Maximum Entropy estimation of probability distribution of variables in higher dimensions from lower dimensional data

TL;DR

This paper introduces a maximum entropy method to estimate high-dimensional probability distributions from lower-dimensional data, useful when direct inference is impossible due to dimensionality constraints.

Contribution

It develops a novel MaxEnt approach that estimates unknown distributions without explicitly calculating Lagrange multipliers, applicable to both discrete and continuous cases.

Findings

Method successfully estimates distributions in simulated examples.

Approach avoids explicit Lagrange multiplier calculations.

Validates the approach with theoretical and practical examples.

Abstract

A common statistical situation concerns inferring an unknown distribution Q(x) from a known distribution P(y), where X (dimension n), and Y (dimension m) have a known functional relationship. Most commonly, n<m, and the task is relatively straightforward. For example, if Y1 and Y2 are independent random variables, each uniform on [0, 1], one can determine the distribution of X = Y1 + Y2; here m=2 and n=1. However, biological and physical situations can arise where n>m. In general, in the absence of additional information, there is no unique solution to Q in those cases. Nevertheless, one may still want to draw some inferences about Q. To this end, we propose a novel maximum entropy (MaxEnt) approach that estimates Q(x) based only on the available data, namely, P(y). The method has the additional advantage that one does not need to explicitly calculate the Lagrange multipliers. In this paper we develop the approach, for both discrete and continuous probability distributions, and demonstrate its validity. We give an intuitive justification as well, and we illustrate with examples.