##
**Advances in pattern theory.**
*(English)*
Zbl 0673.62071

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.

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

### 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 |