  As a tool for capturing irregular temporal dependencies (rather than
resorting to binning temporal observations to construct time series), Hawkes
processes with exponential decay have seen widespread adoption across many
application domains, such as predicting the occurrence time of the next
earthquake or stock market spike. However, practical applications of Hawkes
processes face a noteworthy challenge: There is substantial and often
unquantified variance in decay parameter estimations, especially in the case of
a small number of observations or when the dynamics behind the observed data
suddenly change. We empirically study the cause of these practical challenges
and we develop an approach to surface and thereby mitigate them. In particular,
our inspections of the Hawkes process likelihood function uncover the
properties of the uncertainty when fitting the decay parameter. We thus propose
to explicitly capture this uncertainty within a Bayesian framework. With a
series of experiments with synthetic and real-world data from domains such as
"classical" earthquake modeling or the manifestation of collective emotions on
Twitter, we demonstrate that our proposed approach helps to quantify
uncertainty and thereby to understand and fit Hawkes processes in practice.
