Superresolution in the maximum entropy approach to invert Laplace transforms

Abstract

The method of maximum entropy has proven to be a rather powerful way to solve the inverse problem consisting of determining a probability density $f_S(s)$ on $[0,\infty)$ from the knowledge of the expected value of a few generalized moments, that is, of functions $g_i(S)$ of the variable $S.$ A version of this problem, of utmost relevance for banking, insurance, engineering and the physical sciences, corresponds to the case in which $S \geq 0$ and $g_i(s)=\exp(-\alpha_i s),$ th expected values $E[\exp-\alpha_i S)]$ are the values of the Laplace transform of $S$ the points $\alpha_i$ on the real line. Since inverting the Laplace transform is an ill-posed problem, to devise numerical tecniques that are efficient is of importance for many applications, specially in cases where all we know is the value of the transform at a few points along the real axis. A simple change of variables transforms the Laplace inversion problem into a fractional moment problem on $[0,1].$ It is remarkable that the maximum entropy procedure allows us to determine the density on $[0,1]$ with high accuracy. In this note, we examine why this might be so.