Description of random fields by means of one-point finite-conditional distributions
- 51 Downloads
The aim of this note is to investigate the relationship between strictly positive random fields on a lattice ℤ ν and the conditional probability measures at one point given the values on a finite subset of the lattice ℤ ν . We exhibit necessary and sufficient conditions for a one-point finite-conditional system to correspond to a unique strictly positive probability measure. It is noteworthy that the construction of the aforementioned probability measure is done explicitly by some simple procedure. Finally, we introduce a condition on the one-point finite conditional system that is sufficient for ensuring the mixing of the underlying random field.
KeywordsRandom field one-point conditional distribution mixing properties
MSC2010 numbers82C70 60G60
- 4.S. Yu. Dashyan, B. S. Nahapetian, “Inclusion-exclusion description of random fields”, Journal of Contemporary Mathematical Analysis (Armenian Academy of Sciences), 30(6), 50–61 (1995).Google Scholar
- 5.R. L. Dobrushin, “The problem of uniquenessof a Gibbsian random field and the problem of phase transitions”, Funktsional. Anal. i Prilojen., 2(4), 44–57 (1968).Google Scholar
- 11.N. Shental, A. Zomet, T. Hertz, Y. Weiss, “Learning and inferring image segmentations using the GBP typical cut algorithm”, in 2 (Proceedings of ICCV, October 13–16, 2003).Google Scholar