I am reading Chapter 5 of PRML. Some symbols don't seem to be clear to me.
In page 243, for the chain rule for partial derivative $\dfrac{\partial E_n}{\partial w_{ji}}=\dfrac{\partial E_n}{\partial a_j}\dfrac{\partial a_j}{\partial w_{ji}}$ (equation (5.50)), a notation is defined as $\delta_j\equiv\dfrac{\partial E_n}{\partial a_j}$ (equation (5.51)). In my understanding $\delta_j$ is the first part in the chain rule. However in equation (5.54), the book mentioned
As we have seen already, for the output units, we have $$\delta_k=y_k-t_k$$
Question 1: $y_k-t_k$ is the error on output unit $k$, which is simply the difference between the $k$th output unit value and the corresponding target value. But, from the definition of the notation $\delta_k$, we should have $$\delta_k=\dfrac{\partial \frac{1}{2}(y_k-t_k)^2}{\partial a_k}=(y_k-t_k)\dfrac{\partial y_k}{\partial a_k}=(y_k-t_k)\dfrac{\partial h(a_k)}{\partial a_k}$$ where $h(a_k)$ is the activation function. So why in the book $\delta_k=y_k-t_k$??
In page 242, Section 5.3. Error Backpropagation,
Consider a simple linear model where the outputs $y_k$ are linear combinations of the input variables $x_i$ so that $y_k=\sum_iw_{ki}x_i$. For a particular input pattern $n$, the error function is $E_n=\dfrac{1}{2}\sum_k(y_{nk}-t_{nk})^2$, where $y_{nk}=y_k(\boldsymbol{x_n},\boldsymbol{w})$. So the gradient of this error function with respect to a weight $w_{ij}$ is given by $$\frac{\partial E_n}{\partial w_{ji}}=(y_{nj}-t_{nj})x_{ni}$$ which can be interpreted as a ‘local’ computation involving the product of an ‘error signal’ $y_{nj} − t_{nj}$ associated with the output end of the link $w_{ji}$ and the variable $x_{ni}$ associated with the input end of the link.
Question 2: I am not clear with the structure of this neural network. The one in the book is a two-layer neural network with linear activation, is it?
