NONLINEAR LEAST SQUARES AND BAYESIAN INFERENCE

Many data analysis problems in metrology involve fitting a model to measurement data. Least squares methods arise in both the parameter estimation and Bayesian contexts in the case where the data vector is associated with a Gaussian distribution. For nonlinear models, both classical and Bayesian formulations involve statistical distributions that, in general, cannot be completely specified in analytical form. In this paper, we discuss how approximation methods and simulation techniques, including Markov chain Monte Carlo, can be applied in both a classical and Bayesian setting to provide parameter estimates and associated uncertainty matrices. While in the case of linear models, classical and Bayesian methods lead to the same type of calculations, in the nonlinear case the differences in the approaches are more apparent and sometimes quite significant.