Advances in pattern theory
type: Item
instance of: scholarly article
title: Advances in pattern theory
zbMATH Open document ID: 0673.62071
author: Ulf Grenander
published in: The Annals of Statistics
publication date: 1989-01-01T00:00:00Z
review text: This paper surveys work in statistical pattern theory due principally to the author and his collaborators. The work concerns patterns formulated in a very general sense, as graphs \(c=\sigma (g_ 1,...,g_ n)\) based on vertices which are generators \(g_ i\). The generators are thought of as interacting via the graph \(\sigma\) by sending messages to neighbours. A Gibbs probability measure is constructed on the graph according to how intercommunicating messages agree. Three examples of this abstract framework are given: context-free grammars, global shape models for three-dimensional objects, and networks of computing modules. The Gibbs measure is used as a prior in a Bayesian formulation, in which partial observation and corruption by noise create the inferential problem. Solution involves simulation of the Markov process representing the posterior, and stochastic relaxation. Several technical issues are discussed, including identifiability problems in parameter estimation, use of pseudo-likelihood, limiting behaviour and rates of convergence.
reviewed by: Wilfrid S. Kendall
Mathematics Subject Classification ID: 62M05
Mathematics Subject Classification ID: 60B99
Mathematics Subject Classification ID: 68T99
Mathematics Subject Classification ID: 62P99
Mathematics Subject Classification ID: 62F15
Mathematics Subject Classification ID: 62M99
zbMATH DE Number: 4102336
zbMATH Keywords: limit theorems for Markov processes on graphs
zbMATH Keywords: asymptotic
zbMATH Keywords: efficiencies
zbMATH Keywords: parallel logic under uncertainty for complex systems
zbMATH Keywords: image processing
zbMATH Keywords: estimation of acceptor functions
zbMATH Keywords: statistical pattern theory
zbMATH Keywords: Gibbs probability measure
zbMATH Keywords: context-free grammars
zbMATH Keywords: global shape models
zbMATH Keywords: networks of computing modules
zbMATH Keywords: prior
zbMATH Keywords: partial observation
zbMATH Keywords: noise
zbMATH Keywords: Markov process
zbMATH Keywords: posterior
zbMATH Keywords: stochastic relaxation
zbMATH Keywords: identifiability
zbMATH Keywords: pseudo-likelihood
zbMATH Keywords: limiting behaviour
zbMATH Keywords: rates of convergence
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1214/aos/1176347002
OpenAlex ID: W2036999441
DOI: 10.1214/AOS/1176347002
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description: scientific article; zbMATH DE number 4102336