Statistical Methods for Parameter Estimation

In this chapter, the CIP is formulated in a statistical framework. Byusing the Bayesian inference theory, we cast the CIP asaninformation transfer problem, in which the prior information andtheinformation transferred from state observations are combinedtoreduce uncertainty in the estimated parameters. The priorinformationis modeled using a probability density function (PDF)called the priorPDF and the inverse solution is also a PDF known asthe posteriorPDF. Because it is a PDF, the inverse solution is alwaysexistent andunique but with uncertainty. When the posterior PDF is ina relativelysimple form, point estimates of the unknown parameterscan bereadily obtained by solving an optimization problem, just as wehavedone in the deterministic framework. When the posterior PDFhas acomplex multimodal shape, however, the non-uniquenessandinstability issues associated with the inverse solution arise again.Forsuch cases, Monte Carlo sampling methods provide powerfultoolsfor learning the posterior PDFs without requiring theiractualfunctional forms be known. Two popular Markov Chain MonteCarlo(MCMC) algorithms are introduced. The application of MCMCforinverse solution and global optimization is also discussed.