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What is an n-gram language model?
An n-gram is a sequence of n consecutive tokens. An order-n language model estimates the probability of the next token from as many as n − 1 preceding tokens. It learns those conditional probabilities from counts in a training corpus. The model’s vocabulary, sentence boundaries and treatment of out-of-vocabulary (OOV) tokens are also part of its definition. Stanford’s n-gram language-model chapter covers these fundamentals.
| Model | Context used to predict the next token |
|---|---|
| Unigram | No preceding token |
| Bigram | One preceding token |
| Trigram | Two preceding tokens |
For example, a trigram model estimating the next word after “read the” uses the two-token context “read the.” Its context is fixed in length; unlike a model that conditions on an arbitrarily long history, it does not directly use every preceding token.
What does perplexity measure?
Perplexity summarizes the probabilities a model assigns to the actual next tokens in a test sequence. Let N be the number of scored tokens, and let p(wi | contexti) be the probability assigned to the actual token at position i. Average negative log probability is cross-entropy. With base-2 logarithms, it is measured in bits per token:
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H(W) = −(1/N) Σ log2 p(wi | contexti)
Perplexity is the exponentiated cross-entropy:
PP(W) = 2H(W)
Equivalently, it is the inverse geometric mean of the probabilities assigned to the observed tokens. If cross-entropy is calculated with natural logarithms instead, use e as the exponent base. Stanford’s textbook chapter and Princeton’s course text explain the measure.
How to interpret a score
Perplexity can be understood as an effective branching factor: the number of equally likely next-token choices that would yield the same average surprise. NLP course notes describe it as “the model’s effective branching factor, the size of the uniform distribution that would be equally surprised.” NLP course notes, “Perplexity: measuring a language model”.
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A lower score means the model assigned greater probability to the evaluated sequence, under that evaluation setup. It does not, by itself, prove that the model is more useful in an application or produces better text.
Why smoothing changes perplexity
A basic count-based estimate can assign probability zero to an n-gram that never appeared in training. If that n-gram occurs in test text, its zero probability makes the sequence probability zero and perplexity infinite. Smoothing prevents this failure by reserving or reallocating probability mass for unseen events.
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Common smoothing approaches
- Additive smoothing: Adjust counts so an event with zero observed count receives nonzero probability.
- Interpolation: Combine estimates from higher- and lower-order n-grams, so a longer context can draw on shorter-context evidence.
- Discounting and backoff: Reduce probability assigned to observed events and use lower-order evidence when a higher-order n-gram is missing.
These choices, along with their tuning, affect the resulting probabilities and perplexity. Assess them on held-out data rather than selecting a method by its training score; a method that fits its training counts closely is not necessarily better at predicting unseen text. The textbook chapter describes additive and lower-order methods.
How to compare perplexity scores fairly
A score is meaningful only with its evaluation setup. For a fair comparison, check that both models are tested on the same held-out text and use compatible conventions for every token they score.
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- Text and preprocessing: Use the same held-out corpus and preprocessing.
- Tokenization and unit: Align tokenization. A word-level score and a subword-level score use different units, so their raw perplexity values are not directly comparable.
- Vocabulary and OOV handling: Check how each model represents or masks words outside its vocabulary.
- Boundaries and scored tokens: Align sentence start/end markers and identify exactly which tokens count toward N.
- Scoring convention: Confirm the log base, cross-entropy calculation and per-token normalization. Report the corpus and scoring convention with any score.
These conditions matter because a change in tokenization or the number of scored tokens can change the measured unit or value without demonstrating a better underlying model. Princeton’s course text discusses perplexity, while NLTK’s language-model API documentation documents its implementation conventions.
Independent reader supportYour contribution helps us test, update, and keep practical guides available for everyone.Calculating perplexity with NLTK
NLTK documents a perplexity(text_ngrams) method and defines its result as 2 raised to the text cross-entropy. The method takes n-grams as input; check the documentation for your installed NLTK version to understand its exact input conventions and how vocabulary masking affects scoring. NLTK language-model API.
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Do not compare an NLTK result with another implementation until you have aligned the evaluated text, token sequence, boundary treatment, vocabulary behavior and scored-token count.
Quick Recap
What a perplexity result can—and cannot—tell you
- It can tell you how well a model assigns probability to a specified held-out sequence under stated scoring conventions.
- It cannot establish by itself which model is universally better, how useful a model will be in a real task, or whether its generated text will be higher quality.
- It is not a context-free benchmark: a score needs its corpus, tokenizer, vocabulary and scoring setup to be interpretable.
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