# Pattern theory

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For "pattern theory" in pain science, see [Pain#Pattern Theory](/source/Pain#Pattern_Theory). For other meanings of "pattern", see [Pattern (disambiguation)](/source/Pattern_(disambiguation)).

**Pattern theory**, formulated by [Ulf Grenander](/source/Ulf_Grenander), is a mathematical [formalism](/source/Formalism_(mathematics)) to describe knowledge of the world as [patterns](/source/Pattern). It differs from other approaches to [artificial intelligence](/source/Artificial_intelligence) in that it does not begin by prescribing [algorithms](/source/Algorithm) and machinery to recognize and classify patterns; rather, it prescribes a vocabulary to articulate and recast the pattern concepts in precise language. Broad in its mathematical coverage, Pattern Theory spans [algebra](/source/Algebra) and [statistics](/source/Statistics), as well as local topological and global entropic properties.

In addition to the new algebraic vocabulary, its [statistical](/source/Statistics) approach is novel in its aim to:

- Identify the [hidden variables](/source/Latent_variable) of a [data set](/source/Data_set) using real world data rather than artificial stimuli, which was previously commonplace.
- Formulate prior distributions for hidden variables and models for the observed variables that form the vertices of a [Gibbs](/source/Gibbs_sampling)-like graph.
- Study the randomness and variability of these graphs.
- Create the basic classes of [stochastic](/source/Stochastic) models applied by listing the deformations of the patterns.
- Synthesize (sample) from the models, not just analyze signals with them.

The [Brown University](/source/Brown_University) Pattern Theory Group was formed in 1972 by Ulf Grenander.[1] Many mathematicians are currently working in this group, noteworthy among them being the [Fields Medalist](/source/Fields_Medal) [David Mumford](/source/David_Mumford).[2] Mumford regards Grenander as his "guru"[3] in Pattern Theory.

## See also

- [Abductive reasoning](/source/Abductive_reasoning)
- [Algebraic statistics](/source/Algebraic_statistics)
- [Computational anatomy](/source/Computational_anatomy)
- [Formal concept analysis](/source/Formal_concept_analysis)
- [Grammar induction](/source/Grammar_induction)
- [Image analysis](/source/Image_analysis)
- [Induction](/source/Mathematical_induction)
- [Lattice theory](/source/Lattice_theory)
- [Spatial statistics](/source/Spatial_statistics)

## References

1. ["Ulf Grenander's Home Page"](https://www.dam.brown.edu/pattern/ug.html). January 22, 2009. [Archived](https://web.archive.org/web/20090122102751/http://www.dam.brown.edu/pattern/ug.html) 2009-01-22 at the Wayback Machine.

1. Mumford, David (2002-12-01). "Pattern theory: the mathematics of perception". [arXiv:math/0212400v1](https://arxiv.org/abs/math/0212400v1)

1. ["| Vision Introductory Comments"](https://www.dam.brown.edu/people/mumford/vision/introvision.html). *www.dam.brown.edu*. Retrieved 2025-12-13.

## Further reading

- 2007. [Ulf Grenander](/source/Ulf_Grenander) and Michael Miller *Pattern Theory: From Representation to Inference*. Oxford University Press. Paperback. (ISBN 9780199297061)
- 1994. Ulf Grenander *General Pattern Theory*. Oxford Science Publications. (ISBN 978-0198536710)
- 1996. Ulf Grenander *Elements of Pattern Theory*. Johns Hopkins University Press. (ISBN 978-0801851889)

## External links

- [Pattern Theory Group](https://www.brown.edu/research/projects/pattern-theory/) at Brown University
- [Pattern Theory: Grenander's Ideas and Examples](http://video.google.com/videoplay?docid=-8276853449498438102&ei=WmXGSbLUJ9OK-Qah07zGDQ) - a video lecture by David Mumford
- [Pattern Theory and Applications](http://wiki.nosdigitais.teia.org.br/Pattern_Theory_and_Applications) - graduate course page with material by a Brown University alumnus

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