In the last chapter we showed that classification with generative models is based on building simple probability models. In particular, we build class-conditional density functions Pr(x|w = k) over the observed data x for each value of the world state w.

In Chapter 3 we introduced several probability distributions that could be used for this purpose, but these were quite limited in scope. For example, it is not realistic to assume that all of the complexities of visual data are well described by the normal distribution. In this chapter, we show how to construct complex probability density functions from elementary ones using the idea of a hidden variable.

As a representative problem we consider face detection; we observe a 60 × 60 RGB image patch, and we would like to decide whether it contains a face or not. To this end, we concatenate the RGB values to form the 10800 × 1 vector x. Our goal is to take the vector x and return a label w ϵ {0,1} indicating whether it contains background (w =0) or a face (w = 1). In a real face detection system, we would repeat this procedure for every possible subwindow of an image (Figure 7.1).

We will start with a basic generative approach in which we describe the likelihood of the data in the presence/absence of a face with a normal distribution. We will then extend this model to address its weaknesses.