The term optimal filtering traditionally refers to a class of methods that can be used for estimating the state of a time-varying system which is indirectly observed through noisy measurements. The term optimal in this context refers to statistical optimality. Bayesian filtering refers to the Bayesian way of formulating optimal filtering. In this book we use these terms interchangeably and always mean Bayesian filtering.

In optimal, Bayesian, and Bayesian optimal filtering the state of the system refers to the collection of dynamic variables such as position, velocity, orientation, and angular velocity, which fully describe the system. The noise in the measurements means that they are uncertain; even if we knew the true system state the measurements would not be deterministic functions of the state, but would have a distribution of possible values. The time evolution of the state is modeled as a dynamic system which is perturbed by a certain process noise. This noise is used for modeling the uncertainties in the system dynamics. In most cases the system is not truly stochastic, but stochasticity is used for representing the model uncertainties.

Bayesian smoothing (or optimal smoothing) is often considered to be a class of methods within the field of Bayesian filtering. While Bayesian filters in their basic form only compute estimates of the current state of the system given the history of measurements, Bayesian smoothers can be used to reconstruct states that happened before the current time.

This chapter provides a brief presentation of the philosophical and math¬ematical foundations of Bayesian inference. The connections to classical statistical inference are also briefly discussed.

Philosophy of Bayesian inference

The purpose of Bayesian inference (Bernardo and Smith, 1994; Gelman et al., 2004) is to provide a mathematical machinery that can be used for modeling systems, where the uncertainties of the system are taken into account and the decisions are made according to rational principles. The tools of this machinery are the probability distributions and the rules of probability calculus.

If we compare the so-called frequentist philosophy of statistical analysis to Bayesian inference the difference is that in Bayesian inference the prob¬ability of an event does not mean the proportion of the event in an infinite number of trials, but the uncertainty of the event in a single trial. Because models in Bayesian inference are formulated in terms of probability dis¬tributions, the probability axioms and computation rules of the probability theory (see, e.g., Shiryaev, 1996) also apply in Bayesian inference.

Connection to maximum likelihood estimation

Consider a situation where we know the conditional distribution p(yk | θ) of conditionally independent random variables (measurements) y1:T = {y1,…, yT}, but the parameter θ ∊ ℝd is unknown.

The aim of this book is to give a concise introduction to non-linear Kalman filtering and smoothing, particle filtering and smoothing, and to the related parameter estimation methods. Although the book is intended to be an introduction, the mathematical ideas behind all the methods are carefully explained, and a mathematically inclined reader can get quite a deep understanding of the methods by reading the book. The book is purposely kept short for quick reading.

The book is mainly intended for advanced undergraduate and graduate students in applied mathematics and computer science. However, the book is suitable also for researchers and practitioners (engineers) who need a concise introduction to the topic on a level that enables them to implement or use the methods. The assumed background is linear algebra, vector calculus, Bayesian inference, and MATLAB® programming skills.

As implied by the title, the mathematical treatment of the models and algorithms in this book is Bayesian, which means that all the results are treated as being approximations to certain probability distributions or their parameters. Probability distributions are used both to represent uncertainties in the models and for modeling the physical randomness. The theories of non-linear filtering, smoothing, and parameter estimation are formulated in terms of Bayesian inference, and both the classical and recent algorithms are derived using the same Bayesian notation and formalism. This Bayesian approach to the topic is far from new. It was pioneered by Stratonovich in the 1950s and 1960s – even before Kalman's seminal article in 1960.