Log-Likelihood of a Sequence

When calculating the probability of a long sequence of words, the standard approach involves multiplying many conditional probabilities, each of which is a value between 0 and 1. This product is often converted into a sum by applying the logarithm to each term. What is the primary computational reason for this transformation?

A language model calculates the probability of a sequence of tokens, $\mathbf{x} = (x_0, x_1, \dots, x_m)$, using the product of conditional probabilities: $\text{Pr}(\mathbf{x}) = \prod_{j=0}^{m} \text{Pr}(x_j|\mathbf{x}_{<j})$. To improve numerical stability and simplify calculations, this product is converted into a sum by taking the logarithm. Which of the following expressions correctly represents the log-probability, $\log \text{Pr}(\mathbf{x})$?

Calculating Sequence Log-Probability