Stieltjes and Hamburger Reduced Moment Problem When MaxEnt Solution Does Not Exist

Pier Luigi Novi Inverardi,Aldo Tagliani

doi:10.3390/math9040309

Abstract

For a given set of moments whose predetermined values represent the available information, we consider the case where the Maximum Entropy (MaxEnt) solutions for Stieltjes and Hamburger reduced moment problems do not exist. Genuinely relying upon MaxEnt rationale we find the distribution with largest entropy and we prove that this distribution gives the best approximation of the true but unknown underlying distribution. Despite the nice properties just listed, the suggested approximation suffers from some numerical drawbacks and we will discuss this aspect in detail in the paper.

Highlights

For a given set of moments whose predetermined values represent the available information, we consider the case where the Maximum Entropy (MaxEnt) solutions for Stieltjes and Hamburger reduced moment problems do not exist
In the context of testable information that is, when a statement about a probability distribution whose truth or falsity is well-defined, the principle of maximum entropy states that the probability distribution which best represents the current state of knowledge is the one with largest entropy
Due to the non-uniqueness of the recovered density, the best choice among the competitors may be done by invoking the Maximum Entropy (MaxEnt) principle [2] which consists in maximizing the Shannon-entropy

Summary

Problem Formulation and MaxEnt Rationale

On the other hand, when the existence conditions for f M are not satisfied, the nonexistence of the MaxEnt solution in Stieltjes and Hamburger reduced moment problem poses a series of interesting and important questions about how to find an approximant of the unknown density f least committed to the information not given to us (still obeying to Jaynes’ Principle). This problem is addressed the present paper. In Appendix A. the existence conditions of MaxEnt distributions in Stieltjes and Hamburger case are shortly reviewed

Stieltjes Case

Hamburger Case

Numerical Aspects

Discussion and Conclusions