Scientists often summarize how well data pin down model parameters with a simple Gaussian approximation. This paper shows that such Gaussian or “Fisher” approximations can miss or even get the direction of parameter shifts wrong when the true statistical shape is not Gaussian. The authors present a framework that tracks how systematic mismatches between data and model move and reshape parameter posteriors beyond that simple approximation.

The key technical tool is DALI, the Derivative Approximation for Likelihoods. DALI is a higher-order expansion of the likelihood that keeps more information about curvature and nonlinearity than the Fisher (Gaussian) approximation. Using DALI the authors derive analytic formulas for the maximum a posteriori (MAP) point—the best-fit parameters under the posterior—and for the posterior mean (the average parameter value under the posterior). They also give a semi-analytic “response expansion” that shows how the posterior shifts as the amplitude of a systematic mismatch grows.

The framework was tested in a controlled nonlinear problem with two parameters. The authors validated their analytic predictions against direct numerical calculations and against Markov Chain Monte Carlo (MCMC) sampling, a brute-force numerical method. They report results up to a covariance-weighted data-space mismatch labelled d_data = 5, a measure of systematic size relative to the data noise. At d_data = 5 the Fisher (Gaussian) prediction gave the wrong direction of shift for one parameter and differed from the numerical result by a large Fisher-metric distance of 23.1. By contrast the analytic DALI MAP prediction stayed within 0.091 in that metric, with componentwise fractional errors of 0.04% and 0.46%. The response expansion reproduced the posterior mean with errors of 0.02% and 0.18%, and the DALI posterior closely matched the sampled nonlinear posterior’s displacement and deformation.