  The Fisher information matrix (FM) plays an important role in forecasts and
inferences in many areas of physics. While giving fast parameter estimation
with the Gaussian likelihood approximation in the parameter space, the FM can
only give the ellipsoidal posterior contours of parameters and lose the
higher-order information beyond Gaussianity. We extend the FM in
gravitational-wave (GW) data analysis using the Derivative Approximation for
LIkelihoods (DALI), a method to expand the likelihood while keeping it positive
definite and normalizable at every order, for more accurate forecasts and
inferences. When applied to the two real GW events, GW150914 and GW170817, DALI
can reduce the difference between FM approximation and the real posterior by 5
times in the best case. The calculation time of DALI and FM is at the same
order of magnitude, while obtaining the real full posterior will take several
orders of magnitude longer. Besides more accurate approximations, higher-order
correction from DALI provides a fast assessment on the FM analysis and gives
suggestions for complex sampling techniques which are computationally
intensive. We recommend using the DALI method as an extension to the FM method
in GW data analysis to pursue better accuracy while still keeping the speed.
