Multi-Neuron Forward Pass

We have the linear scores for the scripts answering all three questions. But as we saw earlier, raw numbers on their own don't tell us anything because they don't have a standardized scale. A studio executive cannot greenlight a script based on $+7.5$ or $-8.5$.

To make actual decisions, we need to know: what is the likelihood of a hit? What is the probability of winning an Oscar?

Just as we did for a single neuron in Topic 2, we convert these unbounded linear scores into calibrated probabilities between $0$ and $1$ using the Sigmoid activation function ($\sigma$).

Passing each neuron's linear score through this function completes the multi-neuron forward pass:

Element-Wise Activation ($a = \sigma(z)$)

Because each neuron evaluates its own question, we apply the activation function to each element individually (element-wise):

Applying the Sigmoid function to all the linear scores produces the activation vector ($a$), representing the predicted probability for each question:

Applying Sigmoid to Both Scripts

Now we pass our linear score vectors through the Sigmoid function to calculate probabilities:

• 🎬 'Die Hard in Space':

  $$a = \sigma\left(\begin{bmatrix} +7.5 \\ -8.5 \\ +10.0 \end{bmatrix}\right) \approx \begin{bmatrix} \mathbf{0.999} \\ \mathbf{0.0002} \\ \mathbf{0.9999} \end{bmatrix}$$
  • Hit ($99.9\%$): High commercial confidence.
  • Oscar ($0.02\%$): Practically zero awards potential.
  • International ($>99.9\%$): Overwhelming foreign market confidence.

• 🎬 'The Notebook 2':

  $$a = \sigma\left(\begin{bmatrix} -3.5 \\ +0.5 \\ 0.0 \end{bmatrix}\right) \approx \begin{bmatrix} \mathbf{0.029} \\ \mathbf{0.622} \\ \mathbf{0.500} \end{bmatrix}$$
  • Hit ($2.9\%$): Low commercial appeal.
  • Oscar ($62.2\%$): Solid awards potential.
  • International ($50.0\%$): Lands directly on the neutral tipping point.

Three Independent Classifications

To turn probabilities into classifications, we apply the standard 0.50 decision threshold ($P \ge 0.50 \to \text{Yes}$, $P < 0.50 \to \text{No}$).

Because each neuron has its own weights, bias, and Sigmoid output, these three classifications are completely independent:

• Independent outcomes: A script can be classified as a hit, an award winner, and an overseas success—or none of them.
• No interference: One neuron's classification never restricts or alters another neuron's classification.