  The principle of maximum entropy, as introduced by Jaynes in information
theory, has contributed to advancements in various domains such as Statistical
Mechanics, Machine Learning, and Ecology. Its resultant solutions have served
as a catalyst, facilitating researchers in mapping their empirical observations
to the acquisition of unbiased models, whilst deepening the understanding of
complex systems and phenomena. However, when we consider situations in which
the model elements are not directly observable, such as when noise or ocular
occlusion is present, possibilities arise for which standard maximum entropy
approaches may fail, as they are unable to match feature constraints. Here we
show the Principle of Uncertain Maximum Entropy as a method that both encodes
all available information in spite of arbitrarily noisy observations while
surpassing the accuracy of some ad-hoc methods. Additionally, we utilize the
output of a black-box machine learning model as input into an uncertain maximum
entropy model, resulting in a novel approach for scenarios where the
observation function is unavailable. Previous remedies either relaxed feature
constraints when accounting for observation error, given well-characterized
errors such as zero-mean Gaussian, or chose to simply select the most likely
model element given an observation. We anticipate our principle finding broad
applications in diverse fields due to generalizing the traditional maximum
entropy method with the ability to utilize uncertain observations.
