Fisher and DALI posterior approximations for testing gravity with gravitational waves

We investigate the accuracy and computational efficiency of posterior approximations for gravitational-wave (GW) parameter estimation based on Fisher Matrix and its higher-order extension, the Derivative Approximation for LIkelihoods (DALI), in the context of tests of General Relativity using parameterized deviations in the inspiral waveforms (the TIGER formalism). We adopt the $\mathtt{\text{IMRPhenomD}}$ waveform and compare the exact posterior sampling to the following approximations in increasing order of complexity: the traditional Fisher Matrix, the Fisher sampling (singlet-DALI), the doublet-DALI and the triplet-DALI. We find that both singlet and doublet-DALI are significantly faster than the exact posterior by factors between 20 and 1000, depending on the inspiral SNRs – larger gains correspond to larger SNRs. These gains are considerably larger for more complete waveforms, such as $\mathtt{\text{IMRPhenomHM}}$. To evaluate the accuracy of the approximations, we use Jensen-Shannon divergence. We conclude that both singlet and doublet-DALI systematically improves over the standard Fisher Matrix approximation, while triplet-DALI shows large variability, limiting its practical advantage. We also introduce a code with a new implementation of Fisher and DALI analysis (Symbolic DALI, $\mathtt{\text{SymDALI}}$) which computes derivatives symbolically via $\mathit{\text{Wolfram}}\mathit{\text{Language}}$.