Recall the definition of value of
information in Section VPI-section.
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Prove that the value of information is nonnegative and order independent.
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Explain why it is that some people would prefer not to get some information—for example, not wanting to know the sex of their baby when an ultrasound is done.
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A function
$f$ on sets is submodular if, for any element$x$ and any sets$A$ and$B$ such that$A\subseteq B$ , adding$x$ to$A$ gives a greater increase in$f$ than adding$x$ to$B$ :$$A\subseteq B \Rightarrow (f(A \cup {x}) - f(A)) \geq (f(B\cup {x}) - f(B))\ .$$ Submodularity captures the intuitive notion of diminishing returns. Is the value of information, viewed as a function$f$ on sets of possible observations, submodular? Prove this or find a counterexample.