Little Bits of Artificial Intelligence

1. Real-World Setup Recap (with math symbols)

Let’s say we want to judge a cake using a neural network.

We have:

• Inputs:
  • x₁: Sweetness
  • x₂: Moistness

1. Input Layer → Hidden Layer 1 (Basic Transformation)

Let’s say Hidden Layer 1 has 2 neurons (just like 2 mini-judges).

Each neuron takes both inputs, applies weights (w), adds a bias (b), and then passes the signal through an activation function f.

Neuron 1:
z₁ = x₁·w₁₁ + x₂·w₂₁ + b₁a₁ = f(z₁)

Neuron 2:
z₂ = x₁·w₁₂ + x₂·w₂₂ + b₂a₂ = f(z₂)

• a₁, a₂ are the outputs from Hidden Layer 1
• These are like “judgment summaries” from expert judges based on combinations of sweetness and moistness.

Let’s plug in some real numbers for better understanding:

• Inputs: x₁ = 7 (Sweetness), x₂ = 6 (Moistness)
• Weights:
  • Neuron 1: w₁₁ = 0.5, w₂₁ = 0.4, b₁ = 1
  • Neuron 2: w₁₂ = -0.3, w₂₂ = 0.8, b₂ = 0.5
• Activation function: Let’s use ReLU (Rectified Linear Unit):
  f(z) = max(0, z)

Now calculate:

z₁ = 7·0.5 + 6·0.4 + 1 = 3.5 + 2.4 + 1 = 6.9
a₁ = max(0, 6.9) = 6.9

z₂ = 7·(-0.3) + 6·0.8 + 0.5 = -2.1 + 4.8 + 0.5 = 3.2
a₂ = max(0, 3.2) = 3.2

So now, Hidden Layer 1 outputs:
[a₁, a₂] = [6.9, 3.2]

2. Hidden Layer 1 → Hidden Layer 2 (Deeper Thinking)

Now let’s say Hidden Layer 2 has 1 neuron (a high-level judge who evaluates previous judgments).

z₃ = a₁·w₃₁ + a₂·w₃₂ + b₃a₃ = f(z₃)

Let’s use:

• w₃₁ = 0.6, w₃₂ = 0.9, b₃ = -1
• Inputs: a₁ = 6.9, a₂ = 3.2

Calculate:

z₃ = 6.9·0.6 + 3.2·0.9 – 1 = 4.14 + 2.88 – 1 = 6.02
a₃ = max(0, 6.02) = 6.02

3. Final Output (Cake Score)

Output Layer is a single neuron giving final score:

score = a₃*w₄ + b₄

Assume:

• w₄ = 1.2, b₄ = 0.3

score = 6.02 · 1.2 + 0.3 = 7.524

Final Cake Score = 7.524 / 10

4. So How Does “Ensemble Effect” Happen?

• Layer 1 mixes inputs differently (different weights) → basic interpretations.
• Layer 2 combines Layer 1 outputs → higher abstraction.
• Output Layer converts that into one final signal.

Each layer adds complexity, filtering, and intelligence. They ensemble their judgment step-by-step like a decision pipeline.