Comparison of maximum entropy and Laguerre inverse moment problem algorithms in the context of Stieltjes summation problem

Abstract

The reconstruction of a probability distribution function (PDF) given a finite string of moments, also known as the inverse moment problem, is central to the Stieltjes summation problem. We compare two solutions to the moment problem: Mead's maximum entropy algorithm (MEA) which invokes the principle of maximum entropy and the popular Laguerre polynomial algorithm. We compare the methods based on two criteria. The first is their accuracy in reproducing the actual moments ⟨x^k⟩. The second concern their accuracy in solving Stieltjes summation problems, specifically in approximating the ground-state energy of the quartic anharmonic oscillator. The results show that MEA reproduces the actual moments better than the Laguerre algorithm. Furthermore, MEA also performs better in approximating energy with a small number of input moments, while the Laguerre algorithm excels when a longer string of moments is available. It has great potential in solving the criticism in the current solution to the Stieltjes summation and continuation problem.