# A Tale of Three Probabilistic Families: Discriminative, Descriptive and Generative Models

arXiv: https://arxiv.org/abs/1810.04261 · HTML: https://mlsanity.com/paper/1810.04261/

## Key facts

- Cited by 9 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 9 Oct 2018.

## Built on by (most cited first, 9 of 9)

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- [Toward a `Standard Model' of Machine Learning](https://mlsanity.com/paper/2108.07783/) (2021)
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- [On the Complexity of Bayesian Generalization](https://mlsanity.com/paper/2211.11033/) (2022)
- [Robust Audio Anomaly Detection](https://mlsanity.com/paper/2202.01784/) (2022)
- [On Discriminative Probabilistic Modeling for Self-Supervised Representation Learning](https://mlsanity.com/paper/2410.09156/) (2024)
- [Deriving discriminative classifiers from generative models](https://mlsanity.com/paper/2201.00844/) (2022)
- [Targeted control of fast prototyping through domain-specific interface](https://mlsanity.com/paper/2506.11070/) (2025)

## Abstract (arXiv)

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.