Consider a real$-$valued function $h(x).$ You were trying to estimate this function using a regression model on the dataset $D$ consisting of

$(x,Y)$ pairs. Let the output of the training procedure be another function given by $y(x$;$D).$ This new function $y(x$;$D)$ depends on $D$ since it

was obtained by regressing on $D.$

To evaluate how well $y(x$;$D)$ generalizes we are interested in computing the expected error $E_D[(y(x$;$D)-h(x){)}^2],$ where the expectation

is over all datasets of the same size as $D,$ each obtained by i$.$i$.$d$.$ sampling from the underlying joint distribution of $x$ and $Y.$ Show that this

expected error decomposes into a bias term plus a variance term that you have seen in lecture slides named $\text{'}2$ dsp regression.pdf$\text{'}.$

Hint: Write

$(y(x$;$D)-h(x){)}^2=(y(x$;$D)-E_D[y(x$;$D)]+E_D[y(x$;$D)]-h(x){)}^2=(y(x$;$D)-E_D[y(x$;$D)]{)}^2+(E_D[y(x$;$D)]-h(x){)}^2$

$+2(y(x$;$D)-E_D[y(x$;$D)])(E_D[y(x$;$D)]-h(x))$

And take expectation over $D$ on both sides

show all work and explain steps.