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.
Edited to add: The idea behind Von Neumann Morgenstern Expected Utility Theory is that optimizing your expected utility does not imply the same choices as optimizing the expected payoff. If you maximize for E(X) your utility function is risk neutral, if you maximize for E(log X) your utility function is risk averse etc. If maximizing these two expectations always implied identical choices, it would not be possible to define risk aversion.
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.
Edited to add: The idea behind Von Neumann Morgenstern Expected Utility Theory is that optimizing your expected utility does not imply the same choices as optimizing the expected payoff. If you maximize for E(X) your utility function is risk neutral, if you maximize for E(log X) your utility function is risk averse etc. If maximizing these two expectations always implied identical choices, it would not be possible to define risk aversion.