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Statistical Methods for Fuzzy Data by Reinhard Viertl

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Chapter 17

a. For discrete quantity X ~ p(x|θ), θ ∈ Θ with finite observation space M = {x1,…,xm} and continuous parameter space Θ, for fuzzy a posteriori density π*(·|D*) the fuzzy numbers are fulfilling the following. Since p(·|θ) is a classical discrete probability distribution and π*(·|D*) a fuzzy probability density, the fuzzy values

Unnumbered Display Equation

The fuzzy sum is a fuzzy interval whose characterizing function has δ-cuts and

Unnumbered Display Equation

By is a fuzzy probability density on and by Section 14.1 p*(·|D*) generates a fuzzy discrete distribution on M.

b. In order to prove that f*(·|D*) is a fuzzy probability density we have to prove the existence of a classical probability density g(·) on MX which fulfills

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