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A neural language model uses a neural network to assign a probability distribution to language sequences. An autoregressive model factorizes the probability of words $w_1,\dots ,w_T$ asTraining by maximum likelihood estimation rewards the observed next words; generation repeatedly samples from the conditional distribution. If a two-word sequence has first-word probability $0.2$ and conditional second-word probability $0.3$, its joint probability is $0.06$.

Yoshua Bengio and coauthors' 2003 model jointly learned a distributed representation of words and a neural probability predictor. Sharing representation parameters allowed similar contexts to inform one another, rather than treating every observed word sequence as an unrelated event.^[ref]

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.

A word embedding represents a word by a vector, usually learned so that relationships in the data become useful relationships between vectors. A vocabulary of $V$ words can be stored as a matrix with $V$ rows and $d$ columns: selecting a word selects one row. Unlike a separate indicator for each word, a dense representation can share predictive information across words.

For example, training a neural language model can place words occurring in similar contexts near one another, making evidence about “cat” useful when predicting contexts involving “dog”. This is statistical similarity, not a guarantee of synonymy: opposite words can occur in similar contexts too. The 2003 neural probabilistic model learned these vectors jointly with next-word probabilities.^[ref]

Yoshua Bengio coauthored a 2003 neural language model that learns word embeddings jointly with a probability model for word sequences. The aim is to share statistical information among related contexts rather than assign unrelated parameters to every observed sequence.^[ref]

For example, a learned vector for a word can contribute to predictions in many contexts. Updating that vector through one training example can therefore affect predictions elsewhere, providing a mechanism for generalization. This is an explicit computational use of a distributed representation, not a claim that nearby vectors always encode the same linguistic relationship. Bengio also coauthored Deep Learning by Ian Goodfellow and others.