I am reading PRML by Bishop, I encounter the following variational calculus problem.
I want to get a function $y(\textbf{x})$ minimize the average loss given by following: $$ \mathbb{E}[L] = \iint \{y(\mathbf{x})-t\}^2p(\mathbf{x},t) \,\,\mathrm d x \mathrm dt $$ I know the Euler-Lagrange equation, but I don't know how the get the following condition: $$ \frac{\delta\mathbb E[L]}{\delta y(\mathbf x)} = 2\int\{y(\mathbf x)-t\}p(\mathbf x, t) \,\mathrm dt =0 $$
Consider the functional given by $$L[y] = \int f(y(\mathbf{x}),y'(\mathbf{x}); \mathbf{x}) \text{d}\mathbf{x} \text{ .}$$
Since you are familiar with the E-L equations, you know that a the variation of $L$ will be
$$ \frac{\delta L}{\delta y} = \dfrac{\partial f}{\partial y} - \dfrac{\text{d}}{\text{d}\mathbf{x}}\bigg[ \dfrac{\partial f}{\partial y'} \bigg] \text{ .}$$
Using your equation, we see that the function you are integrating is in fact
$$ f(y(\mathbf{x}),y'(\mathbf{x}); \mathbf{x}) = \int \{y(\mathbf{x})-t\}^2p(\mathbf{x},t) \text{d}t \text{ .}$$
Since clearly $$\dfrac{\partial f}{\partial y'} = 0$$
for your case, we just have that
$$ \frac{\delta L}{\delta y} = \dfrac{\partial}{\partial y}\int \{y(\mathbf{x})-t\}^2p(\mathbf{x},t) \text{d}t = 2\int\{y(\mathbf{x})-t\} p(\mathbf{x},t) \text{d}t $$
with the underlying assumption that the bounds of integration aren't dependent on $y$.
A stationary variation thus implies that
$$ \frac{\delta L}{\delta y} = 2\int\{y(\mathbf{x})-t\} p(\mathbf{x},t) \text{d}t = 0\text{ .} $$