Abstract
In this paper, we present a method to solve analytically the simplest Entropiece Inversion Problem (EIP). This theoretical problem consists in finding a method to calculate a Basic Belief Assignment (BBA) from the knowledge of a given entropiece vector which quantifies effectively the measure of uncertainty of a BBA in the framework of the theory of belief functions. We give an example of the calculation of EIP solution for a simple EIP case, and we show the difficulty to establish the explicit general solution of this theoretical problem that involves transcendental Lambert’s functions.
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Notes
- 1.
For notation convenience, we denote by m or \(m(\cdot )\) any BBA defined implicitly on the FoD \(\varTheta \), and we also denote it as \(m^\varTheta \) to explicitly refer to the FoD when necessary.
- 2.
m is Bayesian BBA if it has only singletons as focal elements, i.e. \(m(\theta _i)>0\) for some \(\theta _i \in \varTheta \) and \(m(X)=0\) for all non-singletons X of \(2^\varTheta \).
- 3.
Once the binary values are converted into their digit value with the most significant bit on the left (i.e. the least significant bit on the right).
- 4.
We always omit the 1st component \(s(\emptyset )\) of entropiece vector \(\textbf{s}(m)\) which is always equal to zero and not necessary in our analysis.
- 5.
Lambert’s W-function is implemented in MatlabTM as lambertw function.
- 6.
If the two masses values are admissible, that is if \(m_1(A\,\cup \, B)\in [0,1]\) and if \(m_2(A\,\cup \, B)\in [0,1]\). If one of them is non-admissible it is eliminated.
- 7.
Using lambertw MatlabTM function.
- 8.
We use the formal notation \(\log (0)\) even if \(\log (0)\) is \(-\infty \) because in our derivations we have always a \(0\log (0)\) product which is equal to zero due to L’Hôpital’s rule [4].
References
Shafer, G.: A Mathematical Theory of Evidence. Princeton University Press, Princeton (1976)
Dezert, J.: An effective measure of uncertainty of basic belief assignments. In: Fusion 2022 International Conference on Proceedings, ISIF Editor, Linköping, Sweden (2022)
Dezert, J., Tchamova, A.: On effectiveness of measures of uncertainty of basic belief assignments, information & security journal. Int. J. (ISIJ) 52, 9–36 (2022)
Bradley, R.E., Petrilli, S.J., Sandifer, C.E.: L’Hôpital’s analyse des infiniments petits (An annoted translation with source material by Johann Bernoulli), Birkhäuser, p 311 (2015)
Corless, R.M., Gonnet, G.H., Hare, D.E.G., Jeffrey, D.J., Knuth, D.E.: On the Lambert \(W\) function. Adv. Comput. Math. 5, 329–359 (1996)
Lambert \(W\) function (2022). https://en.wikipedia.org/wiki/Lambert_W_function. Accessed 1 Dec 2022
Shannon, C.E.: A mathematical theory of communication, in [9] and in The Bell System Technical Journal, 27, 379–423 & 623–656 (1948)
Dezert, J., Tchamova, A., Han, D.: Measure of information content of basic belief assignments. In: Le Hégarat-Mascle, S., Bloch, I., Aldea, E. (eds.) BELIEF 2022. LNCS, vol. 13506, pp. 119–128. Springer, Cham (2022). https://doi.org/10.1007/978-3-031-17801-6_12
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Dezert, J., Smarandache, F., Tchamova, A. (2023). Analytical Solution of the Simplest Entropiece Inversion Problem. In: Simian, D., Stoica, L.F. (eds) Modelling and Development of Intelligent Systems. MDIS 2022. Communications in Computer and Information Science, vol 1761. Springer, Cham. https://doi.org/10.1007/978-3-031-27034-5_15
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DOI: https://doi.org/10.1007/978-3-031-27034-5_15
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