Limit theorems for Markov random fields
Markov Random Fields (MRF's) have been extensively applied in Statistical Mechanics as well as in Bayesian Image Analysis. MRF's are a special class of dependent random variables located at the vertices of a graph whose joint distribution includes a parameter called the temperature. When the number of vertices of the graph tends to infinity, the normalized distribution of statistics based on these random variables converge in distribution. It can happen that for certain values of the temperature, that the rate of growth of these normalizing constants change drastically. This feature is generally used to explain the phenomenon of phase transition as understood by physicist. In this dissertation the author will show that this drastic change in normalizing constants occurs even in the relatively smooth case when all the random variables are Gaussian. Hence any image analytic MRF ought to be checked for such discontinuous behavior before any analysis is performed. Mixed limit theorems in Bayesian Image Analysis seek to replace intensive simulations of MRF's with limit theorems that approximate the distribution of the MRF's as the number of sites increases. The problem of deriving mixed limit theorems for MRF's on a one dimensional lattice graph with an acceptor function that has a second moment has been studied by Chow. A mixed limit theorem for the integer lattice graph is derived when the acceptor function does not have a second moment as for instance when the acceptor function is a symmetric stable density of index 0 < {alpha} < 2.
- Research Organization:
- Florida State Univ., Tallahassee, FL (United States)
- OSTI ID:
- 5511080
- Resource Relation:
- Other Information: Thesis (Ph.D.)
- Country of Publication:
- United States
- Language:
- English
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