arXiv subjects

Matthias C. M. Troffaes

Publications and source records attributed to Matthias C. M. Troffaes.

23 records · Page 2Linked to original sources

arXiv2012

Robust detection of exotic infectious diseases in animal herds: A comparative study of three decision methodologies under severe uncertainty

When animals are transported and pass through customs, some of them may have dangerous infectious diseases. Typically, due to the cost of testing, not all animals are tested: a reasonable selection must be made. How to test effectively whilst avoiding costly disease outbreaks? First, we extend a model proposed in the literature for the detection of invasive species to suit our purpose. Secondly, we explore and compare three decision methodologies on the problem at hand, namely, Bayesian statistics, info-gap theory and imprecise probability theory, all of which are designed to handle severe uncertainty. We show that, under rather general conditions, every info-gap solution is maximal with respect to a suitably chosen imprecise probability model, and that therefore, perhaps surprisingly, the set of maximal options can be inferred at least partly---and sometimes entirely---from an info-gap analysis.

stat.ME↗

arXiv2011

Subtree perfectness, backward induction, and normal-extensive form equivalence for single agent sequential decision making under arbitrary choice functions

We revisit and reinterpret Selten's concept of subgame perfectness in the context of single agent normal form sequential decision making, which leads us to the concept of subtree perfectness. Thereby, we extend Hammond's characterization of extensive form consequentialist consistent behaviour norms to the normal form and to arbitrary choice functions under very few assumptions. In particular, we do not need to assume probabilities on any event or utilities on any reward. We show that subtree perfectness is equivalent to normal-extensive form equivalence, and is sufficient, but, perhaps surprisingly, not necessary, for backward induction to work.

math.ST↗

arXiv2011

Probability boxes on totally preordered spaces for multivariate modelling

A pair of lower and upper cumulative distribution functions, also called probability box or p-box, is among the most popular models used in imprecise probability theory. They arise naturally in expert elicitation, for instance in cases where bounds are specified on the quantiles of a random variable, or when quantiles are specified only at a finite number of points. Many practical and formal results concerning p-boxes already exist in the literature. In this paper, we provide new efficient tools to construct multivariate p-boxes and develop algorithms to draw inferences from them. For this purpose, we formalise and extend the theory of p-boxes using Walley's behavioural theory of imprecise probabilities, and heavily rely on its notion of natural extension and existing results about independence modeling. In particular, we allow p-boxes to be defined on arbitrary totally preordered spaces, hence thereby also admitting multivariate p-boxes via probability bounds over any collection of nested sets. We focus on the cases of independence (using the factorization property), and of unknown dependence (using the Fréchet bounds), and we show that our approach extends the probabilistic arithmetic of Williamson and Downs. Two design problems---a damped oscillator, and a river dike---demonstrate the practical feasibility of our results.

math.PR↗

arXiv2011

On the connection between probability boxes and possibility measures

We explore the relationship between possibility measures (supremum preserving normed measures) and p-boxes (pairs of cumulative distribution functions) on totally preordered spaces, extending earlier work in this direction by De Cooman and Aeyels, among others. We start by demonstrating that only those p-boxes who have 0-1-valued lower or upper cumulative distribution function can be possibility measures, and we derive expressions for their natural extension in this case. Next, we establish necessary and sufficient conditions for a p-box to be a possibility measure. Finally, we show that almost every possibility measure can be modelled by a p-box. Whence, any techniques for p-boxes can be readily applied to possibility measures. We demonstrate this by deriving joint possibility measures from marginals, under varying assumptions of independence, using a technique known for p-boxes. Doing so, we arrive at a new rule of combination for possibility measures, for the independent case.

math.PR↗

arXiv2008

n-Monotone exact functionals

We study n-monotone functionals, which constitute a generalisation of n-monotone set functions. We investigate their relation to the concepts of exactness and natural extension, which generalise the notions of coherence and natural extension in the behavioural theory of imprecise probabilities. We improve upon a number of results in the literature, and prove among other things a representation result for exact n-monotone functionals in terms of Choquet integrals.

math.FA↗