Analytical Forms for Most Likely Matrices Derived from Incomplete Information

TL;DR

This paper derives analytical, closed-form solutions for inferring matrix entries from limited data constraints, extending maximum entropy methods with approximations and applicability to higher dimensions.

Contribution

It provides explicit analytical formulas for most likely matrices under various constraints, reducing reliance on numerical methods and extending to 3D matrices.

Findings

Analytical solutions exhibit properties similar to maximum entropy.

Solutions are valid for matrices of any size.

Approximate solutions use power series expansions.

Abstract

Consider a rectangular matrix describing some type of communication or transportation between a set of origins and a set of destinations, or a classification of objects by two attributes. The problem is to infer the entries of the matrix from limited information in the form of constraints, generally the sums of the elements over various subsets of the matrix, such as rows, columns, etc, or from bounds on these sums, down to individual elements. Such problems are routinely addressed by applying the maximum entropy method to compute the matrix numerically, but in this paper we derive analytical, closed-form solutions. For the most complicated cases we consider the solution depends on the root of a non-linear equation, for which we provide an analytical approximation in the form of a power series. Some of our solutions extend to 3-dimensional matrices. Besides being valid for matrices of arbitrary size, the analytical solutions exhibit many of the appealing properties of maximum entropy, such as precise use of the available data, intuitive behavior with respect to changes in the constraints, and logical consistency.