Abstract
The paper is concerned with decision making under complex uncertainty. We consider the Hodges and Lehmann-criterion relying on uncertain classical probabilities and Walley’s maximality relying on imprecise probabilities. We present linear programming based approaches for computing optimal acts as well as for determining least favorable prior distributions in finite decision settings. Further, we apply results from duality theory of linear programming in order to provide theoretical insights into certain characteristics of these optimal solutions. Particularly, we characterize conditions under which randomization pays out when defining optimality in terms of the Gamma-Maximin criterion and investigate how these conditions relate to least favorable priors.
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- 1.
A further mathematical characterization from the viewpoint of Gamma-Maximinity for certain imprecise probabilities is given in Footnote 3.
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- 3.
For the special case of an \(\varepsilon \)-contamination model (a.k.a. linear-vacuous model) of the form \(\mathcal {M}_{(\pi _0 , \varepsilon )}:= \{(1- \varepsilon ) \pi _0 + \varepsilon \pi : \pi \in \mathcal {P}(\varTheta )\}\), where \(\mathcal {P}(\varTheta )\) denotes the set of all probability measures on \(( \varTheta , 2^{\varTheta })\), \(\varepsilon > 0\) is a fixed contamination parameter and \(\pi _0 \in \mathcal {P}(\varTheta )\) is the central distribution, Gamma-Maximin is mathematically closely related to the Hodges and Lehmann-criterion: For fixed \(X:(\varTheta , 2^{\varTheta }) \rightarrow \mathbb {R}\) we have \(\underline{\mathbb {E}}_{\mathcal {M}_{(\pi _0 , \varepsilon )}}(X) = \min _{\pi \in \mathcal {P}(\varTheta )}((1 - \varepsilon ) \mathbb {E}_{\pi _0}(X)+ \varepsilon \mathbb {E}_{\pi }(X))= (1 - \varepsilon ) \mathbb {E}_{\pi _0}(X)+ \varepsilon \min _{\pi \in \mathcal {P}(\varTheta )}\mathbb {E}_{\pi }(X)=(1 - \varepsilon ) \mathbb {E}_{\pi _0}(X)+ \varepsilon \min _{\theta \in \varTheta }X(\theta )\). Thus, maximizing the lower expectation w.r.t. the \(\varepsilon \)-contamination model is equivalent to maximizing the Hodges and Lehmann-criterion with trust parameter \((1- \varepsilon )\) and prior \(\pi _0\).
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The authors would like to thank the three anonymous referees for their helpful comments and their support.
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Jansen, C., Augustin, T., Schollmeyer, G. (2017). Decision Theory Meets Linear Optimization Beyond Computation. In: Antonucci, A., Cholvy, L., Papini, O. (eds) Symbolic and Quantitative Approaches to Reasoning with Uncertainty. ECSQARU 2017. Lecture Notes in Computer Science(), vol 10369. Springer, Cham. https://doi.org/10.1007/978-3-319-61581-3_30
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