Certain but not Probable? Differentiating Certainty from Probability in LLM Token Outputs for Probabilistic Scenarios (2511.00620v1)

Abstract: Reliable uncertainty quantification (UQ) is essential for ensuring trustworthy downstream use of LLMs, especially when they are deployed in decision-support and other knowledge-intensive applications. Model certainty can be estimated from token logits, with derived probability and entropy values offering insight into performance on the prompt task. However, this approach may be inadequate for probabilistic scenarios, where the probabilities of token outputs are expected to align with the theoretical probabilities of the possible outcomes. We investigate the relationship between token certainty and alignment with theoretical probability distributions in well-defined probabilistic scenarios. Using GPT-4.1 and DeepSeek-Chat, we evaluate model responses to ten prompts involving probability (e.g., roll a six-sided die), both with and without explicit probability cues in the prompt (e.g., roll a fair six-sided die). We measure two dimensions: (1) response validity with respect to scenario constraints, and (2) alignment between token-level output probabilities and theoretical probabilities. Our results indicate that, while both models achieve perfect in-domain response accuracy across all prompt scenarios, their token-level probability and entropy values consistently diverge from the corresponding theoretical distributions.

• The paper’s main contribution is revealing that token-level probability outputs in LLMs consistently diverge from theoretical expectations in controlled probabilistic scenarios.
• It employs experiments with GPT-4.1 and DeepSeek-Chat, using classical prompts like die rolls and coin flips to assess alignment and entropy values.
• Findings imply that standard uncertainty quantification may mislead applications, urging the development of refined methods for strict statistical alignment.

Differentiating Certainty from Probability in LLM Token Outputs

LLMs have become integral in decision-support and knowledge-intensive applications where uncertainty quantification (UQ) is critical. This paper examines the alignment between model certainties—derived from token logits—and theoretical probability distributions in well-defined probabilistic scenarios, employing GPT-4.1 and DeepSeek-Chat to assess their ability to align output probabilities with theoretical expectations.

Uncertainty Quantification in Probabilistic Scenarios

UQ aims to measure confidence in model outputs through derived probabilities and entropy values. However, in probabilistic tasks, such as predicting outcomes of random events, accuracy should also reflect alignment with theoretical probability distributions. This paper evaluates two dimensions of LLM responses: validity concerning scenario constraints and alignment with theoretical outcomes.

Figure 1: Probabilistic scenario prompting and response evaluation design.

Our hypothesis is that token-level certainty should correspond with theoretically expected distributions, an assertion not typically addressed by conventional UQ methods designed for natural language generation tasks.

Methodology and Experiments

Using 10 prompts based on classical probabilistic scenarios, we assessed LLM responses regarding their probability alignment and entropy in tasks such as "rolling a die" or "flipping a coin". The models were queried in both specified and unspecified contexts to evaluate changes in output certainty.

Prompts were designed to ensure LLMs provided only the outcome, thereby isolating token probability evaluations. Responses were closely examined against expected probabilities and entropies to determine alignment.

Key Findings

Both GPT-4.1 and DeepSeek-Chat correctly understood and adhered to the probabilistic scenario constraints, achieving 100% valid response accuracy. However, their token-level probabilities consistently diverged from theoretical expectations across all scenarios. Explicit prompts specifying probabilistic behavior improved alignment somewhat, but disparities persisted, with entropy values notably higher than theoretical expectations.

Figure 2: Difference between the selected token's probability and its theoretical value. The error bars show 1 SEM.

These results indicate that reliance on standard UQ measures might be misleading for tasks requiring consistency with statistically derived probabilities.

Discussion: Implications and Future Research

This analysis raises concerns about using current LLMs in probability-driven tasks requiring precise statistical alignment. The research questions the efficacy of LLMs in environments necessitating calibrated probabilistic outputs, casting doubt on their suitability without additional refinement or calibration.

(Figures 3 and 4 provide qualitative insight into how both models articulate their understanding of probability but fail to reflect it quantitatively in their token probabilities.)

Figure 3: DeepSeek-Chat example dialogue for probabilistic reasoning about prompt scenarios.

Figure 4: GPT-4.1 example dialogue for probabilistic reasoning about prompt scenarios.

Future work should focus on developing UQ methods that factor in distributional alignment to ensure that LLM outputs are not only valid within scenario constraints but also reflect the statistical nature of the task. This could enhance model utility in applications where probabilistic reasoning is critical, like automated testing or simulation.

Conclusion

While LLMs exhibit high response certainty and understanding of scenario constraints, their outputs show significant divergence from expected statistical distributions. These discrepancies highlight a critical gap in the probabilistic calibration of LLM outputs that needs addressing for tasks demanding strict adherence to theoretical models. Addressing these gaps with refined UQ methods could improve their robustness and applicability in decision-support systems dependent on statistical accuracy.