Normal Form Backward Induction for Decision Trees with Coherent Lower Previsions

We examine normal form solutions of decision trees under typical choice functions induced by lower previsions. For large trees, finding such solutions is hard as very many strategies must be considered. In an earlier paper, we extended backward induction to arbitrary choice functions, yielding far more efficient solutions, and we identified simple necessary and sufficient conditions for this to work. In this paper, we show that backward induction works for maximality and E-admissibility, but not for interval dominance and Gamma-maximin. We also show that, in some situations, a computationally cheap approximation of a choice function can be used, even if the approximation violates the conditions for backward induction; for instance, interval dominance with backward induction will yield at least all maximal normal form solutions

Sequential Decision Making For Choice Functions On Gambles

Choice functions on gambles (uncertain rewards) provide a framework for studying diverse preference and uncertainty models. For single decisions, applying a choice function is straightforward. In sequential problems, where the subject has multiple decision points, it is less easy. One possibility, called a normal form solution, is to list all available strategies (specifications of acts to take in all eventualities). This reduces the problem to a single choice between gambles. We primarily investigate three appealing behaviours of these solutions. The first, subtree perfectness, requires that the solution of a sequential problem, when restricted to a sub-problem, yields the solution to that sub-problem. The second, backward induction, requires that the solution of the problem can be found by working backwards from the final stage of the problem, removing everything judged non-optimal at any stage. The third, locality, applies only to special problems such as Markov decision processes, and requires that the optimal choice at each stage (considered separately from the rest of the problem) forms an optimal strategy. For these behaviours, we find necessary and sufficient conditions on the choice function. Showing that these hold is much easier than proving the behaviour from first principles. It also leads to answers to related questions, such as the relationship between the normal form and another popular form of solution, the extensive form. To demonstrate how these properties can be checked for particular choice functions, and how the theory can be easily extended to special cases, we investigate common choice functions from the theory of coherent lower previsions

ISIPTA'07: Proceedings of the Fifth International Symposium on Imprecise Probability: Theories and Applications

B

Credal Networks under Epistemic Irrelevance

A credal network under epistemic irrelevance is a generalised type of Bayesian network that relaxes its two main building blocks. On the one hand, the local probabilities are allowed to be partially specified. On the other hand, the assessments of independence do not have to hold exactly. Conceptually, these two features turn credal networks under epistemic irrelevance into a powerful alternative to Bayesian networks, offering a more flexible approach to graph-based multivariate uncertainty modelling. However, in practice, they have long been perceived as very hard to work with, both theoretically and computationally. The aim of this paper is to demonstrate that this perception is no longer justified. We provide a general introduction to credal networks under epistemic irrelevance, give an overview of the state of the art, and present several new theoretical results. Most importantly, we explain how these results can be combined to allow for the design of recursive inference methods. We provide numerous concrete examples of how this can be achieved, and use these to demonstrate that computing with credal networks under epistemic irrelevance is most definitely feasible, and in some cases even highly efficient. We also discuss several philosophical aspects, including the lack of symmetry, how to deal with probability zero, the interpretation of lower expectations, the axiomatic status of graphoid properties, and the difference between updating and conditioning

Uncertainty in Engineering

This open access book provides an introduction to uncertainty quantification in engineering. Starting with preliminaries on Bayesian statistics and Monte Carlo methods, followed by material on imprecise probabilities, it then focuses on reliability theory and simulation methods for complex systems. The final two chapters discuss various aspects of aerospace engineering, considering stochastic model updating from an imprecise Bayesian perspective, and uncertainty quantification for aerospace flight modelling. Written by experts in the subject, and based on lectures given at the Second Training School of the European Research and Training Network UTOPIAE (Uncertainty Treatment and Optimization in Aerospace Engineering), which took place at Durham University (United Kingdom) from 2 to 6 July 2018, the book offers an essential resource for students as well as scientists and practitioners

Uncertainty in Engineering

This open access book provides an introduction to uncertainty quantification in engineering. Starting with preliminaries on Bayesian statistics and Monte Carlo methods, followed by material on imprecise probabilities, it then focuses on reliability theory and simulation methods for complex systems. The final two chapters discuss various aspects of aerospace engineering, considering stochastic model updating from an imprecise Bayesian perspective, and uncertainty quantification for aerospace flight modelling. Written by experts in the subject, and based on lectures given at the Second Training School of the European Research and Training Network UTOPIAE (Uncertainty Treatment and Optimization in Aerospace Engineering), which took place at Durham University (United Kingdom) from 2 to 6 July 2018, the book offers an essential resource for students as well as scientists and practitioners