Grenander, Ulf Advances in pattern theory. (English) Zbl 0673.62071 Ann. Stat. 17, No. 1, 1-30 (1989). 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. Reviewer: W.S.Kendall Cited in 2 Documents MSC: 62M05 Markov processes: estimation; hidden Markov models 60B99 Probability theory on algebraic and topological structures 68T99 Artificial intelligence 62P99 Applications of statistics 62F15 Bayesian inference 62M99 Inference from stochastic processes Keywords:limit theorems for Markov processes on graphs; asymptotic; efficiencies; parallel logic under uncertainty for complex systems; image processing; estimation of acceptor functions; statistical pattern theory; Gibbs probability measure; context-free grammars; global shape models; networks of computing modules; prior; partial observation; noise; Markov process; posterior; stochastic relaxation; identifiability; pseudo-likelihood; limiting behaviour; rates of convergence × Cite Format Result Cite Review PDF Full Text: DOI