4 Bayesian Estimation

ayesian estimation is a framework for the formulation of statistical inference problems. In the prediction or estimation of a random process from a related observation signal, the Bayesian philosophy is based on combining the evidence contained in the signal with prior knowledge of the probability distribution of the process. Bayesian methodology includes the classical estimators such as maximum a posteriori (MAP), maximum-likelihood (ML), minimum mean square error (MMSE) and minimum mean absolute value of error (MAVE) as special cases. The hidden Markov model, widely used in statistical signal processing, is an example of a Bayesian model. Bayesian inference is based on minimisation of the so-called Bayes’ risk function, which includes a posterior model of the unknown parameters given the observation and a cost-of-error function. This chapter begins with an introduction to the basic concepts of estimation theory, and considers the statistical measures that are used to quantify the performance of an estimator. We study Bayesian estimation methods and consider the effect of using a prior model on the mean and the variance of an estimate. The estimate–maximise (EM) method for the estimation of a set of unknown parameters from an incomplete observation is studied, and applied to the mixture Gaussian modelling of the space of a continuous random variable. This chapter concludes with an introduction to the Bayesian classification of discrete or finite-state signals, and the K-means clustering method. B f(y,θ)