I am trying to show that the Bayes predictor for linear regression with square loss is:
$$h^{\star}(x) = \mathbb{E}[Y|X = x]$$
I found the following slide from here, but don't understand which properties were used to derive the conclusion:
My question is: which properties were to reach the Bayes predictor $h^{\star}(x)$?
For the squared error loss $l(\theta, a)=(\theta-a)^2$, the Bayes estimate of $\theta$ after $X=x$ is observed is given by the value of $a$ which minimises the expected squared error loss $\mathbb{E}[(\theta-a)^2\mid X=x]$. The unique minimiser of this loss is the posterior mean of $\theta$ after observing $X=x$; that is, $\mathbb{E}[\theta\mid X=x]$.
This can be seen from the fact that the value of $d$ which minimises $\mathbb{E}[(X-d)^2]$ is the mean of $X$, since
$$ \mathbb{E}[(X-d)^2] = \mathbb{E}[X^2]-2d\mathbb{E}[X]+d^2 $$
Differentiating and setting the derivative to $0$ gives $d=\mathbb{E}[X]$.
