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Imprecise probability generalizes probability theory to allow for partial probability specifications, and is applicable when information is scarce, vague, or conflicting, in which case a unique probability distribution may be hard to identify. Thereby, the theory aims to represent the available knowledge more accurately. Imprecision is useful for dealing with expert elicitation, because: People have a limited ability to determine their own subjective probabilities and might find that they can only provide an interval. As an interval is compatible with a range of opinions, the analysis ought to...
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On p. 48 in Troffaes and de Cooman's Lower Previsions there's a claim that "It is clear that" the natural extension $\underline{E}_\underline{P}$ of lower prevision $\underline{P}$ dominates (42) it, i.e. that $\underline{E}_\underline{P}(f) \geq \underline{P}(f)$ for all bounded functions ("gamb...

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In Example 11.6 on p. 270 of Hermans and Å kulj's "Stochastic Processes" in Augustin et al.'s Introduction to Imprecise Probabilities, there is a definition of an upper transition operator as $$\overline T=I_{\cal X} \max$$ $\overline T$ is an upper prevision (upper expectation) operator, where f...

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In Miranda and de Cooman's chapter 3, "Structural judgements", in Augustin et al.'s Introduction to Imprecise Probability, example 3.4 on p. 65 shows that independence in the selection (type-2 independence) does not imply strong independence (type-3 independence) for lower previsions. One of the...

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Let $X$ be, for simplicity, a finite set (with the discrete topology). Denote with $M(X)$ the set of probability measures on $X$ endowed with the weak topology. For $\mu\in M(X)$ and a (necessarily measurable) function $f:X\rightarrow[-1,1]$ denote with $E_{\mu}(f)$ the expected value of $f$. F...