Softmax Probability Distribution

The softmax function is one of the most essential transformations in machine learning and deep learning. It converts a vector of arbitrary real numbers (often called logits or raw scores) into a valid probability distribution — a vector of positive numbers that sum to exactly 1.

Mathematical Definition

Given an input vector z = [z₁, z₂, ..., zₙ] containing n real numbers, the softmax function computes the output vector σ(z) where each element is defined as:

$$\sigma(z)i = \frac{e^{z_i}}{\sum{j=1}^{n} e^{z_j}}$$

In this formula:

• e^(zᵢ) is the exponential of the i-th input element
• The denominator is the sum of exponentials of all elements
• The result is a probability between 0 and 1 for each element

Key Properties

1. Output Range: Each output value lies strictly between 0 and 1: 0 < σ(z)ᵢ < 1
2. Normalization: The outputs always sum to exactly 1: Σ σ(z)ᵢ = 1
3. Preserves Ordering: Larger input values produce larger output probabilities
4. Emphasizes Differences: The exponential function amplifies differences between inputs, making the largest value dominate

Numerical Stability Consideration

When implementing softmax, a critical challenge is numerical overflow. For large input values, computing e^z directly can exceed floating-point limits. The solution is to subtract the maximum value from all inputs before exponentiation:

$$\sigma(z)i = \frac{e^{z_i - \max(z)}}{\sum{j=1}^{n} e^{z_j - \max(z)}}$$

This transformation is mathematically equivalent but prevents overflow since the largest exponent becomes e⁰ = 1.

Your Task: Write a Python function that computes the softmax probability distribution for a given list of input scores. Your implementation should handle numerical stability to prevent overflow when dealing with large input values. Return the result as a list of floating-point numbers rounded to 4 decimal places.