Perplexity ($\mathrm{PPL}=\exp [-N^{-1}{\sum }_i\ln q_i]$)

Perplexity measures how much probability a model assigns to observed outcomes, expressed as the inverse geometric mean of those probabilities. For a sequence evaluated by a neural language model, let $q_i$ be the model's conditional probability of the actually observed token given its allowed context. ThenIt is therefore exponentiated mean negative log-likelihood. If cross-entropy is expressed in bits rather than natural-log units, the equivalent expression is $\mathrm{PPL}=2^{H_{\mathrm{bits}}}$.^[ref]

A uniform predictor over $V$ possible tokens assigns each observed token probability $1/V$, giving perplexity $V$. For an unequal two-token example with assigned probabilities $1/2$ and $1/4$,The result need not be an integer or an actual vocabulary size. It is an effective branching measure defined by the likelihood calculation.

On a fixed evaluation representation and dataset, lower perplexity means higher geometric mean probability for the realized outcomes. Assigning zero probability to any realized outcome makes the unsmoothed sequence perplexity infinite. Correctness of the most probable token alone does not determine the metric: the probabilities assigned to all realized outcomes matter.

For an empirical example, the 2003 neural-language-model paper reports test perplexity $268$ for its MLP1 row, compared with $321$ for the five-gram back-off comparison row. These are measurements under the paper's Brown-corpus setup and token-counting conventions.^[ref]

Changing tokenization, vocabulary, punctuation or sentence-boundary treatment changes the outcomes and the divisor $N$. Per-word and per-subword scores therefore cannot be compared as if they measured the same event space. Perplexity is also distinct from BLEU, which compares generated text with references rather than directly averaging the probability of observed tokens.