stat.MLOct 20, 2026

Predicting silent failures of learned inference under restricted access: what measurement physics and prior information explain

Authors: Mario Gianni

Organizations: School of Computer Science and Informatics, University of Liverpool, UK

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Abstract

Learned priors are used to reconstruct regions that measurements barely reach, and to report uncertainty there. They can fail silently: the reported interval excludes the truth while every ground-truth-free diagnostic stays within its noise. We call such errors the blind set and ask whether they can be predicted before the target data are seen. For a correctly specified linear-Gaussian model averaged over the prior, the risk of a silent failure factorises exactly into non-detection and non-coverage, which does not hold for learned systems in general; we build a score that approximates this reference by linearising a VAE prior. In a prospective study on real X-ray CT objects (2DeteCT) under simulated restricted access, with predictions committed by hash before any trial, the score ranked the realised blindness of two learned inference systems (latent Laplace and ensemble Kalman inversion) with Spearman ρ=0.79ρ= 0.79 and 0.84 over 240 pre-specified errors. Simple structural predictors would also have passed these tests. Sequential post-hoc analyses, each specified before its own computation, show that under bounded reach the learned score gave no detectable improvement over a physics-based baseline, in rankings or trial-level log scores. At the weakest attenuation studied, prior information improved rankings over the physics-based baseline, most clearly for ensemble Kalman inversion, but a Gaussian prior did as well and the gain was not detectable at the trial level. The score is not a calibrated probability. Silent failures can be anticipated here, but their structure follows mainly from measurement access and the decision quantity rather than from the learned prior.

Figures & tables

Figure 1: Set-up. (a) A test slice with the hidden disc RR and the within-reach control disc CC . (b) Bounded reach: every ray through RR is removed (dark band in the sinogram). (c) Attenuated access (study A3): no ray is removed, but the noise standard deviation grows with path length, up to ≈\approx 360 σ0\sigma_{0} on the longest paths ( F=128,000F=128{,}000 ).
LevelAnalysesFixed
ConfirmatoryH1, H1b, H2 in the main studybefore any trial
Pre-specified robustnessH1–H6 in A1–A3before each study’s trials, after the main results
Post-hoc, specified before computationseven analysis plans: (1) simpler predictors, bootstraps, attenuation strengths; (2) physics comparison at all strengths; (3) approximations, calibration, adjusted rankings, sensitivity; (4) components of the score; (5) trial-level analysis, linear-Gaussian reference, location control, prior–population check, κ=1\kappa=1 , flagging; (6) detection-law dimension, leave-one-slice-out, flag budgets, physics at κ=1\kappa=1 ; (7) refitted and held-out-slice cross-validationbefore each plan’s computation, after earlier results
Exploratoryties (Kendall τb\tau_{b} ), within-region comparisons, descriptive figuresnot fixed in advance
Table 1: Hierarchy of evidence. Each level was fixed at the time shown; later levels were designed with earlier results known.
HypothesisVAE-Laplace (primary)VAE-EKI (secondary)rule
H1: ρ\rho (score, realised), 240 errors0.785 [0.701, 0.851] met0.844 [0.786, 0.887] met≥0.5\geq 0.5 , CI >0>0
H1b: ρ\rho within RR , 60 errors0.948 [0.905, 0.968] met0.947 [0.910, 0.962] met≥0.3\geq 0.3 , CI >0>0
H2: ρ\rho (VAE) −- ρ\rho (Gaussian)++ 0.334 [0.221, 0.453] met++ 0.343 [0.229, 0.473] metCI >0>0
Table 2: Main study: pre-specified decision criteria (95% bootstrap intervals resampling errors).
Figure 2: Main study: blindness risk score SblindS_{\mathrm{blind}} (computed before any test data) against the realised blind rate of VAE-Laplace, one point per pre-specified error (small vertical jitter; realised rates are multiples of 0.1). (a) Learned score; (b) Gaussian prediction, which assigns lower scores than the learned system’s observed blind rates beyond reach. Neither score is a calibrated probability.
  all errorswithin reach
StudySystemv1v2physicsGauss∣c⊤δ∣|c^{\top}\delta|regionv1 −- physics [2-way]v1physicsv1 −- physics [2-way]
MainLaplace0.790.810.800.450.650.63−- 0.02 [ −- 0.068, 0.061]0.620.66−- 0.04 [ −- 0.141, 0.195]
 EKI0.840.880.870.500.700.66−- 0.03 [ −- 0.059, 0.002]0.720.78−- 0.06 [ −- 0.126, 0.000]
A1Laplace0.600.560.590.440.590.56++ 0.01 [ −- 0.114, 0.153]0.290.27++ 0.02 [ −- 0.278, 0.410]
 EKI0.700.740.740.370.570.52−- 0.04 [ −- 0.076, 0.020]0.610.69−- 0.08 [ −- 0.180, 0.149]
A2Laplace0.750.780.770.470.590.55−- 0.02 [ −- 0.191, 0.196]0.640.69−- 0.04 [ −- 0.391, 0.451]
Table 3: Comparison with simpler predictors (post-hoc; first and second analysis plans, and the fifth for Δρ\Delta\rho under bounded reach). ρ\rho : Spearman correlation with realised blindness (raw rate in Main, excess in A1–A3). Predictors: v1, the pre-specified score; v2, full-diagnostic variant; physics, physics-based baseline; Gauss, Gaussian prediction; ∣c⊤δ∣|c^{\top}\delta| , effect size; region, indicator of the unreached region (A3: deepest ring). Brackets: two-way bootstrap (errors ×\times base slices) 95% intervals; Table 12 gives all three intervals for the differences in ρ\rho , and the deposited analysis files give them for every quantity. “Within reach” excludes errors in the unreached region. n.d.: not defined (within reach at F=512,000F=512{,}000 the EKI outcome is constant).
  AUROCcalibration (v1)log-score gain per trial
StudySystemv1physicsregionslopeinterceptv1 over structurephysics over structurev1 over physics
MainLaplace0.780.790.700.58 [0.37, 1.85]++ 1.03 [ −- 2.07, 2.84]++ 0.077 [0.001, 0.181]++ 0.077 [0.001, 0.186]−- 0.002 [ −- 0.025, 0.016]
 EKI0.810.810.730.74 [0.51, 2.63]++ 1.15 [ −- 1.61, 2.97]++ 0.061 [ −- 0.009, 0.169]++ 0.062 [ −- 0.005, 0.175]−- 0.001 [ −- 0.071, 0.010]
A1Laplace0.950.940.871.14 [0.93, 1.51]−- 0.87 [ −- 1.45, −- 0.38]++ 0.077 [0.036, 0.146]++ 0.078 [0.039, 0.146]−- 0.002 [ −- 0.022, 0.005]
 EKI0.930.930.851.12 [0.81, 1.99]−- 0.28 [ −- 1.19, 0.63]++ 0.073 [ −- 0.015, 0.168]++ 0.073 [ −- 0.021, 0.169]0.000 [ −- 0.019, 0.002]
A2Laplace0.830.840.750.72 [0.56, 1.06]++ 0.21 [ −- 1.04, 1.16]++ 0.089 [0.034, 0.405]++ 0.089 [0.033, 0.404]−- 0.001 [ −- 0.019, 0.013]
Table 4: Trial-level analysis (fifth and seventh post-hoc plans; raw blind indicator of every trial). AUROC of the learned score (v1), the physics-based score and the region indicator. Calibration slope: logistic regression of the blind indicator on logit(score); calibration intercept: the same model with logit(score) as an offset (neither model has base-slice terms, and wide intervals fail to reject calibration rather than show it). Log-score gains: mean per-trial gain in 10-fold cross-validated log score from adding logit(v1) or logit(physics) to a structural model (region indicator and rank of ∣c⊤δ∣|c^{\top}\delta| , with base-slice intercepts), and from adding v1 to the model that already contains physics. Folds hold out errors (all trials of an error, on every base slice), not base slices or objects; in each fold the intercepts and coefficients are refitted on the remaining errors, while the scores and the rank of ∣c⊤δ∣|c^{\top}\delta| are computed from the model alone. The gains therefore measure incremental information for unseen errors on the same base slices. Calibration models are fitted to all trials, not cross-validated. Brackets: two-way bootstrap 95% intervals; for log-score gains every model is refitted in every fold of every replicate (seventh plan, 5,000 replicates). Table 10 gives sensitivity to the fold partition and cross-validation on held-out base slices. ‡ n.i.: not interpretable. At F=512,000F=512{,}000 , where blindness is confined almost entirely to the deepest ring, the refitted intervals are orders of magnitude wider than elsewhere (for example [−5.45,0.01][-5.45,0.01] for EKI, v1 over structure), consistent with refitted models that separate, or nearly separate, the outcomes in some replicates and give a few held-out trials extreme log scores; the values are in the deposited results.
  partial ρ\rho (all errors)stratified ρ\rho
StudySystemv1physicsΔ\Deltav1physicsΔ\Delta
MainLaplace++ 0.58 ∗ [0.009, 0.686]++ 0.62 [ −- 0.037, 0.730]−- 0.04 [ −- 0.141, 0.135]++ 0.34 ∗ [0.018, 0.517]++ 0.34 ∗ [0.071, 0.497]0.00 [ −- 0.198, 0.213]
 EKI++ 0.66 ∗ [0.005, 0.740]++ 0.73 [0.000, 0.780]−- 0.06 [ −- 0.127, 0.015]++ 0.36 ∗ [0.095, 0.527]++ 0.39 ∗ [0.138, 0.537]−- 0.03 [ −- 0.122, 0.020]
A1Laplace++ 0.26 ∗ [0.053, 0.332]++ 0.25 [ −- 0.080, 0.382]++ 0.01 [ −- 0.229, 0.323]++ 0.30 ∗ [0.050, 0.509]++ 0.23 [ −- 0.005, 0.459]++ 0.07 [ −- 0.149, 0.281]
 EKI++ 0.51 [ −- 0.017, 0.594]++ 0.58 [ −- 0.032, 0.659]−- 0.07 [ −- 0.153, 0.060]++ 0.36 ∗ [0.127, 0.555]++ 0.34 ∗ [0.116, 0.543]++ 0.02 [ −- 0.115, 0.158]
A2Laplace++ 0.58 ∗ [0.178, 0.690]++ 0.65 [ −- 0.001, 0.819]−- 0.08 [ −- 0.355, 0.284]++ 0.18 [ −- 0.213, 0.541]++ 0.35 [ −- 0.080, 0.632]−- 0.18 [ −- 0.589, 0.332]
Table 5: Rankings after adjustment for properties of the errors (third post-hoc plan). Partial ρ\rho : all errors, adjusted for the rank of ∣c⊤δ∣|c^{\top}\delta| and the unreached-region indicator. Stratified ρ\rho : within strata of location class ×\times error kind ×\times amplitude. Δ\Delta : v1 −- physics. Brackets: two-way bootstrap 95% intervals; ∗ : interval excludes zero. The within-reach analysis with a distance spline is in Table 9 .
Figure 3: Shape of the blind set (inclusions; excess blind rate, mean and 95% bootstrap interval resampling errors). (a) Bounded reach (A1): an edge at the boundary of the unreached region. (b) Attenuated access (A3, F=128,000F=128{,}000 ): blindness concentrated in the deepest ring and graded outward. Transplants are omitted because their contrast against the base varies from slice to slice. The shapes depend on the chosen decision quantity, the error geometry and locations, and the measurement configuration.
FFSystemH1: ρ≥0.5\rho\geq 0.5H2: Δρ\Delta\rho (VAE −- Gauss)H6: ring 0 −- ring 1H6: ring 1 −- ring 3
32,00032{,}000Laplace0.587 [0.427, 0.713]−- 0.054 [ −- 0.154, 0.044]++ 0.083 [0.000, 0.171]++ 0.046 [ −- 0.004, 0.100]
 EKI0.688 [0.564, 0.786]−- 0.041 [ −- 0.130, 0.045]++ 0.117 [0.021, 0.221]++ 0.125 [0.075, 0.175]
128,000128{,}000Laplace0.466 [0.277, 0.624]−- 0.052 [ −- 0.161, 0.047]++ 0.200 [0.104, 0.304]++ 0.062 [0.025, 0.100]
 EKI0.754 [0.632, 0.847]++ 0.120 [0.015, 0.238]++ 0.331 [0.200, 0.469]++ 0.042 [0.006, 0.079]
512,000512{,}000Laplace0.578 [0.450, 0.683]−- 0.009 [ −- 0.028, 0.000]++ 0.283 [0.133, 0.438]++ 0.004 [0.000, 0.013]
 EKI0.662 [0.539, 0.750]0.000 [0.000, 0.000]++ 0.379 [0.221, 0.542]0.000 [0.000, 0.000]
Table 6: Attenuation-strength sensitivity (post-hoc; thresholds shown but not used as pass/fail). Excess blind rate; 95% bootstrap intervals resampling errors. F=128,000F=128{,}000 is the pre-specified A3 study (20 base slices; as in Table 7 ); the neighbouring strengths use 10 base slices each.
Figure 4: Reliability of the scores against the raw blind rate (third post-hoc plan), for the learned (v1), Gaussian and physics-based predictions; bins of score, observed rate with two-way bootstrap 95% intervals; dashed line: perfect calibration.
Appendix figures & tables7 assets

Supplementary material from the paper’s appendix.

Appendix

StudyHypothesisVAE-LaplaceVAE-EKI
A1H1 ρ≥0.5\rho\geq 0.50.596 [0.492, 0.684] met0.704 [0.607, 0.787] met
(20 slices)H2 Δρ>0\Delta\rho>0++ 0.154 [0.076, 0.236] met++ 0.337 [0.231, 0.449] met
 H4 within reach ρ≥0.3\rho\geq 0.30.290 [0.167, 0.410] not met0.608 [0.486, 0.713] met
 H4 within reach Δρ>0\Delta\rho>0++ 0.354 [0.197, 0.510] met++ 0.746 [0.566, 0.924] met
 H5 Δ\Delta MAE in R<0R<0−- 0.009 [ −- 0.048, 0.030] not met−- 0.121 [ −- 0.153, −- 0.089] met
A2H10.750 [0.621, 0.848] met0.833 [0.730, 0.905] met
Appendix
Table 7: Robustness study: pre-specified decision criteria (excess blind rate; 95% bootstrap intervals resampling errors).
Figure 5: H1 and H2 across the main study (Main: bounded reach, 5 base slices) and the robustness study (A1: same set-up, 20 base slices; A2: second geometry; A3: attenuated access). Filled markers: pre-specified criterion met; open: not met. Bars: 95% bootstrap intervals resampling errors, which understate uncertainty across base slices. Dashed lines: decision thresholds.
StudySystemCITL v1Δ\Delta Brier (v1 −- Gauss)partial ρ\rho : v1Gaussphysicsρ\rho : physics / ρ0\rho_{0}
MainLaplace−- 0.08−- 0.008 [ −- 0.033, 0.018]++ 0.58 [0.01, 0.69]++ 0.04++ 0.620.80 / 0.66
 EKI−- 0.09−- 0.016 [ −- 0.035, 0.004]++ 0.66 [0.00, 0.74]++ 0.07++ 0.730.87 / 0.69
A1Laplace++ 0.04−- 0.005 [ −- 0.015, 0.005]++ 0.26 [0.05, 0.33]++ 0.09++ 0.250.59 / 0.61
 EKI++ 0.02−- 0.013 [ −- 0.024, −- 0.003]++ 0.51 [ −- 0.02, 0.59]−- 0.01++ 0.580.74 / 0.55
A2Laplace−- 0.01−- 0.005 [ −- 0.019, 0.008]++ 0.58 [0.18, 0.69]++ 0.13++ 0.650.77 / 0.59
 EKI−- 0.03−- 0.013 [ −- 0.027, −- 0.002]++ 0.67 [0.12, 0.77]++ 0.09++ 0.660.81 / 0.66
Appendix
Table 8: What the predictor is and is not (third post-hoc analysis plan; all errors). CITL: calibration-in-the-large, mean score minus observed raw blind rate. Δ\Delta Brier: Brier score of the learned prediction (v1) minus that of the Gaussian prediction (negative favours v1). Partial ρ\rho : rank correlation with realised blindness after adjusting for the unreached-region indicator and effect size ∣c⊤δ∣|c^{\top}\delta| (location within the reached region is not adjusted for). ρ0\rho_{0} : physics baseline with no VAE-derived input. Brackets: two-way bootstrap 95% intervals.
StudySystemv1physicsΔ\Delta
MainLaplace++ 0.69 ∗ [0.019, 0.776]++ 0.69 [ −- 0.066, 0.798]0.00 [ −- 0.104, 0.221]
 EKI++ 0.75 ∗ [0.651, 0.814]++ 0.78 ∗ [0.718, 0.832]−- 0.04 [ −- 0.103, 0.016]
A1Laplace++ 0.34 ∗ [0.076, 0.410]++ 0.30 [ −- 0.143, 0.478]++ 0.04 [ −- 0.265, 0.435]
 EKI++ 0.64 [ −- 0.063, 0.727]++ 0.71 [ −- 0.089, 0.782]−- 0.06 [ −- 0.180, 0.177]
A2Laplace++ 0.64 ∗ [0.230, 0.753]++ 0.69 [ −- 0.106, 0.866]−- 0.05 [ −- 0.348, 0.418]
 EKI++ 0.75 ∗ [0.134, 0.828]++ 0.70 [ −- 0.224, 0.888]++ 0.05 [ −- 0.261, 0.479]
Appendix
Table 9: Partial ρ\rho within reach with a location control (fifth post-hoc plan): errors within reach, adjusted for the rank of ∣c⊤δ∣|c^{\top}\delta| and a natural cubic spline in the distance to the unreached disc (A3: to the tube centre). Δ\Delta : v1 −- physics. Brackets: two-way bootstrap 95% intervals; ∗ : interval excludes zero. n.d.: not defined. † n.i.: not interpretable (almost all within-reach outcomes are zero; the estimate lies at an interval endpoint).
  v1 beyond physics, plan-5 foldsheld-out base slices and errors (H3)
StudySystemH1 (refitted)H2 rangephysics over structurev1 over structurev1 beyond physicsM0: H3 / plan 5
MainLaplace−- 0.002 [ −- 0.025, 0.016]−- 0.005 to 0.000++ 0.175 [ −- 0.306, 3.098]++ 0.176 [ −- 0.424, 3.152]−- 0.020 [ −- 1.103, 0.025]−- 0.94 / −- 0.23
 EKI−- 0.001 [ −- 0.071, 0.010]−- 0.003 to 0.000++ 1.191 [ −- 0.320, 4.163]++ 1.197 [ −- 0.520, 3.994]−- 0.194 [ −- 2.401, 0.010]−- 2.66 / −- 0.13
A1Laplace−- 0.002 [ −- 0.022, 0.005]−- 0.002 to 0.000++ 0.055 [0.007, 0.156]++ 0.056 [0.008, 0.155]−- 0.001 [ −- 0.020, 0.003]−- 0.20 / −- 0.17
 EKI0.000 [ −- 0.019, 0.002]−- 0.001 to 0.000++ 0.297 [ −- 0.004, 0.939]++ 0.298 [ −- 0.003, 0.938]0.000 [ −- 0.092, 0.012]−- 0.67 / −- 0.12
A2Laplace−- 0.001 [ −- 0.019, 0.013]−- 0.001 to 0.000++ 0.058 [0.022, 0.278]++ 0.058 [0.021, 0.283]−- 0.003 [ −- 0.069, 0.007]−- 0.38 / −- 0.25
Appendix
Table 10: Refitted cross-validation (seventh post-hoc plan; log-score gains in nats per trial). H1: gain from adding v1 to the model with physics, plan-5 folds (by error, base-slice intercepts), refitted two-way intervals. H2: range of that gain over 101 random fold partitions. H3: errors and base slices held out together (5 error folds ×\times 5 slice groups formed by CT block; in Main each group is one slice), common intercept; gains of physics and v1 over the structural model, and of v1 beyond physics, with refitted two-way intervals. M0: mean log score of the structural model under H3 / under the plan-5 design. Held-out slices come from the same sample tubes as the training slices. In the main study, with five base slices, the H3 intervals are very wide and the estimates unstable; they should not be read as evidence about performance on new images. ‡ n.i.: not interpretable (refitted intervals at F=512,000F=512{,}000 ; see Table 4 ).
  detectionnon-coverage CITLswapsjoint vs product
StudySystemρ\rhoCITLCITL MCposteriorsamplingGG [2-way]A−Hdet⁡A{-}H_{\det}A−HcovA{-}H_{\mathrm{cov}}mean diff.ρ\rho
MainLaplace0.83−- 0.03++ 0.37−- 0.11−- 0.16++ 0.16 [ −- 0.17, 0.26]−- 0.01++ 0.35++ 0.0030.951
 EKI0.83−- 0.03++ 0.37−- 0.12−- 0.17++ 0.18 [ −- 0.13, 0.26]−- 0.02++ 0.38++ 0.0030.937
A1Laplace0.94−- 0.08++ 0.33++ 0.02−- 0.02−- 0.01 [ −- 0.14, 0.05]++ 0.01++ 0.09−- 0.0020.941
 EKI0.94−- 0.08++ 0.32++ 0.02−- 0.03++ 0.12 [ −- 0.15, 0.19]−- 0.03++ 0.280.0000.931
A2Laplace0.93−- 0.07++ 0.38−- 0.06−- 0.11++ 0.14 [ −- 0.10, 0.23]−- 0.01++ 0.31−- 0.0020.939
Appendix
Table 11: Components of the score (fourth post-hoc analysis plan; all errors; raw rates). Detection: Spearman ρ\rho of the analytic detection component with the realised alarm rate, and calibration-in-the-large (CITL, mean component value minus realised rate; average level only) of the analytic and Monte-Carlo (MC) components. Non-coverage: CITL of the analytic component (calibrated posterior spread) and of the MC component (sampling variability of the linearised estimator). Swaps (a sensitivity analysis, not a causal decomposition): G=ρ(A)−ρ(M)G=\rho(A)-\rho(M) , the gap in ranking realised blindness between the analytic and MC products, with two-way bootstrap 95% interval; ρ(A)−ρ(Hdet⁡)\rho(A)-\rho(H_{\det}) and ρ(A)−ρ(Hcov)\rho(A)-\rho(H_{\mathrm{cov}}) , the loss from swapping in the MC detection or coverage component. Joint vs product: mean over errors of the realised blind rate minus the product of realised non-coverage and non-detection rates, and Spearman ρ\rho between the two. Error-only, two-way and block intervals for every entry are in the deposited analysis files.
   v2 −- physicsv2 −- v1
StudySystemErrorsest.error-onlytwo-wayblockest.two-wayblock
MainLaplaceall0.002[ −- 0.011, 0.011][ −- 0.015, 0.019][ −- 0.015, 0.020]0.020[ −- 0.051, 0.064][ −- 0.052, 0.064]
 Laplacewithin reach0.000[ −- 0.029, 0.020][ −- 0.036, 0.059][ −- 0.035, 0.060]0.044[ −- 0.173, 0.134][ −- 0.171, 0.134]
 EKIall0.004[ −- 0.001, 0.009][0.000, 0.009][0.000, 0.009]0.031[0.001, 0.063][0.001, 0.063]
 EKIwithin reach0.009[ −- 0.002, 0.021][ −- 0.001, 0.019][ −- 0.001, 0.020]0.069[0.000, 0.134][0.000, 0.132]
A1Laplaceall−- 0.028[ −- 0.057, −- 0.001][ −- 0.062, 0.020][ −- 0.064, 0.019]−- 0.036[ −- 0.144, 0.067][ −- 0.146, 0.074]
Appendix
Table 12: Post-hoc differences in ρ\rho with all three 95% intervals (error-only; two-way, errors ×\times base slices; block, errors ×\times 10-slice CT blocks). 5,000 replicates. Studies: Main, main study (bounded reach, 5 base slices); A1, same set-up with 20 base slices; A2, second geometry; A3, attenuated access at strength FF . Predictors: v1, pre-specified; v2, full-diagnostic. Rows for F=32,000F=32{,}000 and 512,000512{,}000 are from the second post-hoc analysis plan; within reach at F=512,000F=512{,}000 the EKI outcome is constant, so its ρ\rho is not defined (n.d.).
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