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In common implementations of Large Language Models (LLMs), the log-probability of the input sequence does not need to be computed. Instead, the model directly computes the conditional log-probability of the output sequence given the input. This is done by summing the log-probabilities of each individual output token. The formula is:

$$ \log \Pr(\mathbf{y}|\mathbf{x}) = \sum_{i=1}^{n} \log \Pr(y_i|\mathbf{x},\mathbf{y}_{<i}) $$

In this notation, $$[\mathbf{x},\mathbf{y}_{<i}]$$ represents the context used for predicting the token $$y_i$$. Furthermore, the expression $$\Pr(y_i|\mathbf{x},\mathbf{y}_{<i})$$ is a common literature shorthand used to denote $$\Pr(y_i|[\mathbf{x},\mathbf{y}_{<i}])$$.

Direct Computation of Output Sequence Log-Probability in LLMs

In text generation, candidate sequences are evaluated based on their log-probability. Given an input `x`, the quality of a potential output sequence `y` (composed of tokens `y1...yi`) is measured by `log Pr(y|x)`. Using log-probabilities is a standard practice because it simplifies computation; the joint probability of a sequence, which is a product of conditional probabilities, becomes a sum of log-probabilities. This conversion is more numerically stable and computationally easier to manage.

Sequence Evaluation using Log-Probability

Analyze the junior engineer's hypothesis. Is their reasoning correct for how standard generative language models compute the score for an output sequence? Explain why or why not.

Debugging a Language Model's Output Score

A developer is using a large language model to evaluate the quality of a generated sentence (the output) in response to a user's prompt (the input). The developer's colleague suggests that to get a complete score, they must calculate both the log-probability of the input and the log-probability of the output. Explain why this suggestion is incorrect for this specific task and what the model actually needs to compute.

Rationale for Log-Probability Calculation in Generative Models

The fundamental computational task of an autoregressive language model is to model the conditional probability of the next token given the preceding context, `Pr(y_i|x, y_{<i})`. A key requirement for practical implementation is that this computation must be performed in an efficient manner.

Core Computational Task in Autoregressive Generation

The process of computing the log-probability of an output sequence given an input, $$\log \Pr(\mathbf{y}|\mathbf{x})$$, in a Transformer language model involves several sequential operations. First, the input $$\mathbf{x}$$ and output $$\mathbf{y}$$ are concatenated. For each position $$i'$$, the corresponding token embedding is processed through a stack of Transformer layers, where self-attention networks update the KV cache and compute attention outputs. If the position corresponds to a generated token, the model utilizes a Softmax layer to determine the token's prediction probability. Finally, the total log-probability is calculated by summing these individual token log-probabilities over the entire generated sequence.

Step-by-Step Sequence Log-Probability Computation

An engineer is using a generative language model to decide which of two possible sentences is a more likely completion for the input prompt 'Once upon a time,'. The model can compute various log-probability scores. To select the better completion, which of the following scores should the engineer compare for each candidate sentence?

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