Bias Vector and Multi-Neuron Linear Step

We have raw scores for each question. But just as an industry slump creates a baseline bias affecting a movie's chance of becoming a hit, outside conditions can affect each question differently:

What if ticket sales are in a slump? Audiences are staying home, making a hit harder to achieve (a hurdle).
What if awards voters are skeptical of commercial scripts? The Academy starts with heavy doubt, requiring extraordinary evidence (a steep hurdle).
What if foreign markets are hungry for Hollywood movies? Audiences abroad readily buy tickets for major studio releases, giving them an automatic advantage (a head start).

A drop in domestic ticket sales doesn't change the Academy's skepticism, nor does it affect overseas demand for Hollywood releases. Because these background forces pull in completely different directions, a single shared bias cannot work.

Each neuron requires its own independent baseline bias. Adding each neuron's baseline bias to its raw score produces the final linear score ( $z = Wx + b$ ):

The Multi-Neuron Linear Step: From Input to Linear Score — The multi-neuron linear step: matrix multiplication followed by vector addition.

The Bias Vector ( $b$ )

To model these conditions, the studio assigns a baseline bias to each question:

Hit Bias: $b_{\text{Hit}} = -2.0$
Oscar Bias: $b_{\text{Oscar}} = -4.0$
International Bias: $b_{\text{Intl}} = +1.0$

Our matrix multiplication ( $Wx$ ) gave us a 3-dimensional column vector of raw scores. To add each bias to its matching score in a single step, we bundle the three biases into a matching column vector: the Bias Vector ( $b$ ). Standard vector addition then adds each bias to its respective score position-by-position:

b = \begin{bmatrix} b_{\text{Hit}} \\ b_{\text{Oscar}} \\ b_{\text{Intl}} \end{bmatrix} = \begin{bmatrix} -2.0 \\ -4.0 \\ +1.0 \end{bmatrix}

Because Position is Identity, the row order guarantees that each baseline offset automatically pairs with its corresponding question: Row 1 to Hit, Row 2 to Oscar, Row 3 to International.

The Linear Step ( $z = Wx + b$ )

Combining matrix multiplication ( $Wx$ ) with vector addition ( $+ b$ ) executes the linear step for all three neurons at once:

z = Wx + b

Writing this out in full vector notation:

\begin{bmatrix} w_{\text{Hit}} \cdot x \\ w_{\text{Oscar}} \cdot x \\ w_{\text{Intl}} \cdot x \end{bmatrix} + \begin{bmatrix} b_{\text{Hit}} \\ b_{\text{Oscar}} \\ b_{\text{Intl}} \end{bmatrix} = \begin{bmatrix} (w_{\text{Hit}} \cdot x) + b_{\text{Hit}} \\ (w_{\text{Oscar}} \cdot x) + b_{\text{Oscar}} \\ (w_{\text{Intl}} \cdot x) + b_{\text{Intl}} \end{bmatrix} = \begin{bmatrix} z_{\text{Hit}} \\ z_{\text{Oscar}} \\ z_{\text{Intl}} \end{bmatrix} = z

The result is the linear score vector ( $z$ ).

The mathematics has not changed from Topic 2. Each row still computes the exact single-neuron linear equation: $(w \cdot x) + b$ . The matrix-vector formula $z = Wx + b$ simply executes all three equations in a single unified step.

Evaluating Our Two Scripts

Now we compute the linear step for both scripts using our Weight Matrix $W$ and Bias Vector $b$ :

🎬 'Die Hard in Space': Linear Step

z = Wx + b = \begin{bmatrix} +9.5 \\ -4.5 \\ +9.0 \end{bmatrix} + \begin{bmatrix} -2.0 \\ -4.0 \\ +1.0 \end{bmatrix} = \begin{bmatrix} +7.5 \\ -8.5 \\ +10.0 \end{bmatrix}

Neuron	Raw Score (Wx)	Baseline Bias (b)	Linear Score (z = Wx + b)
Hit	+9.5	−2.0	+7.5
Oscar	−4.5	−4.0	−8.5
International	+9.0	+1.0	+10.0

🎬 'The Notebook 2': Linear Step

z = Wx + b = \begin{bmatrix} -1.5 \\ +4.5 \\ -1.0 \end{bmatrix} + \begin{bmatrix} -2.0 \\ -4.0 \\ +1.0 \end{bmatrix} = \begin{bmatrix} -3.5 \\ +0.5 \\ 0.0 \end{bmatrix}

Neuron	Raw Score (Wx)	Baseline Bias (b)	Linear Score (z = Wx + b)
Hit	−1.5	−2.0	−3.5
Oscar	+4.5	−4.0	+0.5
International	−1.0	+1.0	0.0

Comparing the Outcomes

'Die Hard in Space' stays a strong commercial hit (+7.5) and international spectacle (+10.0), but voter skepticism drives its Oscar score deep into negative territory (-8.5).
'The Notebook 2' misses the commercial bar (-3.5), but its script values overcome the Academy's hurdle to stay positive (+0.5), while foreign distribution pulls it to the neutral break-even point (0.0).

But raw scores like $+7.5$ and $-8.5$ don't directly answer our questions. In Lesson 3, we will pass this vector through an activation function to convert these scores into probabilities and classify our scripts.