Question

Let p=(p1,p2,…,pm) be a probability distribution over m possible outcomes. The entropy of p is a measure of how much randomness there is in the outcome. It is defined as

pi Inp i= 1

where ln denotes natural logarithm. We wish to ascertain whether F(p) is a convex function of p. As usual, we begin by computing the Hessian.

A) Consider the specific point p=(1/m,1/m,…,1/m). What is the (1,1) entry of the Hessian at this point? Your answer should be a function of m.

B) Continuing, what is the (1,2) entry of the Hessian at this specific point?

Answer 1

dly 万hm de 1 0 0 dene