A Maximum Entropy Method with a Priori Maximum Likelihood Constraints

Abstract

Implementations of the maximum entropy method for data reconstruction have almost universally used the approach of maximizing the statistic S - lambda chi(2), where S is the Shannon entropy of the reconstructed distribution and chi(2) is the usual statistical measure associated with agreement between certain properties of the reconstructed distribution and the data. We develop here an alternative approach which maximizes the entropy subject to the set of constraints that chi(2) be at a minimum with respect to the reconstructed distribution. This in turn modifies the fitting statistic to be S - lambda .del chi(2) where lambda is now a vector. This new method provides a unique solution to both the well-posed and ill-posed problem, provides a natural convergence criterion which has previously been lacking in other implementations of maximum entropy, and provides the most conservative (least informative) data reconstruction result consistent with both maximum entropy and maximum likelihood methods, thereby mitigating against overinterpretation of reconstruction results. A spectroscopic example is shown as a demonstration.