A note on the approximation of Shenoy's expectation operator using probabilistic transforms

ABSTRACT

Recently, a new way of computing an expected value in the Dempster–Shafer theory of evidence was introduced by Prakash P. Shenoy. Up to now, when they needed the expected value of a utility function in D-S theory, the authors usually did it indirectly: first, they found a probability measure corresponding to the considered belief function, and then computed the classical probabilistic expectation using this probability measure. To the best of our knowledge, Shenoy's operator of expectation is the first approach that takes into account all the information included in the respective belief function. Its only drawback is its exponential computational complexity. This is why, in this paper, we compare five different approaches defining probabilistic representatives of belief function from the point of view, which of them yields the best approximations of Shenoy's expected values of utility functions.

KEYWORDS:

• Expectation
• belief function
• probabilistic transform
• commonality function
• utility
• ambiguity
• Choquet integral

Disclosure statement

No potential conflict of interest was reported by the authors.

Notes

1 Notice that, in correspondence with Shenoy (2018), we consider only normal basic assignments (i.e. basic assignments for which Equality 1(1) $\sum _{\mathsf{a}\in {\mathsf{2}}^{{\mathsf{\Omega }}_{\mathsf{X}}}}m\left(\mathsf{a}\mathsf{\right)}\mathsf{=}\mathsf{1.}$(1) holds true), and therefore, the exclusion of the empty set from $2^{{\mathrm{\Omega }}_X}$ simplifies some of the formulas.

2 $\mathbb{R}$ denotes the set of real numbers.

3 Though these assertions are not exactly in this wording among those proven in Section 4.3 of Shenoy (2019), one can easily get them from statements 3 (Expected value of a function of X) and 7 (Bounds on expected value), respectively, using the following simple modification: Shenoy considers real-valued state space, i.e. ${\mathrm{\Omega }}_X\subset \mathbb{R}$. Therefore, to show the validity of the above-presented statements (1) and (2), it is enough to consider ${\hat{\mathrm{\Omega }}}_X=\left\{g\right(x):x\in \mathrm{\Omega }\}$ with $\hat{m}\left(y\right)=\sum _{x\in {\mathrm{\Omega }}_X:g\left(x\right)=y}m\left(x\right)$. Recall that this transformation is correct because we extend the considered utility function for all subsets of ${\mathrm{\Omega }}_X$ in the same way as Shenoy defines his function $v_m$.

4 We always take $0\log \left(\frac{0}{0}\right)=0$.

Additional information

Funding

The research was supported by a grant from Grantová Agentura České Republiky – GA ČR No. 19-06569S (the first two authors).

Notes on contributors

R. Jiroušek

Radim Jiroušek graduated in mathematics from Charles University (Prague) in 1969. He received his PhD and DrSc degrees from the Czech Academy of Sciences in 1979 and 1993, respectively. Though he has retired in 2016, he still keeps research contacts with the Faculty of Management which is located in Jindřichův Hradec, and with the Institute of Information Theory and Automation of the Czech Academy. Currently, he serves also as the Editor in Chief of the Kybernetika journal.

V. Kratochvíl

Václav Kratochvíl obtained his PhD degree in Artificial Intelligence from the Czech Technical University in Prague. He spent one year at Concordia University in Montréal, Canada, where he worked as a postdoc in the team of Vašek Chvátal. He serves currently as a research associate at the Institute of Information Theory and Automation of the Academy of Sciences of the Czech Republic and at the Faculty of Management, University of Economics, Jindřichův Hradec. His researcher interests are Bayesian Networks, Belief Functions, and various problems in algorithmization.

J. Rauh

Johannes Rauh graduated in Physics and Mathematics at the University of Würzburg. He did his PhD at the Max Planck Institute for Mathematics in the Sciences in Leipzig, to which he returned as a postdoctoral researcher after having worked at the Leibniz University Hannover and York University. He also has a part-time position with the Federal Institute for Quality Assurance and Transparency in Healthcare (IQTIG) at Berlin. His research interest lies in geometric and algebraic methods in information theory, statistics and machine learning.