I am following Bengio's Learning Deep Architectures for AI and at page 28 there is a phrase that confuses me:

$a(x)$ is the discriminant function or an unnormalized conditional log-probability, just like the free energy

From equation (13) in page 22 it is quite clear why it is an unnormalized log-probability:

$P(x) = \frac{e^{-FreeEnergy(x)}}{\sum_x e^{-FreeEnergy(x)}} $

But I don't understand the conditional part. Conditional on what? $P(x)$ is marginalized over the hidden variables, not conditioned. The same question goes for $a(x)$.

This part of the document doesn't seem to be that important for my purposes and I understand that $a(x)$ plays the same role as the free energy in a classification approach but it is in situations like these that I discover some conceptual mistakes I've made, therefore the question.


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