High-Confidence Performance Estimations

• High-confidence performance estimation is a framework that employs statistical methods to provide explicit confidence intervals and error bounds on performance metrics under uncertainty.
• It leverages methodologies such as polynomial chaos expansion, bootstrap sampling, and Bayesian predictive intervals to efficiently propagate uncertainty.
• Its applications span software engineering, quantum computing, and reinforcement learning, ensuring robustness and safety in high-stakes decision-making.

High-confidence performance estimations refer to statistical and computational methods that provide guarantees—often in the form of coverage probabilities or quantile intervals—about the reliability or robustness of performance metrics under uncertainty. These metrics range from classical response times in software engineering to model accuracy under domain shift, complex confusion-matrix statistics in machine learning without ground truth, and rigorous bounds in high-stakes applications such as quantum computing and safety-constrained RL. High-confidence methodologies typically quantify the likelihood that a reported performance metric lies within a specified range, despite input, model, or process uncertainty, and do so with explicit error rates and computational efficiency.

1. Principles of High-Confidence Estimation

High-confidence estimation is grounded in probabilistic uncertainty quantification and concentration measures. If the underlying process contains stochastic elements—unknown parameters, random seeds, configuration options, or noisy measurement—the desired goal is to state

$\Pr(M \in [L, U]) \geq 1-\alpha$

where $M$ is a scalar or vector-valued performance metric, $[L, U]$ is a formally constructed confidence interval, and $\alpha$ is a user-chosen risk level. Applications span software system robustness (Aleti et al., 2018), empirical model monitoring (Kivimäki et al., 11 Jul 2024), Bayesian predictive intervals (Ha et al., 2022), and ratio-effect size intervals under non-determinism (Kalibera et al., 2020).

Key requirements include:

• Explicit control of confidence (coverage) levels, possibly at extreme (99.9%) thresholds.
• Correct propagation of input uncertainty (e.g., parameter ranges, unknown workloads).
• Computational efficiency: preference for methods that avoid exhaustive simulation in favor of surrogate models or analytic intervals.
• Interpretability: intervals or bounds must offer actionable guarantees (e.g., “response time ≤ 200ms with probability ≥ 0.99”).

2. Methodologies for High-Confidence Estimation

2.1 Polynomial Chaos Expansion (PCE) in Robust Performance

PCE constructs orthogonal polynomial surrogates over input uncertainties ($\boldsymbol{\theta}$) and projects the performance metric $Y(\boldsymbol{\theta})$ onto this basis: $$Y(\boldsymbol{\xi})\approx \sum_{i=0}^P a_i\,\Phi_i(\boldsymbol{\xi})$$ Coefficients $a_i$ are computed via projection or regression on sampled input vectors, requiring only $M\approx\binom{d+p}{p}$ model evaluations. Mean, variance, quantiles, and robustness probabilities are computed analytically or by sampling the PCE surrogate, yielding high-confidence bounds (e.g., $\mathbb{P}(Y \le D_{\max}) \geq \beta$) with orders-of-magnitude speedups over Monte Carlo (Aleti et al., 2018).

2.2 Confidence Interval Construction in Model and Data Evaluation

• Parametric (Student-t) Intervals: Sample mean and standard deviation from $N$ test cases yield

$$[\bar{M} - t_{1-\alpha/2,N-1} \cdot \text{SEM},~\bar{M} + t_{1-\alpha/2,N-1} \cdot \text{SEM}]$$

where $\text{SEM} = s/\sqrt{N}$ (Jurdi et al., 2023).

• Bootstrap: Nonparametric resampling to empirically obtain confidence intervals for arbitrary distributions of metrics; robust for small or skewed samples (Jurdi et al., 2023). For quantiles, exact binomial-tails or order-statistic bootstrap schemes guarantee conservative coverage even for small $n$ (Lehmann et al., 28 Jan 2025).
• Poisson-Binomial Intervals: For batch model accuracy, the binomial approximation is replaced by Poisson–binomial, leveraging the batch’s confidence scores to produce exact coverage intervals (Kivimäki et al., 11 Jul 2024, Kivimäki et al., 8 May 2025).
• Hierarchical and Fieller CIs for Ratios: For performance changes aggregated over n-way random effects, both parametric (Fieller’s theorem) and non-parametric hierarchical bootstrap quantify uncertainty in ratios, fully accounting for all sources of variability (Kalibera et al., 2020).

2.3 Bayesian and Ensemble Predictive Intervals

• Bayesian neural networks provide posterior predictive distributions over configuration settings, decomposing aleatoric and epistemic uncertainty for robust interval reporting per configuration (Ha et al., 2022).
• Credible intervals (calibrated via Platt-like rescaling and ensemble averaging) ensure the empirical coverage closely matches the nominal rates.

2.4 Semi-parametric Bootstrap in Quantum Benchmarking

• For high-confidence certification in the Quantum Volume (QV) test, a semi-parametric bootstrap simulates circuit-level and shot-level heavy-output frequencies, achieving sharp coverage control and substantial circuit economies versus binomial–Gaussian approximations (Baldwin et al., 2021).

2.5 High-confidence Safety in RLHF

• RLHF with high-confidence safety constraints employs pessimistic cost-constrained optimization (inflated empirical mean + $K(\delta)\cdot\hat{\sigma}$) followed by a held-out $t$-test to certify $\mathbb{P}(\text{cost} \leq \tau) \geq 1-\delta$ (Chittepu et al., 9 Jun 2025).

3. High-Confidence Estimation Without Ground Truth

In deployment settings where ground truth is delayed or unavailable, confidence-based estimators guarantee performance monitoring:

• Average Confidence (AC): Under perfect calibration and conditional independence,

$$\widehat{\mathrm{Acc}}_{AC} = \frac{1}{N}\sum_{i=1}^N c_i$$

is unbiased and consistent for true batch accuracy; Poisson–binomial CIs around AC yield sharp, valid interval coverage (Kivimäki et al., 11 Jul 2024, Kivimäki et al., 8 May 2025).

• CBPE (Confidence-Based Performance Estimator): Treats confusion-matrix counts as random variables under calibrated confidence, deriving full distributions and HDI intervals for metrics like precision, recall, F$_1$ (Kivimäki et al., 8 May 2025).
• Post-hoc Models and Class-Specific Calibration: Regression models (NN/XGBoost) or per-class temperature/difference/thresholded calibrations provide high-confidence F$_1$, recall, and Dice estimates under domain or class-shift (Zhang et al., 2021, Li et al., 2022).
• Relative Confidence Estimation (Tournament-based): Rank aggregation methods (Elo, Bradley-Terry) on pairwise model confidence preferences yield calibrated, discriminative scores with improved selective classification AUC (Shrivastava et al., 3 Feb 2025).

4. Efficient Search and Query Strategies for High-Confidence Mistake Detection

Black-box adversarial distance search systematically identifies high-confidence errors by exploiting the discrepancy between minimal adversarial perturbation required and historical distribution of perturbation sizes at given confidence levels. The Standardized Discovery Ratio (SDR) quantifies whether errors are discovered at a rate expected by model confidence; SDR $\gg$ 1 signals systematic overconfidence, enabling targeted error discovery within a minimal label budget (Bennette et al., 2020).

5. Theory and Asymptotics of Fixed-width and High-confidence Intervals

Two-stage nonparametric algorithms for fixed-width intervals guarantee asymptotic coverage $1-\alpha$ and first-/second-order efficiency whether confidence or precision is held fixed. Under Random CLT and consistency of variance estimators, the intervals remain valid even in “high-confidence” ($\alpha \to 0$) regimes typical of critical applications and big data (Chang et al., 2019).

6. Domain-specific Applications and Validation

• Software Performance Engineering: PCE yields rapid, accurate estimation for queuing/network models, runtime surrogates, and system-level simulations, validated empirically at $\geq 97\%$ accuracy and $>$200× speedup (Aleti et al., 2018).
• Real-time Schedulability: Probabilistic regression over sampled worst-case scenarios systematically shrinks WCET ranges with high-confidence, outperforming baseline random-search and guaranteeing near-zero deadline misses in tens of thousands of runs (Lee et al., 2023, Lee et al., 2020).
• Quantum Computing: Bootstrap-certified heavy-output probabilities enable QV definition with sharp coverage and efficient resource usage (Baldwin et al., 2021).
• High-confidence Off-policy RL: Rigorous variance and mean CI construction supports robust deployment decision-making under off-policy uncertainty (Chandak et al., 2021).

7. Practical Guidelines and Limitations

Selection of CI construction—parametric, bootstrap, Poisson–binomial, or Bayesian—must account for sample size, distributional properties (skewness, multimodality), and calibration error. Confidence-based estimators require ongoing calibration auditing (e.g., ECE/ACE), particularly under covariate or concept drift. In multi-class or multi-lingual settings, class-specific calibration and cross-lingual prompt engineering provide measurable improvements in reliability and actionable refinement triggers (Li et al., 2022, Xue et al., 21 Feb 2024).

In all cases, empirical studies demonstrate that high-confidence estimation, when properly constructed and calibrated, yields reliable actionable intervals, tight robustness metrics, and guarantees that are essential for deployment in real-world, high-risk, or mission-critical domains.