  The pattern theory of Grenander is a mathematical framework where patterns
are represented by probability models on random variables of algebraic
structures. In this paper, we review three families of probability models,
namely, the discriminative models, the descriptive models, and the generative
models. A discriminative model is in the form of a classifier. It specifies the
conditional probability of the class label given the input signal. A
descriptive model specifies the probability distribution of the signal, based
on an energy function defined on the signal. A generative model assumes that
the signal is generated by some latent variables via a transformation. We shall
review these models within a common framework and explore their connections. We
shall also review the recent developments that take advantage of the high
approximation capacities of deep neural networks.
