Parameter estimation in the Error-in-Variables models using the Gibbs Sampler

Least squares and maximum likelihood techniques have long been used in parameter estimation problems. However, those techniques provide only point estimates with unknown or approximate uncertainty information. Bayesian inference coupled with the Gibbs Sampler is an approach to parameter estimation that exploits modern computing technology. The estimation results are complete with exact uncertainty information. The Error-in-Variables model (EVM) approach is investigated in this study. In it, both dependent and independent variables contain measurement errors, and the true values and uncertainties of all measurements are estimated. This EVM set-up leads to unusually large dimensionality in the estimation problem, which makes parameter estimation very difficult with classical techniques. In this paper, an innovative way of performing parameter estimation is introduced to chemical engineers. The paper shows that the method is simple and efficient; as well, complete and accurate uncertainty information about parameter estimates is readily available. Two real-world EVM examples are demonstrated: a large-scale linear model and an epidemiological model. The former is simple enough for most readers to understand the new concepts without difficulty. The latter has very interesting features in that a Poisson distribution is assumed, and a parameter with known distribution is retained while other unknown parameters are estimated. The Gibbs Sampler results are compared with those of the least squares.