+* [A tale of three probabilistic families: Discriminative, descriptive, and generative models](https://www.ams.org/journals/qam/2019-77-02/S0033-569X-2018-01528-5/home.html) - ***Quarterly of Applied Mathematics***, 2018. [[All Versions](https://scholar.google.com/scholar?cluster=6129609629126793774)]. [[Preprint](http://www.stat.ucla.edu/~ywu/QAM2018.pdf)]. 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, the authors 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. The authors shall review these models within a common framework and explore their connections, and shall also review the recent developments that take advantage of the high approximation capacities of deep neural networks.
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