  Boolean functions and networks are commonly used in the modeling and analysis
of complex biological systems, and this paradigm is highly relevant in other
important areas in data science and decision making, such as in the medical
field and in the finance industry. Automated learning of a Boolean network and
Boolean functions, from data, is a challenging task due in part to the large
number of unknowns (including both the structure of the network and the
functions) to be estimated, for which a brute force approach would be
exponentially complex. In this paper we develop a new information theoretic
methodology that we show to be significantly more efficient than previous
approaches. Building on the recently developed optimal causation entropy
principle (oCSE), that we proved can correctly infer networks distinguishing
between direct versus indirect connections, we develop here an efficient
algorithm that furthermore infers a Boolean network (including both its
structure and function) based on data observed from the evolving states at
nodes. We call this new inference method, Boolean optimal causation entropy
(BoCSE), which we will show that our method is both computationally efficient
and also resilient to noise. Furthermore, it allows for selection of a set of
features that best explains the process, a statement that can be described as a
networked Boolean function reduced order model. We highlight our method to the
feature selection in several real-world examples: (1) diagnosis of urinary
diseases, (2) Cardiac SPECT diagnosis, (3) informative positions in the game
Tic-Tac-Toe, and (4) risk causality analysis of loans in default status. Our
proposed method is effective and efficient in all examples.
