### Artificial IntelligenceAIMA Exercises

Recall the definition of value of information in Section VPI-section.
1. Prove that the value of information is nonnegative and order independent.
2. 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.
3. 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.

Recall the definition of value of information in Section VPI-section.
1. Prove that the value of information is nonnegative and order independent.
2. 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.
3. 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.

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