Non-Gaussian Parameter Bias: Formalism and Validation

Abstract

We present a framework for propagating systematic data mismatches into parameter biases and posterior deformations beyond the Gaussian Fisher approximation. Using the Derivative Approximation for Likelihoods (DALI), we derive analytic expressions for the maximum a posteriori (MAP) point and posterior mean, including a semi-analytic response expansion in systematic amplitude. We validate the framework against direct numerical calculations and Markov Chain Monte Carlo (MCMC) sampling in a controlled nonlinear two-parameter benchmark up to a covariance-weighted data-space mismatch of $d_{data}=5$. At $d_{data}=5$, Fisher predicts the wrong direction of the shift in one parameter and differs from the numerical result by a Fisher-metric distance of 23.1, whereas the analytic DALI MAP prediction remains within 0.091, with componentwise errors of 0.04% and 0.46%. The response expansion reproduces the posterior mean with errors of 0.02% and 0.18%, while the DALI posterior closely reproduces the displacement and deformation of the fully sampled nonlinear posterior. Mismatches of equal significance but different directions produce markedly different parameter shifts and posterior deformations, while multiple additive mismatches can produce nonadditive parameter responses through nonlinear inference. By reconstructing the posterior under systematic mismatch, the framework captures the nonlinear response and non-Gaussian geometry of parameter bias: how mismatches move and reshape the posterior, why their direction matters, and how their combined effects can be dissected to understand the interplay between multiple systematics.