1 Recall the definition of value of information in Section 6. a. Prove that the value of...

1 Recall the definition of value of information in Section 6.

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a. Prove that the value of information is nonnegative and order independent. b. Explain why it is that some people would prefer not to get some information-for ex- ample, not wanting to know the sex of their baby when an ultrasound is done. c. A function f on sets is submodular if, for any element and any sets A and B such that AC B, adding a to A gives a greater increase in f than adding a to B: ACB (f(AU{x}) f(A)) (f(BU{a})f(B)). Submodularity captures the intuitive notion of diminishing returns. Is the value of in- formation, viewed as a function f on sets of possible observations, submodular? Prove this or find a counterexample.