  Searching for X-ray and gamma-ray bursts, including Gamma-ray bursts (GRBs),
Soft Gamma-ray Repeaters (SGRs) and high energy transients associated with
Gravitational wave (GW) events or Fast radio bursts (FRBs), is of great
importance in the multi-messenger and multi-wavelength era. Although a coherent
search based on the likelihood ratio and Gaussian statistics has been
established and utilized in many studies, this Gaussian-based method could be
problematic for weak and short bursts which usually have very few counts. To
deal with all bursts including weak ones, here we propose the coherent search
in Poisson statistics. We studied the difference between Poisson-based and
Gaussian-based search methods by Monte Carlo simulations, and find that the
Poisson-based search method has advantages compared to the Gaussian case
especially for weak bursts. Our results show that, for very weak bursts with
very low number of counts, the Poisson-based search can provide higher
significance than the Gaussian-based search, and its likelihood ratio (for
background fluctuation) still generally follows the chi2 distribution, making
the significance estimation of searched bursts very convenient. Thus, we
suggest that the coherent search based on Poisson likelihood ratio is more
appropriate in the search for generic transients including very weak ones.
