Probability density function estimation using the MinMax measure

The problem of initial probability assignment which is consistent with the available information about a probabilistic system is called a direct problem. E.T. Jaynes' (1957) maximum entropy principle (MaxEnt) provides a method for solving direct problems when the available information is in the form of moment constraints. On the other hand, given a probability distribution, the problem of finding a set of constraints which makes the given distribution a maximum entropy distribution is called an inverse problem. A method based on the MinMax measure to solve the above inverse problem is presented. The MinMax measure of information, defined by Kapur, Baciu and Kesavan (1995), is a quantitative measure of the information contained in a given set of moment constraints. It is based on both maximum and minimum entropy. Computational issues in the determination of the MinMax measure arising from the complexity in arriving at minimum entropy probability distributions (MinEPD) are discussed. The method to solve inverse problems using the MinMax measure is illustrated by solving the problem of estimating a probability density function of a random variable based on sample data.