A Bayesian-Inspired Framework for Parameter Estimation and Error Quantification

Abstract

This work introduces a novel Bayesian inspired regression method for the simultaneous estimation of model parameters and data uncertainties. The key mathematical result of this framework is an extended least squares objective function. The conventional sum of squared residuals is expanded by adding the logarithms of the computed standard deviations. This approach is particularly useful in cases with strongly varying or parameter-dependent uncertainties. Through five examples, we demonstrate that our extended least squares analysis robustly estimates data uncertainties and quantifies model parameter correlations via the Hessian matrix of the objective function. Finally, in a sixth example from materials science, we applied the method to model the evolution of the dislocation density in martensite during annealing of chromium stainless steel using a Boltzmann function. The approach successfully estimates the values of the parameters, their uncertainties and their correlations. A key outcome of our uncertainty quantification is the derivation of a credible interval for the simulated dislocation densities.