Generating Outputs with AI Models

When a language model decides what comes next, its first output is a list of numbers, one for each candidate token. A natural first reaction is to read each number as the chance that a particular token comes next. The numbers look like scores that rank the options, and a larger number seems to mean a more likely token. Part of that intuition is correct. The belief that the raw numbers already are probabilities, however, is a misconception. The raw numbers are called logits, and they are a different kind of quantity. This article traces what the model actually produces, why logits must be transformed before they can be read as likelihoods, and how the softmax transformation does that work.

For each input the model processes, it produces a logit vector. This is simply an ordered list of numbers. Every position in that list corresponds to one token in the model's vocabulary. If the vocabulary has N tokens, the logit vector has N entries, and each entry is the model's raw score for the token at that same position. The vector is therefore a complete snapshot of the model's current leaning across the entire vocabulary, taken at one moment, for one input.

Because the position of each number is tied to a specific token, the order of the list matters. Reading logit number 2 means reading the score for vocabulary token 2, not some generic value. The diagram above uses three illustrative tokens labeled A, B, and C. Position 1 holds the logit for token A, position 2 holds the logit for token B, and position 3 holds the logit for token C. Keeping this correspondence in mind makes the vector easy to interpret, because each number answers one question: how strongly does the model lean toward one particular token?

Logits share one feature with probabilities: a larger logit corresponds to a higher probability once the scores are converted. The resemblance ends there, because logits differ from probabilities in two specific ways. First, logits can be negative, while probabilities must be non-negative. Second, logits do not sum to one, while the probabilities for a set of options must total one. These two differences explain why the raw output must be transformed before it can be read as a distribution of likelihoods. The transformation is the job of a softmax layer.

To convert logits to probabilities, a softmax layer is often used. Suppose the model has a vocabulary of N tokens and the logit vector is the list of values x_1 through x_N. The probability for token i, written as $p_i$, is computed from the logit at position i using the formula below. The formula has two parts: the numerator exponentiates the logit at position i, and the denominator adds up the exponentiated values of every logit in the vector.

The exponential step is what makes the output usable as probabilities. Exponentiating turns every logit into a positive number, which removes the problem of negative values. Dividing each result by the sum of all of them forces the outputs to total one. Both properties are exactly what probabilities require.

Notice what the transformation did not do. It did not change which token ranks highest. Token A had the largest logit and still has the largest probability. What changed is the scale and the form of the numbers. The raw values were negative-capable and summed to 2.0, while the converted values are all positive and sum to one.