Pii: S0893-9659(01)00136-7

K e y w o r d s C o n v e x hull, Entropy, Hankel matrix, Moment problem. 1. I N T R O D U C T I O N Every probability distribution has some uncertainty associated with it, and its entropy provides a quantitat ive measure of this uncertainty. Partial information given, for instance, in terms of averages about a random variate decreases its entropy. It thus appears interesting to provide an entropy estimate when partial information is given. A viable approach consists in use of the maximum entropy (ME) principle [1], according to which, out of all the probability distributions consistent with a given set of constraints, the one tha t has maximum entropy should be chosen. Such a value of entropy, obtained by the ME principle, therefore, represents an upper bound on the entropy of the underlying distribution. In general, however, the following drawback arises: the entropy is not provided directly in terms of the given averages, but it includes the parameters of the maximum entropy distribution (equations (2.1),(2.2)). Consequently, it is desirable to obtain an entropy estimate in terms of the given averages only. This paper attempts, under special hypotheses, to accomplish that goal by providing an upper and lower bound for absolutely continuous distributions, having assigned the first M algebraic moments (#1 , . , #M). The upper bound (equation (3.8)) will be stated under the most general hypothesis on the underlying distribution, whereas the lower bound (equation 3.14) will be provided under restrictive hypotheses. Indeed, a very sharply peaked distribution has a very low entropy, whereas if the distribution is widely spread, the entropy is higher, owing to the fact that the entropy measures the "uniformity" of a distribution. The absolutely continuous distributions considered here are concentrated on the interval [0, 1] and have their first M moments known. This is the classical reduced Hausdorff moment problem [2] consisting of recovering an unknown probability density in [0, 1] whose first M algebraic moments are known to match the given moments. 0893-9659/02/$ see front matter (~ 2002 Elsevier Science Ltd. All rights reserved. Typeset by ¢ 4 ~ T E X PII: S0893-9659(01)00136-7