact on the results. Another way consists in specifying non-informative priors. However, they are sometimes improper, which is a major barrier for computing the Bayes factor. A way to overcome this issue is to use the so-called intrinsic Bayes factor (IBF) in order to replace the ill-defined Bayes factor when improper priors are used. For computer codes which depend linearly on their parameters, the computation of the IBF is made easier, thanks to some explicit marginalization. In the paper, we present a special case where the IBF is equal to the standard Bayes factor when the right-Haar prior is specified on the code parameters and the scale of the code discrepancy. On simulated data, the IBF has been computed for several prior distributions. A confounding effect between the code discrepancy and the linear code is pointed out. Finally, the IBF is computed for an industrial computer code used for monitoring power plant production.