I have been reading the Deep Learning Book by Ian Goodfellow, and in that, there is a discussion about graphical models like Bayesian belief networks and Markov Random Fields. Here:
One key diﬀerence between directed modeling and undirected modeling is that directed models are deﬁned directly in terms of probability distributions from the start, while undirected models are deﬁned more loosely by clique functions that are then converted into probability distributions
Now, what I understand is that MRFs are undirected, represented with the help of clique functions which may not necessarily give valid probabilities which is why we need the partition function. My question then is for MRFs especially: MRFs are graphs that satisfy the Markov assumptions, which are basically a set of conditional independence assumptions, then why cannot we represent them and the graph directly in terms of probability distribution (instead of cliques)?