I was reading Bishop's Pattern Recognition book, specifically, I was reading his notation for expressing the error just before back propagation. The particular equation I am a little confused about is the following:

![enter image description here](https://i.stack.imgur.com/D5RmA.png)

In that section he discussed how there is an error measure for each data point specifically. I think that is what the subscript means for $E\_n$. In that same section he clearly says what $y\_{nk}$ means:

$$ y\_{nk} = y\_{k}(\bf{x}\_n , \bf{w} )$$

With that information I tried to deduce/infer what $t\_{nk}$ meant. I believe he usually uses $t$ to mean the target value, so I think thats the supervised label we are trying to learn. However, what confuses me is why it requires two subscripts, both an $n$ and $k$. I would assume that $n$ refers to the data point we want to learn from, but $k$ is a bit unclear to me. Also, something that bugs me about this notation that seems unclear is that every output unit $y\_k$ have the same $w$, but that doesn't seem correct to me at all for a neural network. Shouldn't each output be a combination of the inputs? As in:

$$ y\_{nk} = y\_{k}(\bf{x}\_n , \bf{w}\_k )$$

where $\bf{w}\_k$ is for each output?

He seems to fix this in equation 5.45:

![enter image description here](https://i.stack.imgur.com/FmjEm.png)

so he did mean to give a subscript to each weight, right? As in $ y\_{nk} = y\_{k}(\bf{x}\_n , \bf{w} )$?

Does it mean that the data set that we are trying to learn is $x\_n \in \mathbb{R}^{d\_x}$ $t\_n, \in \mathbb{R}^{d\_t}$ and the output of the network is a vector rather than a single number? I am not sure what I am confused about but I am guessing thats it.

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Also, as a reference and add context I will add a bit more form the exact section that I am reading:

![enter image description here](https://i.stack.imgur.com/1wvOI.png)