The log-probability of a generated sequence can be calculated incrementally at each step of the decoding process. For any sequence $\mathbf{y} = y_1...y_i$, the total log-probability $\log \Pr(\mathbf{y}|\mathbf{x})$ is evaluated as the sum of two components. The first component is the accumulated log-probability of the preceding sequence $\mathbf{y}_{<i}$, representing the sum of the log-probabilities on the path from the root to the parent node that was computed in previous steps. The second component is the conditional token prediction log-probability of the current token $y_i$, which is newly computed by the large language model at the current step. The calculation is: $\log \Pr(\mathbf{y}|\mathbf{x}) = \log \Pr(y_1...y_i|\mathbf{x}) = \underbrace{\log \Pr(\mathbf{y}_{<i}|\mathbf{x})}_{\text{accumulated up to the parent node}} + \underbrace{\log \Pr(y_i|\mathbf{x},\mathbf{y}_{<i})}_{\text{newly computed for the current node}}$.