LLM Math Basics 2: Softmax and Probability Interpretation
A walkthrough of how softmax converts raw scores into probability-like values, with formulas, explanations, and examples.
Why Do We Need It?
In LLM papers, raw scores are rarely used as-is — softmax is applied first. It plays a central role both in Attention and in next-token prediction.
After reading this note, the following formula will feel less intimidating.
softmax(x_i) = exp(x_i) / sum_j exp(x_j)
The Core Concept and Formula
Suppose we have an array of scores.
scores = [2.0, 1.0, 0.1]
Softmax applies exp to each score and then divides by the sum of all the exponentiated values.
softmax(x_i) = exp(x_i) / sum_j exp(x_j)
The result has the following properties.
- Every value is greater than 0.
- All values sum to 1.
- Larger scores become more pronounced.
First-Pass Explanation: What the Formula Is Saying
Softmax is a function that decides how much weight to assign to each candidate.
The reason exp is used is to make the gap between large and small scores more distinct.
Because we then divide by the total sum, the result can be interpreted as a probability distribution.
In other words, softmax performs the following transformation.
list of scores -> list of proportions
A Simple Example
Imagine three people gave presentations, and a judge assigned scores.
A: 2.0 points
B: 1.0 points
C: 0.1 points
Softmax converts these scores into "prize distribution ratios."
A: largest share
B: middle share
C: small share
The key point is that A scoring 2.0 does not mean A receives exactly $2. Softmax does not use raw scores directly — it creates relative weights among the candidates.
How to Read It When You Encounter It in Attention
The Attention formula typically looks like this.
Attention(Q, K, V) = softmax(QK^T / sqrt(d_k)) V
The softmax part can be read as follows.
Convert the relevance scores into proportions that say how much to attend to each token.
QK^T is a table of relevance scores, and softmax turns that table into a table of weights.
Those weights are used to blend V, producing the final context vector.
Common Misconceptions
- The output of softmax is not always a "probability of the correct answer."
- The softmax inside Attention is closer to an attention ratio than to a correctness probability.
- Looking at a single score in isolation is not very meaningful. Softmax is about relative comparison among candidates.
Minimum Checkpoint
- Softmax converts scores into proportions that sum to 1.
- Larger scores receive larger weights.
- In Attention, it is the step that decides "which tokens to look at, and by how much."