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Approximations of Belief Functions Using Compositional Models

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Symbolic and Quantitative Approaches to Reasoning with Uncertainty (ECSQARU 2021)

Part of the book series: Lecture Notes in Computer Science ((LNAI,volume 12897))

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Abstract

For applications to practical problems, the paper proposes to use the approximations of belief functions, which simplify their dependence structure. Using an analogy with probability distributions, we represent these approximations in the form of compositional models. As no theoretical apparatus similar to probabilistic information theory exists for belief functions, the problems arise not only in connection with the design of algorithms seeking the optimal approximations but even in connection with a criterion comparing two different approximations. With this respect, the application of the analogy with probability theory fails. Therefore, the paper suggests the employment of simple heuristics easily applicable to real-life problems.

Supported by the Czech Science Foundation – Grant No. 19-06569S.

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Notes

  1. 1.

    The notion reflects the fact that the considered approximation extends the set of conditional independence relations holding for the probability distribution in question [15].

  2. 2.

    Eq. (2) defines the KL divergence if \(\kappa \) dominates \(\pi \), i.e., if for all \(x \in \varOmega \), for which \(\kappa (x) = 0\), \(\pi (x)\) is also 0. Otherwise, the KL divergence is defined to be \(+\infty \).

  3. 3.

    To generate a decomposable model, first, we generate a sequence of sets of variables satisfying running intersection property. Then we generated random basic assignments for given sets of variables and run the perfectization procedure as described in Proposition 2.

  4. 4.

    Most of the characteristics suggested in [8] cannot be used because of their high computational complexity. As said above, only \(H_S\) can be computed for high-dimensional models due to its additivity.

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Acknowledgment

The authors wish to acknowledge that the final version of the paper reflects long discussions with Prakash P. Shenoy.

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Authors and Affiliations

  1. The Czech Academy of Sciences, Institute of Information Theory and Automation, Prague, Czech Republic

    Radim Jiroušek & Václav Kratochvíl

  2. Faculty of Management, Prague University of Economics and Business, Jindřichův Hradec, Czech Republic

    Radim Jiroušek & Václav Kratochvíl

Authors
  1. Radim Jiroušek
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  2. Václav Kratochvíl
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Correspondence to Václav Kratochvíl .

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Editors and Affiliations

  1. Institute of Information Theory and Automation, Prague, Czech Republic

    Jiřina Vejnarová

  2. Insight Centre for Data Analytics, School of Computer Science and IT University College Cork, Cork, Ireland

    Nic Wilson

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Jiroušek, R., Kratochvíl, V. (2021). Approximations of Belief Functions Using Compositional Models. In: Vejnarová, J., Wilson, N. (eds) Symbolic and Quantitative Approaches to Reasoning with Uncertainty. ECSQARU 2021. Lecture Notes in Computer Science(), vol 12897. Springer, Cham. https://doi.org/10.1007/978-3-030-86772-0_26

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  • DOI: https://doi.org/10.1007/978-3-030-86772-0_26

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