An expectation operator for belief functions in the Dempster–Shafer theory^*

* Presented at the 11th Workshop on Uncertainty Processing (WUPES'18), Třeboň, Czech Republic, June 6–9, 2018.

ABSTRACT

The main contribution of this paper is a new definition of expected value of belief functions in the Dempster–Shafer (D–S) theory of evidence. Our definition shares many of the properties of the expectation operator in probability theory. Also, for Bayesian belief functions, our definition provides the same expected value as the probabilistic expectation operator. A traditional method of computing expected of real-valued functions is to first transform a D–S belief function to a corresponding probability mass function, and then use the expectation operator for probability mass functions. Transforming a belief function to a probability function involves loss of information. Our expectation operator works directly with D–S belief functions. Another definition is using Choquet integration, which assumes belief functions are credal sets, i.e. convex sets of probability mass functions. Credal sets semantics are incompatible with Dempster's combination rule, the center-piece of the D–S theory. In general, our definition provides different expected values than, e.g. if we use probabilistic expectation using the pignistic transform or the plausibility transform of a belief function. Using our definition of expectation, we provide new definitions of variance, covariance, correlation, and other higher moments and describe their properties.

KEYWORDS:

• Dempster–Shafer theory of evidence
• expected value
• variance
• covariance
• correlation
• higher moments
• pignistic transformation
• plausibility transformation

Acknowledgments

This work is inspired by recent work on entropy of belief functions in the Dempster-Shafer theory of evidence in collaboration with Radim Jiroušek from The Czech Academy of Sciences (Jiroušek and Shenoy 2018b). Thank you, Radim! Thanks also to Suzanna Emelio for a careful proofreading of this paper. A short version of this paper appeared as Shenoy (2018). Comments and suggestions for improvement of the paper are welcome.

Disclosure statement

No potential conflict of interest was reported by the author.

ORCID

Prakash P. Shenoy http://orcid.org/0000-0002-8425-896X

Notes

* Presented at the 11th Workshop on Uncertainty Processing (WUPES'18), Třeboň, Czech Republic, June 6–9, 2018.

1 The definition of independence here is based on factorization semantics, see, e.g. Shenoy (1994).

Additional information

Funding

This work was supported by Ronald G. Harper Professorship at the University of Kansas (BUSHARPRO).

Notes on contributors

Prakash P. Shenoy

Prakash P. Shenoy is the Ronald G. Harper Distinguished Professor of Artificial Intelligence in Business, University of Kansas at Lawrence. He received a B.Tech. in Mechanical Engineering from the Indian Institute of Technology, Bombay, India, in 1973, and an M.S. and a Ph.D. in Operations Research/Industrial Engineering from Cornell University in 1975 and 1977, respectively. His research interests are in the areas of artificial intelligence and decision sciences. He is the inventor of valuation-based systems, an abstract framework for knowledge representation and inference that includes Bayesian probabilities, Dempster–Shafer belief functions, Spohn's kappa calculus, Zadeh's possibility theory, propositional logic, optimization, solving systems of equations, database retrieval, and other domains. He is also the co-author (with G. Shafer) of the so-called Shenoy–Shafer architecture for finding marginals of joint distributions using local computation. He has published many articles on management of uncertainty in expert systems, decision analysis, and the mathematical theory of games. His articles have appeared in journals such as Operations Research, Management Science, International Journal of Game Theory, Artificial Intelligence, and International Journal of Approximate Reasoning. http://pshenoy.faculty.ku.edu.