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Pattern theory, formulated by Ulf Grenander, is a mathematical formalism to describe knowledge of the world as patterns. It differs from other approaches to artificial intelligence in that it does not begin by prescribing algorithms and machinery to recognize and classify patterns; rather, it prescribes a vocabulary to articulate and recast the pattern concepts in precise language. In addition to the new algebraic vocabulary, its statistical approach is novel in its aim to: * Identify the hidden variables of a 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-like graph. * Study the randomness and variability of these graphs. * Create the basic classes of stochastic models applied by listing the deformations of the patterns. * Synthesize (sample) from the models, not just analyze signals with it. Broad in its mathematical coverage, Pattern Theory spans algebra and statistics, as well as local topological and global entropic properties. The Brown University Pattern Theory Group was formed in 1972 by (Ulf Grenander ). Many mathematicians are currently working in this group, noteworthy among them being the Fields Medalist David Mumford. Mumford regards Grenander as his "guru" in this subject. ==Example: Natural Language Grammar== We begin with an example to motivate the algebraic definitions that follow. If we want to represent language patterns, the most immediate candidate for primitives might be words. However, set phrases, such as “in order to”, immediately indicate the inappropriateness of words as atoms. In searching for other primitives, we might try the rules of grammar. We can represent grammars as finite state automata or context-free grammars. Below is a sample finite state grammar automaton. The following phrases are generated from a few simple rules of the automaton and programming code in pattern theory: :: ''the boy who owned the small cottage went to the deep forest'' :: ''the prince walked to the lake'' :: ''the girl walked to the lake and the princess went to the lake'' :: ''the pretty prince walked to the dark forest'' To create such sentences, rewriting rules in finite state automata act as generators to create the sentences as follows: if a machine starts in state 1, it goes to state 2 and writes the word “the”. From state 2, it writes one of 4 words: prince, boy, princess, girl, chosen at random. The probability of choosing any given word is given by the Markov chain corresponding to the automaton. Such a simplistic automaton occasionally generates more awkward sentences :: ''the evil evil prince walked to the lake'' :: ''the prince walked to the dark forest and the prince walked to a forest and the princess who lived in some big small big cottage who owned the small big small house went to a forest'' From the finite state diagram we can infer the following generators (shown at right) that creates the signal. A generator is a 4-tuple: current state, next state, word written, probability of written word when there are multiple choices. That is, each generator is a state transition arrow of state diagram for a Markov chain. Imagine that a configuration of generators are strung together linearly so its output forms a sentence, so each generator "bonds" to the generators before and after it. Denote these bonds as 1a,1b,2a,2b,…12a,12b. Each numerical label corresponds to the automaton's state and each letter "a" and "b" corresponds to the inbound and outbound bonds. Then the following bond table (left) is equivalent to the automaton diagram. For the sake of simplicity, only half of the bond table is shown—the table is actually symmetric. As one can tell from this example, and typical of signals that are studied, identifying the primitives and bond tables requires some thought. The example highlights another important fact not readily apparent in other signals problems: that a configuration is not the signal that is observed; rather, its image as a sentence is observed. Herein lies a significant justification for distinguishing an observable from a non-observable construct. Additionally, it provides an algebraic structure to associate with hidden Markov models. In sensory examples such as the vision example below, the hidden configurations and observed images are much more similar, and such a distinction may not seem justified. Fortunately, the grammar example reminds us of this distinction. A more sophisticated example can be found in the link grammar theory of natural language. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Pattern theory」の詳細全文を読む スポンサード リンク
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