Chapter 8 - Parameter Estimation With Bayesian Methods

Abstract

Bayesian methods are becoming increasingly popular for estimating parameters of complex mathematical models because the Bayesian approach provides a coherent framework for dealing with uncertainty. To start with, the principle is a prior probability distribution of the model parameters describing our belief about the parameter values before we use the set of measurements. The Bayesian methods then tell us how to update this belief using the measurements to give the posterior parameter density. In the Bayesian approach, the parameters are defined as random variables and the prior and posterior parameter distributions represent our belief about parameter values before and after using observed data to improve estimates. This approach has several advantages: (i) parameters can be estimated from different types of information (data, literature, expert knowledge); (ii) the posterior probability distribution can be used to implement uncertainty analysis methods; and (iii) the posterior probability distribution can be used for optimizing decisions in the face of uncertainty. This chapter presents the basic principles of the Bayesian approach and describes several algorithms to calculate posterior parameter distributions. These algorithms are illustrated in several applications on yield and soil carbon estimation.