I am studying a linear regression example for machine learning. It makes the following definition:
As the name implies, linear regression solves a regression problem. In other words, the goal is to build a system that can take a vector $\mathbf{x} \in \mathbb{R}^n$ as input and predict the value of a scalar $y \in \mathbb{R}$ as its output. The output of linear regression is a linear function of the input. Let $\hat{y}$ be the value that our model predicts $y$ should take on. We define the output to be
$$\hat{y} = \mathbf{w}^T \mathbf{x}$$
where $\mathbf{w} \in \mathbb{R}^n$ is a vector of paramters.
Parameters are values that control the behaviour of the system. In this case, $w_i$ is the coefficient that we multiply by feature $x_i$ before summing up the contributions from all the features. We can think of $\mathbf{w}$ as a set of weights that determine how each feature affects the prediction. If a feature $x_i$ receives a positive weight $w_i$, then increasing the value of that feature increases the value of our prediction $\hat{y}$.
It then says the following:
It is worth noting that the term linear regression is often used to refer to a slightly more sophisticated model with one additional parameter -- an intercept term $b$. In this model
$$\hat{y} = \mathbf{w}^T \mathbf{x} + b,$$
so the mapping from parameters to predictions is still a linear function but the mapping from features to predictions is now an affine function. This extension to affine functions means that the plot of the model's predictions still looks like a line, but it need not pass through the origin. Instead of adding the bias parameter $b$, one can continue to use the model with only weights but augment $\mathbf{x}$ with an extra entry that is always set to $1$. The weight corresponding to the extra $1$ entry plays the role of the bias parameter.
The intercept term $b$ is often called the bias parameter of the affine transformation. This terminology derives from the point of view that the output of the transformation is biased toward being $b$ in the absence of any input. This term is different from the idea of a statistical bias, in which a statistical estimation algorithm’s expected estimate of a quantity is not equal to the true quantity.
This is the part that I am interested in:
This terminology derives from the point of view that the output of the transformation is biased toward being $b$ in the absence of any input.
Can someone please elaborate on this? How is the transformation biased towards being $b$ "in the absence of any input"?
Thank you.