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62) Here again, this total Renyi entropy will be relevant whenever we shall have to take account of the spatial representation of the random variable. 9. 1. Statement of the principle PROBLEM. Suppose we know that a system has a set of possible states Xj, i = 1,2, ... ,m ,with unknown probabilities p(Xj), and we learn that the distribution P satisfies some constraints, for instance given values of some expectations L;p(xj)f(xj), or the range of variation of these values. Suppose that we need to define a distribution P* which is, in some sense, the best estimate of P given these constraints.

Statement of the principle PROBLEM. Suppose we know that a system has a set of possible states Xj, i = 1,2, ... ,m ,with unknown probabilities p(Xj), and we learn that the distribution P satisfies some constraints, for instance given values of some expectations L;p(xj)f(xj), or the range of variation of these values. Suppose that we need to define a distribution P* which is, in some sense, the best estimate of P given these constraints. How should we proceed to determine p*? MAXIMUM ENTROPY PRINCIPLE.

Pm;h) of which the value should not be modified by any permutation on the set (P],P2, ... ,Pmj. ) is continuous with respect to P],P2, ... ,Pm and h, except possibly at h = o. ) is an increasing function of h, (E4) The following relation is satisfied, that is 22 CHAPTER 2 O(Pl%'·"'Pjqj,. 62) Here again, this total Renyi entropy will be relevant whenever we shall have to take account of the spatial representation of the random variable. 9. 1. Statement of the principle PROBLEM. Suppose we know that a system has a set of possible states Xj, i = 1,2, ...

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