Statistics with Imprecise Probabilities—A Short Survey

This chapter aims at surveying and highlighting in an introductory way some challenges and big opportunities a paradigmatic shift to imprecise probabilities could induce in statistical modelling. Working with an informal understanding of imprecise probabilities, we discuss the concepts of model imprecision and data imprecision as the two main types of imprecision in statistical modelling. Then we provide a short survey of some major developments, methodological questions and applications of imprecise probabilistic models under model imprecision in the context of different inference schools and summarize some recent developments in the area of data imprecision.

Overview of Stochastic Model Updating in Aerospace Application Under Uncertainty Treatment

This chapter presents the technique route of model updating in the presence of imprecise probabilities. The emphasis is put on the inevitable uncertainties, in both numerical simulations and experimental measurements, leading the updating methodology to be significantly extended from deterministic sense to stochastic sense. This extension requires that the model parameters are not regarded as unknown-but-fixed values, but random variables with uncertain distributions, i.e. the imprecise probabilities. The final objective of stochastic model updating is no longer a single model prediction with maximal fidelity to a single experiment, but rather the calibrated distribution coefficients allowing the model predictions to fit with the experimental measurements in a probabilistic point of view. The involvement of uncertainty within a Bayesian updating framework is achieved by developing a novel uncertainty quantification metric, i.e. the Bhattacharyya distance, instead of the typical Euclidian distance. The overall approach is demonstrated by solving the model updating sub-problem of the NASA uncertainty quantification challenge. The demonstration provides a clear comparison between performances of the Euclidian distance and the Bhattacharyya distance, and thus promotes a better understanding of the principle of stochastic model updating, as no longer to determine the unknown-but-fixed parameters, but rather to reduce the uncertainty bounds of the model prediction and meanwhile to guarantee the existing experimental data to be still enveloped within the updated uncertainty space.

Reliability

This chapter introduces key concepts for quantification of system reliability. In addition, basics of statistical inference for reliability data are explained, in particular, the derivation of the likelihood function.

Introduction to Bayesian Statistical Inference

We present basic concepts of Bayesian statistical inference. We briefly introduce the Bayesian paradigm. We present the conjugate priors; a computational convenient way to quantify prior information for tractable Bayesian statistical analysis. We present tools for parametric and predictive inference, and particularly the design of point estimators, credible sets, and hypothesis tests. These concepts are presented in running examples. Supplementary material is available from GitHub.

Introduction to the Theory of Imprecise Probability

The theory of imprecise probability is a generalization of classical ‘precise’ probability theory that allows modeling imprecision and indecision. This is a practical advantage in situations where a unique precise uncertainty model cannot be justified. This arises, for example, when there is a relatively small amount of data available to learn the uncertainty model or when the model’s structure cannot be defined uniquely. The tools the theory provides make it possible to draw conclusions and make decisions that correctly reflect the limited information or knowledge available for the uncertainty modeling task. This extra expressivity however often implies a higher computational burden. The goal of this chapter is to primarily give you the necessary knowledge to be able to read literature that makes use of the theory of imprecise probability. A secondary goal is to provide the insight needed to use imprecise probabilities in your own research. To achieve the goals, we present the essential concepts and techniques from the theory, as well as give a less in-depth overview of the various specific uncertainty models used. Throughout, examples are used to make things concrete. We build on the assumed basic knowledge of classical probability theory.

Imprecise Discrete-Time Markov Chains

I present a short and easy introduction to a number of basic definitions and important results from the theory of imprecise Markov chains in discrete time, with a finite state space. The approach is intuitive and graphical.

Simulation Methods for the Analysis of Complex Systems

Everyday systems like communication, transportation, energy and industrial systems are an indispensable part of our daily lives. Several methods have been developed for their reliability assessment—while analytical methods are computationally more efficient and often yield exact solutions, they are unable to account for the structural and functional complexities of these systems. These complexities often require the analyst to make unrealistic assumptions, sometimes at the expense of accuracy. Simulation-based methods, on the other hand, can account for these realistic operational attributes but are computationally intensive and usually system-specific. This chapter introduces two novel simulation methods: load flow simulation and survival signature simulation which together address the limitations of the existing analytical and simulation methods for the reliability analysis of large systems.

Aerospace Flight Modeling and Experimental Testing

Validation processes for aerospace flight modeling require to articulate uncertainty quantification methods with the experimental approach. On this note, the specific strategies for the reproduction of re-entry flow conditions in ground-based facilities are reviewed. It shows how it combines high-speed flow physics with the hypersonic wind tunnel capabilities.

Sampling from Complex Probability Distributions: A Monte Carlo Primer for Engineers

Models which are constructed to represent the uncertainty arising in engineered systems can often be quite complex to ensure they provide a reasonably faithful reflection of the real-world system. As a result, even computation of simple expectations, event probabilities, variances, or integration over utilities for a decision problem can be analytically intractable. Indeed, such models are often sufficiently high dimensional that even traditional numerical methods perform poorly. However, access to random samples drawn from the probability model under study typically simplifies such problems substantially. The methodologies to generate and use such samples fall under the stable of techniques usually referred to as ‘Monte Carlo methods’. This chapter provides a motivation, simple primer introduction to the basics, and sign-posts to further reading and literature on Monte Carlo methods, in a manner that should be accessible to those with an engineering mathematics background. There is deliberately informal mathematical presentation which avoids measure-theoretic formalism. The accompanying lecture can be viewed at https://www.louisaslett.com/Courses/UTOPIAE/.