Clarification on notation used to present back propagation algorithm in 'The Deep Learning Book'

In the deep learning book (free version is available online) the backpropation algorithm is explained in section 6.5.

I have a question on equation (6.53):

$$\frac{\partial u^{(n)}}{\partial u^{(j)}} = \sum_{i:j\in Pa(u^{(i)})}\frac{\partial u^{(n)}}{\partial u^{(i)}}\frac{\partial u^{(i)}}{\partial u^{(j)}}.$$

Can someone explain the index below the summation?

In the notation section I find: $$Pa_{G}(x)$$ are the parents of $$x$$ in graph $$G$$. Problem: parents is nowhere defined. And searching the internet only gives a definition for rooted trees, that doesn't seem to match with how it is used here (it actually seems to be the exact opposite of that definition).

Furthermore I would expect that $$Pa(u^{(i)})$$ is a list, but here it seemed to be indexed with $$i:j$$ does that mean it is two dimensional.

Or should it be interpreted as: $$\forall i \;\forall j \in Pa(u^{(i)})$$?

I think I get it; should it be read as all $$i$$ such that $$j$$ is a parent of $$i$$?

• Yes: all $i$ s.t. $j$ is a parent of $i$. – gunes Mar 11 at 20:28