Markov Random Fields and Conditional Random Fields

Markov chains provided us with a way to model 1D objects such as contours probabilistically, in a way that led to nice, tractable computations. We now consider 2D Markov models. These are more powerful, but not as easy to compute with. In addition we will consider two additional issues. First, we will consider adding observations to our models. These observations are conditioned on the value of the set of random variables we are modeling. (If we had considered observations with Markov chains, we would have arrived at Hidden Markov Models (HMMS), which are widely used in vision and other fields). Second, we will consider the case in which these observations potentially break our Markov assumptions, because the same observation may depend on multiple random variable, making the variables themselves dependent on each other. In such a case, we may retrieve the desired Markov properties when we condition on the observations.