A Derivation of Entropy and the Maximum Entropy Criterion in the Context of Decision Problems

In this paper the entropy functional is derived in the context of decision making under uncertainty. It is shown that the mathematical description of a decision problem can be partitioned into two parts: a probability distribution p and a convex set LN(A) that describes the remaining structure of the problem. The question of selecting a general representative form for this convex set is explored, and a set of assumptions is proposed that specifies a form for such a representative set LN. Corresponding to this set LN, the entropy of the probability p can be interpreted as an information measure with respect to decision problems. In addition, it is shown that if this set LN is substituted for the set LN(A) in a given decision problem, the use of the maximum entropy criterion for probability selection corresponds to a minimax solution to the representative form of the problem. To the extent that the set LN resembles the set LN(A) in a given decision problem, the use of the maximum entropy criterion for probability selection corresponds to a minimax criterion for decision making.