Chapter 14: Fundamental concepts from probability theory

In this chapter, we cover the background concepts from probability theory needed in the remainder of the book. Our discussion includes some basic probability theory as well as more advanced topics such as random fields and probability measures on functions spaces. We begin by covering some basics regarding probability spaces and random variables in Section 14.1. In our study of infinite-dimensional Bayesian inverse problems, we need to be comfortable working with function-valued random variables. However, much insight can be gained by first considering the simpler cases of real- and vector-valued random variables, which are discussed in Section 14.2 and Section 14.3, respectively. We take on the study of function-valued random variables when discussing random fields in Section 14.4. In Section 14.5, we discuss the notion of Radon–Nikodym derivatives, which play a key role in our discussions of Bayesian inversion in a function space. We then consider measures on Hilbert spaces in Section 14.6. This is followed by a brief discussion of the notions of distance between probability measures on Hilbert spaces in Section 14.7.