Exercise 16.23 [nonnegative-VPI-exercise]
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.