Do LLMs Know What They Know? Measuring Metacognitive Efficiency with Signal Detection Theory
Authors: Jon-Paul Cacioli
Organizations: Independent Researcher Melbourne, Australia
Abstract
Standard evaluation of LLM confidence relies on calibration metrics (ECE, Brier score) that conflate how much a model knows (Type-1 accuracy) with how well its confidence signal tracks that knowledge (Type-2 metacognitive sensitivity). We apply Signal Detection Theory (SDT) to decompose these capacities, treating token-level normalised log-probability as a graded confidence variable and answer correctness as the state to be discriminated. We characterise the Type-2 ROC of this signal, including its unequal-variance structure via z-ROC analysis, and -- because the meta-d' efficiency ratio is not well defined for open-ended QA, which lacks a two-alternative Type-1 decision -- quantify metacognitive efficiency with a model-free information measure, normalised metacognitive information (meta-I_2r). Applied to four LLMs (Llama-3-8B-Instruct, Mistral-7B-Instruct-v0.3, Llama-3-8B-Base, Gemma-2-9B-Instruct) across 224,000 factual QA trials, we find: (1) metacognitive information varies more than two-fold across models and co-varies inversely with accuracy -- the least accurate model has the most informative confidence -- though with four models this ordering cannot be separated from an error-difficulty confound, so we report it as coupling, not decoupling; (2) the confidence signal has model-specific unequal-variance structure (z-ROC slopes 0.81 to 1.18) invisible to calibration metrics; (3) metacognitive information is domain-specific, strongest in Arts & Literature for every model; (4) temperature dissociates Type-1 accuracy from metacognitive information, which stays stable while accuracy shifts. All estimates carry permutation nulls and bootstrap confidence intervals. Pre-registered; code and data public.
Confidence-weighted routing, selective abstention, and ensemble weighting all assume that a model's stated confidence is informative about its capability on the question being asked. They presume functional metacognition, the capacity to assess one's own capabilities, without exercising them. Aggregate calibration is well studied, with mixed results, but the underlying structure of elicited confidence is less well understood. We decompose binary confidence judgements from 20 frontier Large Language Models (LLMs) across six benchmarks using tetrachoric factor analysis paired with pairwise calibration, asking whether two models that differ in confidence also differ in performance. On factual recall and information retrieval benchmarks the cross-model confidence matrix is approximately rank-one and a single dominant factor captures most of the latent variance. Models retrieving facts share an item-level difficulty axis and differ mainly in their decision thresholds along it. Across all benchmarks the relationship between confidence and performance collapses once items that all models agree on are removed. Inter-model pairwise calibration is small even where statistically significant, and what remains shrinks to nothing once base-rate differences along the shared factor are controlled for. Mathematical reasoning is the apparent exception, but this turns out to be a confound where reasoning models answer questions about their confidence by trying to solve them in their chain of thought, bypassing the sub-symbolic self-knowledge we seek to measure. We find no evidence for significant verbalised individuated metacognition in any tested domain.
The possibility of LLM self-awareness and even sentience is gaining increasing public attention and has major safety and policy implications, but the science of measuring them is still in a nascent state. Here we introduce a novel methodology for quantitatively evaluating metacognitive abilities in LLMs. Taking inspiration from research on metacognition in nonhuman animals, our approach eschews model self-reports and instead tests to what degree models can strategically deploy knowledge of internal states. Using two experimental paradigms, we demonstrate that frontier LLMs introduced since early 2024 show increasingly strong evidence of certain metacognitive abilities, specifically the ability to assess and utilize their own confidence in their ability to answer factual and reasoning questions correctly and the ability to anticipate what answers they would give and utilize that information appropriately. We buttress these behavioral findings with an analysis of the token probabilities returned by the models, which suggests the presence of an upstream internal signal that could provide the basis for metacognition. We further find that these abilities 1) are limited in resolution, 2) emerge in context-dependent manners, and 3) seem to be qualitatively different from those of humans. We also report intriguing differences across models of similar capabilities, suggesting that LLM post-training may have a role in developing metacognitive abilities.
Verbalized confidence, in which LLMs report a numerical certainty score, is widely used to estimate uncertainty in black-box settings, yet the confidence scale itself (typically 0--100) is rarely examined. We show that this design choice is not neutral. Across six LLMs and three datasets, verbalized confidence is heavily discretized, with more than 78% of responses concentrating on just three round-number values. To investigate this phenomenon, we systematically manipulate confidence scales along three dimensions: granularity, boundary placement, and range regularity, and evaluate metacognitive sensitivity using meta-d′. We find that a 0--20 scale consistently improves metacognitive efficiency over the standard 0--100 format, while boundary compression degrades performance and round-number preferences persist even under irregular ranges. These results demonstrate that confidence scale design directly affects the quality of verbalized uncertainty and should be treated as a first-class experimental variable in LLM evaluation.