Using Bayesian inference for parameter estimation when the system response and experimental conditions are measured with error and some variables are considered as nuisance variables

Parameter estimation is generally based upon the maximum likelihood approach and often involves regularization. Typically it is desired that the results be unbiased and of minimum variance. However, it is often better to accept biased estimates that have minimum mean square error. Bayesian inference is an attractive approach that achieves this goal and incorporates regularization automatically. More importantly, it permits us to analyse experiments in which both the system response and the independent variables (time, sensor position, experimental conditions, etc) are corrupted by noise and in which the model includes nuisance variables. This paper describes the use of Bayesian inference for an apparently simple experiment which is, in fact, fundamentally difficult and is compounded by a nuisance variable. By presenting this analysis we hope that members of the inverse community will see the value of applying Bayesian inference.