Can I get an example? Say, X is a random positive real number. For which distribution which parameters that maximize E(X) will not maximize E(log(X))?
As a trivial example, let’s say you are choosing between distribution A and distribution B.
In distribution A, X=100 with probability 0.5, and X=epsilon with probability 0.5
In distribution B, X=10 with probability 1
The average value of X under distribution A is 50, whereas the average value of X under distribution B is 10. If you want to maximize E(X) you will therefore choose distribution A
The average value of log X under distribution A is negative infinity, whereas the average value of log X under distribution B is 1. If you want to maximize E(log X) you will choose distribution B.
As a trivial example, let’s say you are choosing between distribution A and distribution B.
In distribution A, X=100 with probability 0.5, and X=epsilon with probability 0.5
In distribution B, X=10 with probability 1
The average value of X under distribution A is 50, whereas the average value of X under distribution B is 10. If you want to maximize E(X) you will therefore choose distribution A
The average value of log X under distribution A is negative infinity, whereas the average value of log X under distribution B is 1. If you want to maximize E(log X) you will choose distribution B.