Pattern Theory - Statistics

Statistics

Grenander’s Pattern Theory treatment of Bayesian inference in seems to be skewed towards on image reconstruction (e.g. content addressable memory). That is given image I-deformed, find I. However, Mumford’s interpretation of Pattern Theory is broader and he defines PT to include many more well-known statistical methods. Mumford’s criteria for inclusion of a topic as Pattern Theory are those methods "characterized by common techniques and motivations", such as the HMM, EM algorithm, dynamic programming circle of ideas. Topics in this section will reflect Mumford's treatment of Pattern Theory. His principle of statistical Pattern Theory are the following:

• Use real world signals rather than constructed ones to infer the hidden states of interest.
• Such signals contain too much complexity and artifacts to succumb to a purely deterministic analysis, so employ stochastic methods too.
• Respect the natural structure of the signal, including any symmetries, independence of parts, marginals on key statistics. Validate by sampling from the derived models by and infer hidden states with Bayes’ rule.
• Across all modalities, a limited family of deformations distort the pure patterns into real world signals.
• Stochastic factors affecting an observation show strong conditional independence.

Statistical PT makes ubiquitous use of conditional probability in the form of Bayes theorem and Markov Models. Both these concepts are used to express the relation between hidden states (configurations) and observed states (images). Markov Models also captures the local properties of the stimulus, reminiscent of the purpose of bond table for regularity.

The generic set up is the following: Let s = the hidden state of the data that we wish to know. i = observed image. Bayes theorem gives

p (s | i ) p(i) = p (s, i ) = p (i|s ) p(s)

To analyze the signal (recognition): fix i, maximize p, infer s.

To synthesize signals (sampling): fix s, generate i's, compare w/ real world images

The following conditional probability examples illustrates these methods in action: