Nonlinear Filters

4Bayesian Paradigm and Optimal Nonlinear Filtering

4.1 Introduction

Immanuel Kant proposed the two concepts of the noumenal world and the phenomenal world. While the former is the world of things as they are, which is independent of our modes of perception and thought, the latter is the world of things as they appear to us, which depends on how we perceive things. According to Kant, everything about the noumenal world is transcendental that means it exists but is not prone to concept formation by us [43].

Following this line of thinking, statistics will aim at interpretation rather than explanation. In this framework, statistical inference is built on probabilistic modeling of the observed phenomenon. A probabilistic model must include the available information about the phenomenon of interest as well as the uncertainty associated with this information. The purpose of statistical inference is to solve an inverse problem aimed at retrieving the causes, which are presented by states and/or parameters of the developed probabilistic model, from the effects, which are summarized in the observations. On the other hand, probabilistic modeling describes the behavior of the system and allows us to predict what will be observed in the future conditional on states and/or parameters [44].

Bayesian paradigm provides a mathematical framework in which degrees of belief are quantified by probabilities. It is the method of choice for dealing with uncertainty in measurements. Using the Bayesian ...