Trajectory-Level Uncertainty Quantification

• Trajectory-level uncertainty quantification is the systematic estimation of uncertainty over entire multi-step reasoning paths, surpassing token-level metrics.
• It leverages both token-level confidence and answer-level consistency to compute aggregate credibility scores that enhance model calibration and output selection.
• Empirical results, such as with the STaR system, indicate significant improvements in pass@1 accuracy and robust out-of-domain performance.

Trajectory-level uncertainty quantification denotes the systematic estimation of uncertainty over complete model-generated reasoning paths (“trajectories”), rather than individual prediction tokens or outputs. In the context of LLM–based table reasoning or structured reasoning tasks, it provides a principled framework for model calibration, output selection, and stability enhancements by leveraging the structure and internal dynamics of generated multi-step solutions. Recent work, such as STaR (Slow-Thinking for Table Reasoning), operationalizes trajectory-level uncertainty quantification as an essential pillar for interpretable and robust table question answering and general reasoning reliability (Zhang et al., 14 Nov 2025).

1. Theoretical Foundations and Motivation

Trajectory-level uncertainty quantification is motivated by two core observations. Classical token-level confidence mechanisms (e.g., log-probabilities, entropy) are insufficient to detect over-confident reasoning errors often exhibited by LLMs, particularly in multi-step tasks where local confidence may not correlate with global correctness. Moreover, complex reasoning errors tend to occur at the trajectory (solution path) level, requiring models to aggregate information over entire reasoning chains.

Human cognition in analytical tasks, such as table reasoning, involves explicit error monitoring across multiple reasoning steps and self-verification of the overall strategy, not just at a single step. Trajectory-level approaches thus seek to emulate this holistic, pathwise self-assessment, contrasting with purely local or output-only confidence estimates (Zhang et al., 14 Nov 2025).

2. Formal Methodology: Computing Trajectory-Level Scores

Systems employing trajectory-level uncertainty quantification generate $K$ independent trajectories (“rollouts”) $R = \{r_1, ..., r_K\}$ for a given input (table and question). For each trajectory $y = (t_1, ..., t_n)$, two central metrics are computed:

• Token-Level Confidence:

$$\text{logprob}(y) = \frac{1}{n} \sum_{i=1}^n \log p(t_i \mid t_{<i}, x)$$

$$\text{entropy}(y) = -\frac{1}{n} \sum_{i=1}^n \sum_{v \in V} p(v \mid t_{<i}, x) \log p(v \mid t_{<i}, x)$$

This quantifies how probable and deterministic the LLM's generative process was for $y$ as a whole.

• Answer-Level Consistency:

Extract the final answer $a_i$ from each $r_i$ and group trajectories by answer. For answer $a$, define $C(a) = |G_a|$ as the count of trajectories that produce $a$.

• Fusion into Credibility Score:

Normalize consistency and confidence scores over all answer groups, then compute the final trajectory-level score for each answer $a$:

$$S(a) = 0.25\,\hat{C}(a) + 0.20\,\overline{c}(a) + 0.55\,\hat{c}(a)$$

where $\hat{C}(a)$, $\overline{c}(a)$, $\hat{c}(a)$ represent normalized consistency count, mean confidence, and max confidence among trajectories yielding $a$, respectively (Zhang et al., 14 Nov 2025).

3. Training Pipelines Leveraging Trajectory-Level UQ

Modern LLM-based table reasoning systems such as STaR employ a hybrid training pipeline:

• Curriculum Reinforcement Learning:

A two-stage difficulty-aware reinforcement learning curriculum splits data by difficulty and employs composite rewards combining structural compliance, partial credit, and exact answer matching. Enhanced Grouped Relative Policy Optimization (GRPO) is utilized for robust gradient estimation without KL-penalty and with asymmetric clipping. The reward function

$$R(y) = 0.2\,R_\text{format}(y) + 0.3\,R_\text{partial}(y) + 0.5\,R_\text{complete}(y)$$

evaluates trajectories holistically.

• Inference-Time Trajectory Sampling:

At inference, multiple trajectories are sampled per instance using temperature-controlled decoding, and among answer groups, the highest scoring trajectory according to the trajectory-level uncertainty metric $S(a)$ is selected.

• Bridging Pass@k and Pass@1 Performance:

By converting high pass@k (i.e., correct answer exists among $k$ sampled outputs) into consistently high pass@1 through trajectory-level UQ, empirical robustness is improved (Zhang et al., 14 Nov 2025).

4. Empirical Results and Practical Impact

Implementation of trajectory-level uncertainty quantification yields substantial and measurable gains in stability and accuracy. For example, STaR achieves the following impacts:

• Stability Improvement:

On WTQ, the exact-match pass@1 improves from 76.45% to 81.73% (+5.28%) when trajectory-level UQ is employed. On TabMWP, the improvement is +6.79% (Zhang et al., 14 Nov 2025).

• Calibration:

The scoring algorithm corrects for overconfident but incorrect solution paths by penalizing low consistency or low token-confidence, thereby improving the reliability of final output selection.

• Model Generalization:

Out-of-domain transfer performance remains robust: e.g., TabMWP 97.36% and TabFact 92.05% with trajectory-level UQ, even when training is restricted to in-domain datasets (Zhang et al., 14 Nov 2025).

Dataset	w/o UQ Pass@1	w/ UQ Pass@1	Δ
WTQ	76.45	81.73	+5.28
TabMWP	68.10	74.89	+6.79

5. Distinction from Output-Level and Token-Level Approaches

Trajectory-level uncertainty quantification differs fundamentally from output-only or token-local confidence estimation. Token-level log-probabilities and entropies provide local signal but fail to aggregate cross-token dependencies or error accumulation within a reasoning chain. Output-level (scalar) uncertainty metrics disregard the internal solution path and cannot discriminate between independently erroneous or partially correct chains.

Recent agent-style prompting and explicit chain-of-thought modeling, as in STaR-SQL and Seek-and-Solve, illustrate that success in complex tasks is increasingly dependent on trajectory assessment and selection, not just on verifying final predictions (He et al., 19 Feb 2025, Jiang et al., 9 Sep 2024).

6. Relation to Other Reasoning and Verification Paradigms

Trajectory-level uncertainty quantification complements and extends self-consistency voting, majority voting, and reward model (verifier) ranking as in outcome-supervised models (He et al., 19 Feb 2025). In STaR-SQL, a learned verifier scores complete rationale+SQL chains using outcome labels; in STaR, consistency and aggregated confidence jointly select the best trajectory. These approaches collectively integrate explicit reasoning chains and multi-trajectory analysis for both interpretability and reliability.

A plausible implication is that trajectory-level UQ can unify the strengths of both explicit rationale tracing and implicit probabilistic calibration, making it a generalized solution for model stability in reasoning tasks.

7. Limitations and Ongoing Directions

Although trajectory-level UQ significantly improves reliability, current implementations depend on multi-sample inference, incurring proportional compute overhead. Scalability to large $K$ and complex tasks is an active concern. Additionally, manual prompt engineering for structured chain-of-thought remains a bottleneck for certain paradigms (Jiang et al., 9 Sep 2024). Automatic integration of verifier modules at each reasoning step, and extension to multimodal or streaming environments, remain open research directions.

In sum, trajectory-level uncertainty quantification provides a formal and effective framework for model reliability in multi-step reasoning, as confirmed by empirical analyses and ablation studies in recent LLM-driven table and text-to-SQL reasoning systems (Zhang et al., 14 Nov 2025, He et al., 19 Feb 2025, Jiang et al., 9 Sep 2024).