In large language model inference, the optimal output sequence $\hat{\mathbf{y}}$ is the one that maximizes the conditional log-probability given the input $\mathbf{x}$. This objective, expressed as finding the argument that maximizes $\log \text{Pr}(\mathbf{y}|\mathbf{x})$, can be decomposed using the chain rule of probability. The total log-probability of the output sequence is equivalent to the sum of the conditional log-probabilities of each individual token $y_i$. This is expressed as:

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

In this formula, $\mathbf{x}$ represents the entire input sequence and $\mathbf{y}_{<i}$ represents all previously generated output tokens. A more explicit representation of the conditional probability term is $\text{Pr}(y_i|x_0, ..., x_m, y_1, ..., y_{i-1})$, where the input sequence is $(x_0, ..., x_m)$ and the preceding output is $(y_1, ..., y_{i-1})$. This formulation is the mathematical basis for autoregressive generation.