The previous example is quite impressive and seems to be complex. In the following sections we are going to see how to deal with such complex problems and how to perform inference on them, whatever the model is. In practice, we will see that things are not as idyllic as they seem to be and there are a few restrictions. Moreover, as we saw in the first chapter, when one solves the problem of inference, one has to deal with the NP-hard problem, which leads to algorithms that have an exponential time complexity.
Nevertheless there are dynamic programming algorithms that can be used to achieve a high degree of efficiency in many problems of inference.
We recall that inference means the computing of a posterior distribution of a subset of variables, given observed values of another subset of the variables of the model. Solving this problem in general means we can choose any disjoint subsets.
Let be the set of all the variables in the graphical model and let Y and E be two disjoint...