Bayes statistics is a theory of fundamental importance in estimation and decision making. According to this theory, when both the prior distribution and the likelihood function of a pattern are known, the best that can be estimated from these sources of knowledge is the Bayes labeling. The maximum a posterior (MAP) solution, as a special case in the Bayes framework, is sought in many vision works.
The MAP-MRF framework is advocated by Geman and Geman (1984) and others [Geman and McClure 1985 ; Derin and Elliott 1987 ; Geman and Graffigne 1987 ; Dubes and Jain 1989 ; Besag 1989 ; Szeliski 1989 ; Geman and Gidas 1991]. Since the paper of [Geman and Geman 1984], numerous vision problems have formulated in this framework. This section reviews related concepts and derives involved probabilistic distributions and energies in MAP-MRF labeling. For more detailed materials on Bayes theory, the reader is referred to books like [Therrien 1989].