Perplexity is a measurement used in natural language processing (NLP) to quantify how well a probability model predicts a sample. It is commonly used to evaluate language models, which are models that can predict the probability distribution of words in a language or the likelihood of a sequence of words. Perplexity can be thought of as a measure of the uncertainty of a language model, with lower perplexity indicating a better model that is more certain of its predictions. # Mathematical Definition Perplexity of a language model on a given text is defined as the inverse probability of the test set, normalised by the number of words. For a test set $W = (w_1, w_2, \ldots, w_N)$ (where $W$ is the sequence of $N$ words), the perplexity $\text{PP}(W)$ is defined as: $\text{PP}(W) = P(w_1, w_2, \ldots, w_N)^{-\frac{1}{N}} $ In practice, because models use the chain rule to decompose the joint probability into conditional probabilities, and due to numerical underflow issues with multiplying many small probabilities, we usually work with log probabilities. Therefore, perplexity is often computed as: $\text{PP}(W) = 2^{-\frac{1}{N} \sum_{i=1}^{N} \log_2 P(w_i | w_1, \ldots, w_{i-1})}$ # Interpretation - **Lower Perplexity**: Indicates that the probability distribution predicted by the model assigns high probability to the test data it's evaluating. In other words, the model is more "sure" about its predictions. Lower perplexity is generally better, as it suggests the model is more accurate in predicting the next word in a sequence. - **Higher Perplexity**: Suggests that the model is less certain about its predictions, often because the true distribution of the data is much more spread out or because the model is not well-trained or well-suited to the data. # Usage Perplexity is especially useful in comparing different language models. When developing or training language models, one of your goals might be to minimize the perplexity on a held-out test set. However, it's important to be cautious about overfitting, where a model might achieve very low perplexity on the training data but perform poorly on unseen data. # Limitations While perplexity is a useful measure, it has limitations. It assumes the model is probabilistic and can output well-calibrated probability estimates for sequences of words, which might not always be the case, especially with some neural network architectures. Additionally, perplexity alone might not fully capture a model's usefulness for tasks like translation, summarization, or generation, where factors like fluency, coherence, and alignment with human judgments are also important.