  Modern high-dimensional point process data, especially those from
neuroscience experiments, often involve observations from multiple conditions
and/or experiments. Networks of interactions corresponding to these conditions
are expected to share many edges, but also exhibit unique, condition-specific
ones. However, the degree of similarity among the networks from different
conditions is generally unknown. Existing approaches for multivariate point
processes do not take these structures into account and do not provide
inference for jointly estimated networks. To address these needs, we propose a
joint estimation procedure for networks of high-dimensional point processes
that incorporates easy-to-compute weights in order to data-adaptively encourage
similarity between the estimated networks. We also propose a powerful
hierarchical multiple testing procedure for edges of all estimated networks,
which takes into account the data-driven similarity structure of the
multi-experiment networks. Compared to conventional multiple testing
procedures, our proposed procedure greatly reduces the number of tests and
results in improved power, while tightly controlling the family-wise error
rate. Unlike existing procedures, our method is also free of assumptions on
dependency between tests, offers flexibility on p-values calculated along the
hierarchy, and is robust to misspecification of the hierarchical structure. We
verify our theoretical results via simulation studies and demonstrate the
application of the proposed procedure using neuronal spike train data.
