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In sampling-based decoding methods like top-k or top-p, after a restricted vocabulary $$\overline{V}_i$$ is selected, the probabilities of the tokens within this set are rescaled to form a new, valid probability distribution. The renormalized probability, $$\overline{\text{Pr}}$$, of a token $$y_i$$ is calculated by dividing its original conditional probability by the sum of the original probabilities of all tokens $$y_j$$ in the restricted set $$\overline{V}_i$$. This is expressed as: $$ \overline{\text{Pr}}(y_i|\mathbf{x}, \mathbf{y}_{<i}) = \frac{\text{Pr}(y_i|\mathbf{x}, \mathbf{y}_{<i})}{\sum_{y_j \in \overline{V}_i} \text{Pr}(y_j|\mathbf{x}, \mathbf{y}_{<i})} $$

Probability Renormalization Formula for Restricted Vocabulary Sampling

An engineer is building a text generation system. At a specific step, the system has calculated the following probabilities for the top 3 most likely next words: 'the' (0.5), 'a' (0.2), and 'one' (0.1). The engineer's code directly uses these probabilities to sample the next word from this restricted set. However, they notice their program sometimes fails to select any word at all for this step. Based on the principles of probability distributions for sampling, identify the fundamental error in the engineer's approach and explain why it leads to the observed failure.

Debugging a Sampling Algorithm

A language model is generating text. At one step, it considers a restricted set of three possible next tokens: 'the' (original probability 0.6), 'a' (original probability 0.2), and 'an' (original probability 0.1). The probabilities for this set are then rescaled to form a new distribution. Now, imagine a different scenario where a fourth token, 'one' (original probability 0.05), is also included in the restricted set along with the original three. How does the inclusion of the token 'one' affect the new, rescaled probability of the token 'the'? Explain your reasoning without performing the full calculation.

Impact of Vocabulary Set Size on Renormalized Probabilities

A language model predicts the probabilities for the next word in a sequence. The top four candidates are: 'happy' (0.4), 'sad' (0.2), 'angry' (0.1), and 'joyful' (0.05). A decoding method is applied that restricts the possible choices to only the top three candidates ('happy', 'sad', 'angry'). After the probabilities for this smaller set are rescaled to form a new, valid probability distribution, what is the new probability for the word 'sad'?

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