  Uncertainty quantification in Large Language Models (LLMs) is crucial for
applications where safety and reliability are important. In particular,
uncertainty can be used to improve the trustworthiness of LLMs by detecting
factually incorrect model responses, commonly called hallucinations.
Critically, one should seek to capture the model's semantic uncertainty, i.e.,
the uncertainty over the meanings of LLM outputs, rather than uncertainty over
lexical or syntactic variations that do not affect answer correctness. To
address this problem, we propose Kernel Language Entropy (KLE), a novel method
for uncertainty estimation in white- and black-box LLMs. KLE defines positive
semidefinite unit trace kernels to encode the semantic similarities of LLM
outputs and quantifies uncertainty using the von Neumann entropy. It considers
pairwise semantic dependencies between answers (or semantic clusters),
providing more fine-grained uncertainty estimates than previous methods based
on hard clustering of answers. We theoretically prove that KLE generalizes the
previous state-of-the-art method called semantic entropy and empirically
demonstrate that it improves uncertainty quantification performance across
multiple natural language generation datasets and LLM architectures.
