  Quantification of the significance of a candidate multi-messenger detection
of cosmic events is an emerging need in the astrophysics and astronomy
communities. In this paper we show that a model-independent optimal search does
not exist, and we present a general Bayesian method for the optimal
model-dependent search, which is scalable to any number and any kind of
messengers, and applicable to any model. In the end, we demonstrate it through
an example for a joint gravitational wave, high-energy neutrino, short
gamma-ray burst event search; which has not been examined heretofore.
