ABSTRACT
The term “pattern theory” was coined by Ulf Grenander to distinguish his approach to the analysis of patterned structures in the world from “pattern recognition.” In this book, we use it in a rather broad sense to include the statistical methods used in analyzing all “signals” generated by the world, whether they be images, sounds, written text, DNA or protein strings, spike trains in neurons, or time series of prices or weather; examples from all of these appear either in Grenander’s book Elements of Pattern Theory [94] or in the work of our colleagues, collaborators, and students on pattern theory. We believe the work in all these areas has a natural unity: common techniques and motivations. In particular, pattern theory proposes that the types of patterns (and the hidden variables needed to describe these patterns) that are found in one class of signals will often be found in the others and that their characteristic variability will be similar. Hence the stochastic models used to describe signals in one field will crop up in many other signals. The underlying idea is to find classes of stochastic models that can capture all the patterns we see in nature, so that random samples from these models have the same “look and feel” as the samples from the world itself. Then the detection of patterns in noisy and ambiguous samples can be achieved by the use of Bayes’ rule, a method that can be described as “analysis by synthesis.”
We can express this approach to signals and their patterns in a set of five basic principles, the first three of which are as follows:
