Perplexity-based Performance (PP) Measure

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A Perplexity-based Performance (PP) Measure is an intrinsic performance measure that is based on a perplexity function.

  • Context:
    • It can (typically) quantifies how well a probability model predicts a sample, often used in the context of Language Model Evaluation.
    • It can reflects the level of uncertainty a model has when predicting subsequent elements in a sequence.
    • It can (typically) measure the likelihood of a Language Model generating a given text using probabilities assigned to sequences of words, which is quantified using a Perplexity Score.
    • It can (often) be calculated as the exponential of the entropy or the average negative log-likelihood of a probabilistic model.
    • It can serve as a benchmark to compare different Statistical Models or Machine Learning Algorithms in terms of their efficiency in handling and predicting language data.
    • It can range from being very high, indicating a model with poor predictive performance, to being very low, suggesting a model that effectively predicts text sequences with high accuracy.
    • It can be influenced by the size and diversity of the dataset used to train the Statistical Model.
    • It can be an input to a Perplexity Measuring Task, and its calculation is often represented by the mathematical expression: [math]\displaystyle{ 2^{\it{Entropy}} = 2^{-\Sigma \ p \log p} }[/math].
    • ...
  • Example(s):
  • Counter-Example(s):
    • Generalized Perplexity (Derived from Rényi Entropy): The exponential of Rényi entropy leads to a generalized form of perplexity, which serves as a performance measure by indicating how well a model with the entropy characteristics defined by \( \alpha \) can predict new data.
    • Cross-Entropy and Its Exponential**: Cross-entropy measures the difference between two probability distributions. The exponential of the negative cross-entropy is a performance measure that evaluates how similar one probability distribution is to another, akin to how perplexity quantifies the surprise of a model in predictive scenarios.
    • q-Exponential (Derived from Tsallis Entropy): In statistical mechanics, the q-exponential function related to Tsallis entropy helps define ensembles that describe systems' behaviors, serving as a tool to measure how systems deviate from the expected normative behaviors based on classical statistical mechanics.
    • Exponential of Topological Entropy: This measures the "chaoticness" of a system by quantifying the exponential growth rate of distinguishable orbits, thus providing a performance measure of the system's dynamical complexity and unpredictability.
    • Exponential of Von Neumann Entropy: This can be interpreted as the effective number of quantum states that contribute to the state of the system, offering a performance measure of the quantum system's complexity and state diversity.
    • Accuracy and Precision, which are metrics used in classification tasks, not suitable for measuring uncertainty or randomness in sequence prediction.
    • Extrinsic Performance Measures such as Word Error Rate for Automatic Speech Recognition or BLEU Score for Automated Machine Translation, which focus on external validation rather than model's intrinsic capabilities.
    • ...
  • See: Entropy, Cross-Entropy, Information Theory, Predictive Modeling, Entropy Measure, Empirical Analysis.

References

2024

  • GPT-4
    • As an intrinsic performance measure, perplexity evaluates the effectiveness of a probabilistic model in language processing and other statistical applications. It reflects how well a model predicts a sample and is particularly useful in models where predictions involve likelihood estimations of sequential data, such as in language modeling.

      From this perspective, perplexity quantifies how "surprised" a model is when encountering new data; a lower perplexity indicates that the model is less surprised by the new data, implying better predictive performance. In practical terms, perplexity measures the weighted average branching factor of a language model. A lower branching factor (or lower perplexity) means that the model has a more confident prediction of the next item in the sequence.

    • For language models, the perplexity is formally defined as: [math]\displaystyle{ \text{PP}(P) = 2^{-\frac{1}{N} \sum_{i=1}^{N} \log_2 p(x_i)} }[/math] where \( N \) is the number of tokens in the test corpus, and \( p(x_i) \) is the probability the model assigns to the \( i \)-th token in the sequence. This definition showcases perplexity as a measure of prediction power, balancing between model simplicity (avoiding overfitting) and complexity (capturing the nuances in the data).

      In essence, perplexity as an intrinsic performance measure assesses a model’s ability to efficiently use the information it has learned to make accurate predictions about unseen data, which is crucial for determining the effectiveness of models in real-world tasks.

2020

2019

2018

2016

  • https://www.slideshare.net/alopezfoo/edinburgh-mt-lecture-11-neural-language-models
    • QUOTE: Given: [math]\displaystyle{ \bf{w}, \it{p}_{\text{LM}} }[/math]; [math]\displaystyle{ \text{PPL} = 2 \frac{1}{\bf{w}} \log_w \it{p}_{\text{LM}} (\bf{w}) }[/math]; [math]\displaystyle{ 0 \le \text{PPL} \le \infty }[/math]
    • Perplexity is a generation of the notion of branching factor: How many choices to I have at each position?
    • State-of-the-art English LMs have a PPL of ~100 word choices per position
    • A uniform LM has a perplexity of [math]\displaystyle{ |\Sigma| }[/math]
    • Humans do much better … and bad models can do even worse than uniform!

2017

  • https://web.stanford.edu/class/cs124/lec/languagemodeling2017.pdf
    • QUOTE: The best language model is one that best predicts an unseen test setGives the highest P(sentence).
      • Perplexity is the inverse probability of the test set, normalized by the number of words:
        [math]\displaystyle{ \text{PP}(\bf{w}) = \it{p} (w_1,w_2, ..., w_n)^{-\frac{1}{N}} = \sqrt[N]{\frac{1}{ \it{p}(w_1,w_2, ..., w_n)}} }[/math]
      • Chain rule: [math]\displaystyle{ \text{PP}(\bf{w}) = \sqrt[N]{ \Pi^{N}_{i=1} \frac{1}{ \it{p}(w_i | w_1,w_2, ..., w_n)}} }[/math]
      • For bigrams: [math]\displaystyle{ \text{PP}(\bf{w}) = \sqrt[N]{ \Pi^{N}_{i=1} \frac{1}{ \it{p}(w_i | w_{i-1})}} }[/math]
    • Minimizing perplexity is the same as maximizing probability
    • Lower perplexity = better model
      Training 38 million words, test 1.5 million words, WSJ: Unigram=162 ; Bigram=170 ; Trigram = 109.

2009

1977

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