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PGM seems to me harder than neural nets, but the trend in the last couple of years is to include probabilities in neural nets, so they're hot.

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I used "Probabilistic Graphical Models" By Koller/Friedman

[0] https://www.amazon.com/Probabilistic-Graphical-Models-Princi...

Not an expert but PGMs are mostly a superset of NN, so it's kinda understandable.

What is the connection other than "they both have graphs somewhere in them"?

I sort of see what you mean since NNs transport information across a graph in a straightforward (non-loopy) way, and PGMs can propagate information in crazy (neverending loops) ways when doing belief propagation...

Not just graphs but also probability.

From my limited understanding, pure NNs are not able to express confidence in predictions. By adding probability to the NN, we can have both predictions and confidence scores. In practice, noise is being injected, or connections dropped out randomly, then predictions averaged over multiple runs.

Sure they can, many output probabilities: http://deeplearning.net/tutorial/mlp.html#mlp

They are not.

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