Model Instability Score (MIS) Metrics

• Model Instability Score (MIS) is a suite of statistical metrics that measures fluctuations in model predictions due to sampling variability, training randomness, and distribution shifts.
• It employs bootstrap resampling, KL divergence analysis, and entropy evaluations to assess prediction reliability in clinical risk, ranking, and code generation contexts.
• MIS provides actionable insights for internal validity, fairness audits, and operational safety, guiding improvements in model robustness and consistency.

The Model Instability Score (MIS) is a suite of formal statistical metrics developed to quantify the sensitivity of predictive models to sampling variability, distributional perturbations, training stochasticity, and model-building choices. MIS rigorously measures how much a model’s outputs—at the prediction, ranking, or operational level—vary under small changes in data or training procedures, with particular importance for clinical risk models, machine learning classification tasks, and code generation systems. The metric enables practitioners to assess internal validity, robustness to retraining, fairness across subgroups, and the reliability of individualized predictions or operational behaviors.

1. Formal Definitions of Model Instability Score

The precise definition of MIS varies by context. In the framework developed for clinical risk prediction models, MIS is the mean absolute deviation of predicted risks for each individual when the model is refitted on multiple bootstrap resamples of the development data (Riley et al., 2022). Given a dataset of size $N$, original model predictions $\hat{y}_i$, and $B$ bootstrap predictions $\hat{y}_i^{(b)}$ for individual $i=1,...,N$, the definitions are:

• Individual-level instability:

$$\text{MIS}_i = \frac{1}{B} \sum_{b=1}^B |\hat{y}_i^{(b)} - \hat{y}_i|$$

• Overall Model Instability Score:

$$\text{MIS} = \frac{1}{N} \sum_{i=1}^N \text{MIS}_i = \frac{1}{NB} \sum_{i=1}^N \sum_{b=1}^B |\hat{y}_i^{(b)} - \hat{y}_i|$$

Variants of MIS also exist for model parameters under distributional shifts, risk stratification, neural network classification, and program behavior (Gupta et al., 2021, Kaplan et al., 2016, Lopez-Martinez et al., 2022, Datta et al., 2023, Rajput et al., 3 Jan 2026). A unifying feature is that MIS quantifies fluctuations in model-derived quantities under re-fitting, re-training, or data perturbation.

2. Methodologies for MIS Evaluation

MIS computation protocols depend on the modeling context:

• Bootstrap-based approach (clinical models):

1. Fit the original model to data $D$ using method $M$.
2. For $b=1,...,B$, sample bootstrap dataset $D^{(b)}$, refit, and predict on original $D$ to obtain $\{\hat{y}_i^{(b)}\}_{i=1}^N$.
3. Compute $\text{MIS}_i$ and aggregate.
4. Visualize via prediction instability plots, calibration instability plots, and MAPE plots (Riley et al., 2022).

• Distributional stability (s-value):

Measure the maximum change in parameter $\theta$ over all alternative distributions $P$ within a fixed Kullback-Leibler divergence ball around the empirical distribution $P_0$ (Gupta et al., 2021):

$$\text{MIS}(\theta; \delta, P_0) = \sup_{P: D_{\mathrm{KL}}(P\|P_0) \leq \delta} |\theta(P) - \theta(P_0)|$$

• Run-to-run variability (classification, clinical ranking):

Quantify patient- or instance-level predictions’ variability across $N$ independently trained model instances. For clinical risk stratification, use Jaccard index for top-$K$ patient sets and Kendall's $\tau$ for overall ranking correlation (Lopez-Martinez et al., 2022).

• Entropy-based churn (DNNs):

Compute per-sample label entropy across $M$ reruns (multi-run) or over $T$ steps within a single run (single-run) to produce a mean or sum of label entropies as MIS (Datta et al., 2023):

$$\mathrm{LE}_m^{(i)} = -\sum_{k=1}^K p_{i,k}^{(m)} \log p_{i,k}^{(m)}$$

$$\text{MIS}_m(D) = \frac{1}{N}\sum_{i=1}^N \mathrm{LE}_m^{(i)}$$

• Program execution stability:

For LLM-generated code, define the building block as Dynamic Mean Pairwise Distance (DMPD) between Monotonic Peak Profiles (MPPs) of memory usage, aggregated across tasks for a model-level MIS (macro and micro averages) (Rajput et al., 3 Jan 2026).

3. Interpretation and Thresholds

The value of MIS provides direct information about the reliability of model predictions or outputs:

• For binary clinical risk predictions:
  • MIS < 0.01: very low (stable).
  • 0.01 ≤ MIS ≤ 0.05: moderate (caution warranted).
  • MIS > 0.05: high (instability compromises reliability) (Riley et al., 2022).
• In ranking tasks, Jaccard indices >0.85 and Kendall’s $\tau$ >0.95 across runs are heuristics for acceptable stability; lower values flag risk of inconsistent patient selection (Lopez-Martinez et al., 2022).
• In DNN classification, high average label entropy or a substantial fraction of samples with entropy exceeding 0.5 indicates pronounced instability; ensemble models or data-centric mitigation methods substantively reduce MIS (Datta et al., 2023).
• For runtime memory stability in code, typical $\mathrm{MIS_{macro}}$ ranges (e.g., <0.005) guide operational thresholds for CI/CD, but must be set according to application-specific requirements (Rajput et al., 3 Jan 2026).

4. Visualization and Decomposition Techniques

MIS is often contextualized using diagnostic plots and variance decompositions to localize sources and character of instability:

• Prediction Instability Plot:

$x$-axis: original predictions; $y$-axis: bootstrap predictions for each instance. Dense, wide vertical bands identify instances with high instability (Riley et al., 2022).

• Calibration Instability Plot:

Overlays calibration curves from each bootstrap model, visualizing how instability propagates to miscalibration in risk prediction (Riley et al., 2022).

• MAPE Instability Plot:

Plots the distribution of per-individual MIS values against original predicted probabilities, identifying risk thresholds where instability concentrates (Riley et al., 2022).

• Variance Decomposition:

For performance instability (e.g., in NLI/RC), the decomposition:

$$\mathrm{Var}(\text{Acc}) = \text{Independent variance} + \text{Covariance term}$$

localizes whether random per-instance prediction flips or correlated flipping of example clusters dominate instability (Zhou et al., 2020).

• Direction-specific s-value analysis:

Identifies covariates or subsets most threatening to parameter stability under potential dataset shifts (Gupta et al., 2021).

5. Applications and Practical Recommendations

The MIS is deployed in various domains to inform model assessment, resource allocation, and fairness audits:

• Clinical prediction and risk stratification:

MIS quantifies individual-level predictive uncertainty, guides sample size requirements, and supports critical appraisal, risk-of-bias assessment, and external validation planning (Riley et al., 2022, Lopez-Martinez et al., 2022). Subgroup MIS analyses can expose disparities across demographic or clinical subpopulations.

• Machine learning model selection and reporting:

Routine reporting of MIS (and associated decompositions or plots) is advocated for fair comparison, interpretability, and transparency, particularly for analysis datasets or stress-test sets in NLP or vision (Zhou et al., 2020).

• Code generation correctness and operational safety:

In LLM-generated software, model selection, reranking, and acceptance criteria should integrate MIS to minimize real-world operational risks that remain invisible to unit-test-based correctness metrics (Rajput et al., 3 Jan 2026).

• DNN deployment:

Label entropy-based MIS informs localized data-centric stability mitigation strategies that outperform data-agnostic regularizers and approach the benefits of large-scale ensembles at a fraction of the compute cost (Datta et al., 2023).

6. Limitations and Extensions

MIS quantifies model instability due to sample variability, stochastic training, and distributional shift; however, its sensitivity to domain-specific sources of noise, the set of perturbations considered, and computational feasibility can limit its universal application. Acceptable instability thresholds remain context-dependent. Advanced variants, including the $s$-value for distributional robustness and extensions for non-KL divergences or multivariate functional stability, are under active development (Gupta et al., 2021).

Theoretical work indicates that when model instability grows superlinearly with data dimension (i.e., the worst-case log-probability ratio in probabilistic models outpaces $O(N)$), the model becomes degenerate and places almost all mass on a vanishing fraction of configurations, which is both statistically and computationally unfavorable (Kaplan et al., 2016).

7. Summary Table of Representative MIS Formulations

Context	MIS Definition / Metric	Reference
Clinical prediction	Mean abs. deviation of predictions under bootstrapping	(Riley et al., 2022)
Distributional robustness	Max $|\theta(P) - \theta(P_0)|$ over KL ball	(Gupta et al., 2021)
Rank/selection stability	Jaccard index of top-K, avg. Kendall's $\tau$ across runs	(Lopez-Martinez et al., 2022)
DNN output stability	Avg. per-sample multi-/single-run label entropy	(Datta et al., 2023)
Code execution stability	Macro/micro-avg. DMPD of normalized memory usage traces	(Rajput et al., 3 Jan 2026)
Model probabilistic sharpness	Max log-probability ratio per sample size ($\sup_{x,x'} R_N(x,x')/N$)	(Kaplan et al., 2016)

MIS has become a fundamental diagnostic in model development, emphasizing not only average performance but also the repeatability, robustness, and trustworthiness of predictive systems in varied scientific and engineering domains.