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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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Q: Why does the natural extension of a lower prevision dominate it?

MarsOn 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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Q: Notation in this imprecise Markov chain upper transition operator definition

MarsIn Example 11.6 on p. 270 of Hermans and Ċ 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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Q: Simple calculation in imprecise probability urn example

MarsIn 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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Q: Lower Expectation

IamMeeohLet $X$ 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...

 

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