The purpose of this chapter is to provide a modest introduction to the huge and important topics of sampling and inference, which will serve our purpose in succeeding chapters. This is not a stand-alone chapter, indeed it provides many illustrations of the significance of early sections on probability, just as they in turn utilise the preceding linear algebra/matrix results. So what is the present chapter about? The short answer, which will be amplified section by section, is the interpretation of data, having in mind ultimately the interpretation of pixel values in computer images.

We begin with the idea of a sample, a sequence of determinations X1, …, Xn of a random variable X. We seek statistics, i.e. functions f(X1, …, Xn), to help answer questions such as (a) given that a distribution is of a certain type: Poisson, exponential, normal, …, how can we estimate the distribution parameters and with what certainty, (b) given a sample, what is the underlying distribution, again with what certainty? Sections 11.2, 11.3 and 11.4 utilise the methods of Section 11.1.

In Section 11.2 we introduce the Bayesian approach, distinct from the Bayesian Theorem, but ultimately based upon it. The idea is to improve on an imprecise model of a situation or process by utilising every piece of data that can be gathered. The section concludes with the Bayes pattern classsifer, a first step in object/pattern recognition.