Extending the Fisher Information Matrix in Gravitational-wave Data Analysis

The Fisher information matrix (FM) plays an important role in forecasts and inferences in many areas of physics. While giving fast parameter estimation with Gaussian likelihood approximation in the parameter space, the FM can only give the ellipsoidal posterior contours of the parameters and it loses the higher-order information beyond Gaussianity. We extend the FM in gravitational-wave (GW) data analysis by 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 two real GW events, GW150914 and GW170817, DALI can reduce the difference between the FM approximation and the real posterior by 5 times in the best case. The calculation times of DALI and the FM are 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 of the FM analysis and gives suggestions for complex sampling techniques that 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.