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
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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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