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If I have a graph $H$ with nodes $\mathbf{X} \cup \mathbf{Y}$, and a set of factors $\phi_1(D_1), \ldots, \phi_k(D_k)$, where for each $i$, $D_i \not\subset X$, then doesn't this define both a MRF and a CRF?

If we wanted to view this as a Markov Random Field, then we compute $P(\mathbf{X}, \mathbf{Y})$ by taking the product of factors, and then renormalizing.

If we wanted to view this as a Conditional Random Field, then we compute $P(\mathbf{Y} | \mathbf{X})$ by computing $P(\mathbf{X}, \mathbf{Y})$ as in the previous case. Then We compute $P(\mathbf{X})$ by marginalizing over $Y$, computing the marginal as above, and then renormalizing.

Am I missing something important here? Why does the partition function come up in the definition of a CRF? If we just view the graph as a normal MRF, we can always compute the conditional probabilities by marginalizing and then renormalizing...

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