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I'm trying to understand the assumptions of different parameterizations in a Markov network. In this case, I'm trying to understand the assumptions (and effects) that result from parameterizing pairwise potentials with pairwise frequencies (i.e., $\psi_{(a, b)} = p(a, b)$), which the authors use in:

Mitrofanova, A. et al. Prediction of Protein Functions with Gene Ontology and Interspecies Protein Homology Data. IEEE/ACM Trans. Comput. Biol. and Bioinf., 8, 775–784.

Let's say we have a 3 node network (a)-(b)-(c), where the RV are Bernoulli. By chain rule of probability (and independence), in all cases, $p(a, b, c) = p(a) p(b|a) p(c|b)$.

First, if we define $p(a, b, c) = \psi_{a, b}(a,b) \psi_{b, c}(b,c)$, where $\psi_{a, b} = p(a, b)$ and $\psi_{b, c} = p(b, c)$, then it appears that $p(b)$ must be equal to 1 (or 0). What would happen in larger, loopy networks?

Second, if we use the same number of free parameters, constrained so each local potential sums to one, and learned the parameters (but the potentials need not be marginals), then I calculate that $p(b) = \psi_1(b) \psi_2(b)$, where $\psi_1(b) = \Sigma_a\psi_{b,c}$. As long as I don't assume $\psi_1(b) = \psi_2(b) = p(b)$, then is it a valid, useful parameterization? Are there any assumptions?

Third, let's now remove the constraint on the local potentials, and add in a global normalization constant instead. If I compare $p(a) p(b|a) p(c|b)$ to $\frac{1}{Z} p(a,b) p(c,b)$, I can see that $Z = p(b)$. That's good.

If, however, I try to calculate $Z$ by summing over all combinations of $a, b, c$, then I get that Z equals $\Sigma_b{p(b)p(b)}$. What happened?

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