Weights and bias in one neuron

One neuron computes a weighted sum plus bias. For features $\mathbf{x} = (2, 0, 1)$, weights $\mathbf{w} = (0.8,\,-0.5,\,2.0)$ and $b = -1.5$:

$$z = 0.8 \times 2 + (-0.5) \times 0 + 2.0 \times 1 - 1.5 = 2.1$$

The sign of each weight tells the direction: $w_2 = -0.5$ on link count — more links pull $z$ down in the course spam example. $w_3 = 2.0$ on unknown sender is the strongest cue.

Changing $b$ from $-1.5$ to $+1.0$ while keeping $\mathbf{w}$ adds $2.5$ to every $z$: bias shifts the threshold for all emails, even when $\mathbf{x} = \mathbf{0}$. Beginners often treat bias as a fourth weight — but $b$ is always added.

In training code, $\mathbf{w}$ and $b$ are parameters updated by gradient descent. Separating them helps you read a spam model as something inspectable, not a black box. After training, plotting $\mathbf{w}$ next to feature names shows which words the model actually learned to fear — the course’s first “interpretability” win. Lesson two walks through the same $\mathbf{w}$, $\mathbf{x}$, and $b$ numbers on the toy inbox so you can trace every multiplication on paper.