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I understand the modular nature of directed models, and that each node captures a conditional probability. But why do we need undirected models?

As far as I can see they lack intuition in that the factors don't represent any type (conditional/marginal) of probability. Further, a final step of normalization is needed to convert the un-normalized measure into a true probability. So,

a) what could be the motivation behind introducing Markov nets at all.

b) what is a god intuition to understand factors, and their advantages as compared to the CPD's of Bayes nets.

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a) The motivation for undirected graphical models is that they can represent certain independence relationships more succinctly than directed models can. The classic example of this is a square graph on four variables. There is no directed graph on four variables that can capture the same independencies. (Although you can introduce additional variables to get those independencies, and this is true for any undirected graph.)

b) A good intuition for factors is the product of experts perspective: each factor gives a score representing how much it likes the configuration of variables, and the probability is proportional to the product of these scores. Thus if an observation only needs to satisfy a set of conditions, Markov nets (or more generally product-of-experts models) provide a succinct way to express this. The advantages of this approach are discussed in papers by Hinton and Lafferty. (Although as already mentioned, you can get the same effect by adding additional variables to a directed model. So the "advantage" is more in succinctness than any technical difference.)

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  • $\begingroup$ When you say succinctness, do you also mean in terms of inference. If we had a Bayes net with added variables would it be computationally heavier to do inference on? $\endgroup$ Sep 22 '14 at 2:19
  • $\begingroup$ Starting from an undirected model, if you constructed the equivalent directed model then the inference cost would be the same (since the models are equivalent). $\endgroup$
    – Tom Minka
    Sep 22 '14 at 18:06
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1) some useful models cannot be naturally represented as a Bayesian Network, for instance the two classes of probabilities(regularity and special symmetric ones) that don't have any P-Map. Directed models can be simplified based on undirected models(more flexibility in the model parameters).

2) A factor describes the "compatabilities" between different values of the variables in its scope.

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