How can I compute perplexity for candidate and system responses using a language model API?

Below is an explanation of how you might compute perplexity for a given response and how to evaluate it conditionally on a prompt. In language modeling, perplexity is essentially the exponentiation of the (average) cross-entropy loss over tokens. For a given sequence of tokens, the formula is:

  Perplexity = exp[ –(1/N) · Σ₍ᵢ₌₁₎ⁿ log p(tokenᵢ | context) ]

When you want to assess a candidate answer (like "cat") given a prompt (like "what’s an animal with four legs?"), you would typically concatenate the prompt and the candidate, then calculate the conditional probability assigned by the model for the continuation tokens. A lower perplexity indicates that the continuation is more probable under your model, and presumably a better fit.

Below is an example using Python and the Hugging Face Transformers library with a model such as GPT-2. (Note: You would need a model that supports the type of text generation and probability estimation you require.)

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Example Code for Computing Perplexity

PYTHONCopied
import torch
from transformers import GPT2Tokenizer, GPT2LMHeadModel

# Load pre-trained tokenizer and model
model_name = "gpt2"  # You can replace this with another model as needed
tokenizer = GPT2Tokenizer.from_pretrained(model_name)
model = GPT2LMHeadModel.from_pretrained(model_name)
model.eval()

def compute_perplexity(prompt: str, continuation: str) -> float:
    """
    Computes perplexity for the given continuation conditioned on the prompt.
    """
    # Concatenate prompt and continuation as the full text input
    text = prompt + continuation
    encodings = tokenizer(text, return_tensors="pt")
    input_ids = encodings.input_ids

    # Ensure we're not computing gradients
    with torch.no_grad():
        # The model computes the loss by shifting the inputs internally (LM Head)
        outputs = model(input_ids, labels=input_ids)
        loss = outputs.loss

    # Perplexity is exponentiation of the loss
    perplexity = torch.exp(loss).item()
    return perplexity

# Example usage:

prompt = "what's an animal with four legs? "

# Evaluate perplexity of a candidate response "cat"
response_candidate = "cat"
perplexity_cat = compute_perplexity(prompt, response_candidate)
print(f'Perplexity for candidate response "cat": {perplexity_cat:.2f}')

# Evaluate perplexity of the system's response "dog"
response_system = "dog"
perplexity_dog = compute_perplexity(prompt, response_system)
print(f'Perplexity for system response "dog": {perplexity_dog:.2f}')

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Key Points

• Conditional Probability: By concatenating the prompt with the candidate response, the model computes the probability of the full sequence. Implicitly, this means that the probability of the response tokens is conditioned on the prompt.

• Loss Calculation: The model's language modeling head computes the cross-entropy loss over the tokenized input. Taking the exponent of this loss gives the perplexity.

• Interpreting Perplexity: Lower perplexity indicates that the response is more probable (or is a better fit) under the model given the prompt. However, note that perplexity is a measure of probability mass and does not directly equate to semantic appropriateness.

This approach should give you a measure of how “good” a response is in terms of its likelihood under a specific language model. Adjust the model and tokenization based on your research needs and the domain-specific nuances you are investigating.