Sixteenth Mathematical and Statistical Modeling Workshop for Graduate Students

Entropy is a measure of probabilistic uncertainty. In 1957, E.T. Jaynes introduced his idea of the principle of Maximum Entropy. This idea states that by finding a distribution with the maximum entropy, one maximizes the uncertainty constrained to the given information. In essence, this involves using all of the known information, and assuming nothing about the distribution. In this report, we explore the use of the Maximum Entropy Method, (MEM), in contrast and along with other known methods. We apply this to three different problems: the problem of non-transitive dice, the popcorn problem, and parameter estimation for binary stars. The non-transitive dice problem involves finding the face values and frequencies on three dice given the probabilities that each one beats the other. We use MEM coupled with logic to find this information. For the popcorn problem, we wish to estimate the proportion of unpopped kernals of a given set of popcorn using previous experiments. Finally, the aim of the last problem is to estimate the distance between stars and planets with the help of MEM. In all cases, we find that MEM is a very powerful tool for estimating probability distributions. 1 The Non-Transitive Dice Problem: God DOES Play Dice 1.