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Formula for Optimal Output Sequence in LLMs

In language model inference, the optimal output sequence, denoted as y^\hat{\mathbf{y}}, is found by maximizing the conditional log probability of the output sequence given the input sequence x\mathbf{x}. This objective is formally expressed by decomposing the joint probability over the nn output tokens: y^=arg max⁡ylog⁡Pr⁡(y∣x)=arg max⁡y∑i=1nlog⁡Pr⁡(yi∣x0,...,xm,y1,...,yi−1)\hat{\mathbf{y}} = \argmax_{\mathbf{y}} \log \Pr(\mathbf{y} | \mathbf{x}) = \argmax_{\mathbf{y}} \sum_{i=1}^{n} \log \Pr(y_i|x_0,...,x_m,y_1,...,y_{i-1}) In this formulation, the input sequence is represented as x0,...,xmx_0,...,x_m, and the equation models the log probability of predicting subsequent tokens starting from position m+1m+1, rather than position 0{}0.

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Updated 2026-04-19

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Gemini AI
Gemini AI
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Google
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References

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Foundations of Large Language Models

Ch.2 Generative Models - Foundations of Large Language Models

Foundations of Large Language Models Course

Computing Sciences

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