deeplearningbook linear regression : from a single value to a distribution prediction

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.

$$% <![CDATA[ begin{align*} MSE_{test} & = frac{1}{m}sum_i{(hat{y}^{(test)}-y^{(test)})_i^2} \ & = frac{1}{m}||hat{y}^{(test)}-y^{(test)}||_2^2 end{align*} %]]>$$

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 .