  In this work, we introduce a new methodology for inferring the interaction
structure of discrete valued time series which are Poisson distributed. While
most related methods are premised on continuous state stochastic processes, in
fact, discrete and counting event oriented stochastic process are natural and
common, so called time-point processes (TPP). An important application that we
focus on here is gene expression. Nonparameteric methods such as the popular
k-nearest neighbors (KNN) are slow converging for discrete processes, and thus
data hungry. Now, with the new multi-variate Poisson estimator developed here
as the core computational engine, the causation entropy (CSE) principle,
together with the associated greedy search algorithm optimal CSE (oCSE) allows
us to efficiently infer the true network structure for this class of stochastic
processes that were previously not practical. We illustrate the power of our
method, first in benchmarking with synthetic datum, and then by inferring the
genetic factors network from a breast cancer micro-RNA (miRNA) sequence count
data set. We show the Poisson oCSE gives the best performance among the tested
methods anfmatlabd discovers previously known interactions on the breast cancer
data set.
