Linear Regression :
Linear Regression : to predict $y$ from $x$ by outputting a point $\hat{y}$.
a value prediction
Simply , the task of Linear Regression is to output a value given $x$ , so we take the MSE(mean square error) as cost funtion.
\[\begin{align*} MSE_{test} & = \frac{1}{m}\sum_i{(\hat{y}^{(test)}-y^{(test)})_i^2} \\ & = \frac{1}{m}||\hat{y}^{(test)}-y^{(test)}||_2^2 \end{align*}\]In this example , the prediction of the model is a point , for example , $\hat{y}$ is $3.2$ .
a distribution prediction
Once have learnt MLE , we can get a distribution prediction of every $\hat{y}$ value . For example ,we can know the propability of value $3.2$ is $0.901$ and $3.1$ is $0.09$ ,so on . It is not only a value ,but a distribution of $\hat{y}$ .(P133 Instead of producing a single prediction $\hat{y}$ ,we now think of the model as producing a confitional distribution $p(y|x)$). If we want to output a value only , we can select the value $\hat{y}$ of largest probability .