Context-Adaptive Thresholding for Conditionally Representative Monitoring and Classification

1 Introduction

• For monitoring, a threshold is chosen to keep the probability of an alarm under baseline conditions below a prespecified level, controlling the false-alarm rate.The threshold is estimated from nominal data before applying the rule to a sequential stream.Source: §1 · p. 1
• When context Z affects the outcome, representative fairness requires the predicted label's conditional law given Z to match the ground-truth law, which a global threshold is generally insufficient to achieve.The paper describes this as probabilistic calibration given Z.Source: §1 · p. 2
• Under mild conditions, a measurable threshold function matches the conditional probability π(z), and a computationally cheap nonparametric sliding-window estimator is proposed.The paper establishes nonasymptotic error bounds and asymptotic distribution results for the estimator.Source: §1 · p. 2
• Functional sequential central limit theorems support uniform confidence bands, functional tests, and sequential inference such as testing for a change point.The paper also provides a central limit theorem for pointwise asymptotic confidence intervals.Source: §1 · p. 2
• Earlier Z-based thresholds targeted increased sensitivity for Z-defined classes while controlling type-I error (Steland et al. (2024)); this paper instead aligns conditional alarm rates to a reference population or sample.Steland (2026a) extended theory to unknown Ψ and studied an estimator based on the sample quantile function.Source: §1 · pp. 2–3
• 72.3% in-sample accuracy and 72.2% out-of-sample accuracy make the adaptive FICO rule competitive with state-of-the-art classifiers, including transformers.The rule thresholds the FICO score as a function of loan amount.Source: §1 · p. 3

2.1 Conditional calibrated thresholding is always possible

• Under a continuous, strictly increasing predictor CDF, a threshold function can match the conditional event probability even when the outcome is not generated by thresholding the predictor.The threshold is a degree of freedom for aligning the predictor's conditional alarm probability with the outcome's conditional probability.Source: §2.1 · pp. 4–5

2.2 Threshold estimation and decision rules

• The proposed estimator plugs a sliding-window estimate of the conditional event probability into the inverse predictor CDF to estimate the context-dependent threshold.The local estimate averages binary outcomes for observations whose context lies within a fixed window around z.Source: §2.2 · p. 5

2.3 Representative fairness by calibration

• Representative fairness requires the classifier's conditional label distribution given Z to equal the ground-truth distribution, but can transfer historical bias when Z refines a protective variable.The section notes that curated, unbiased samples can avoid this inheritance problem.Source: §2.3 · pp. 6–7
• Threshold adaptation may mitigate difficulties in low-probability context regions by copying the conditional event probability into the classifier rule.The section links sparse representation in Z to sparse regions of classifier outputs.Source: §2.3 · pp. 7–8

2.4 Relation to probabilistic calibration by histogram binning

• The method differs from histogram calibration because it smooths over context and converts estimated label frequencies into thresholds, rather than recalibrating a score as a probability.Their objectives also differ: conditional label-distribution matching versus probability calibration.Source: §2.4 · p. 8

3.1 Nonasymptotic error bounds and consistency

• The exact-moment-generating-function Okamoto and Hoeffding bounds are much tighter than the simpler b1 bound; Figure 1 compares the bounds as sample size varies.The stated comparison uses pB(z) = 0.1, π(z, h) = 0.05, and δ = 0.1.Source: Figure 1 · p. 9, §3.1 · p. 9
• Consistency for the conditional probability holds when npB(z, h) → ∞; with a bounded positive density for Z, this reduces to the usual smoothing condition nh → ∞.At an atom, the effective rate remains of order n^-1/2.Source: §3.1 · p. 10

3.2 Asymptotic normality and confidence interval

• With fixed h and n → ∞, the pointwise CLT results provide asymptotic confidence intervals for π(z, h) and c(z, h).The section fixes h because shrinking bandwidth prevents a non-degenerate weak limit for the empirical process.Source: §3.2 · p. 11

3.3 Empirical process theory

• These empirical-process results support functional inference, including uniform confidence bands.Source: §3 · p. 8, §3.3 · p. 12
• When h → 0 and nh → ∞, any weak limit of the scaled estimator process described in the impossibility result must be zero.The paper notes that this degeneracy carries over from the numerator estimator to its setting.Source: §3.3 · p. 12
• At fixed h, the conditional-probability estimator process has a tight centered Gaussian limit, with sequential weak convergence also stated under the section’s conditions.Source: §3.3 · p. 13
• The threshold estimator process likewise has a tight centered Gaussian limit, including the sequential process under the stated conditions.Source: §3.3 · p. 14

4 Local bandwidth selection with a bias allowance

• The Lepski-type procedure selects the largest candidate bandwidth compatible with all finer estimates, aiming to minimize the error bound.A candidate is accepted when its estimate differs from every finer-bandwidth estimate by no more than the combined allowances.Source: §4 · pp. 14–15
• The selection criterion combines a bias allowance Bh(z) with a noise error bound sh(z).The paper also gives Lh as a simplified bias allowance.Source: §4 · p. 15
• Theorem 4.1 gives a probability-at-least-1 − α error guarantee for the selected bandwidth under its stated assumptions.Source: §4 · p. 15

5 Applications: Uniform confidence bands, testing and change detection

• Uniform confidence bands support inference on an estimator’s functional form and are constructed using a simulated quantile of the standardized process supremum.Source: §5 · pp. 15–16
• A reference threshold function can be tested with a Kolmogorov-Smirnov-type statistic and an asymptotic critical value.Source: §5 · p. 16
• Confidence-band inversion tests a constant-threshold hypothesis by rejecting when no horizontal line lies within the band; the procedure has asymptotic size at most α and may be conservative.Source: §5 · p. 17
• The change-point procedure splits the sample at candidate times, compares estimated thresholds before and after each split, and maximizes a weighted distance over split points and z.Source: §5 · p. 17

6 Real data example and simulation experiments

• The FICO analysis uses 9,860 observations after missing-value removal and retains the dataset as-is despite noted biases.The authors state that a curated dataset should be used to ensure algorithmic fairness distinct from representative fairness; this analysis is illustrative.Source: §6 · p. 18
• 72.3% in-sample accuracy and 72.2% test accuracy were obtained with the manually selected fixed bandwidth.The test accuracy was averaged over 10 runs using a 75%/25% estimation/test split.Source: §6 · p. 18
• No constant threshold lies within the uniform confidence band, and the adapted rule requires higher FICO scores for small loans than for large loans.The band supports non-constancy at the 5% level.Source: §6 · p. 18, Figure 3 · p. 19
• Lepski bandwidth selection gives nearly the same accuracy, true positive rate (73.97%), and true negative rate (70.45%) as the manually selected bandwidth.Its local adaptation also makes the fit and confidence bands smoother in sparse tails; more irregular tail thresholds do not materially improve classification.Source: §6 · p. 18, Figure 4 · p. 20, Figure 5 · p. 20
• The adapted rule is reported on par with tested classifiers and outperforming some, including a neural network, KNN, CART, and linear SVM.The comparison also reports transformer accuracy of 71.5%, which does not improve on the adapted rule.Source: §6 · p. 21

6.1 Simulation

• The simulation reports good confidence-band coverage and classification accuracy exceeding 95% at a sample size of about 500.Coverage, accuracy, true positive rate, and true negative rate were estimated across sample sizes using 1,000 Monte Carlo runs.Source: §6.1 · pp. 21–22, Table 1 · p. 22

7.1 Nonasymptotic Bounds

• Conditioning on the number of observations in a covariate window makes the local conditional-probability estimator an empirical measure of Bernoulli variables with the window-conditional success probability.The proof then applies concentration inequalities to bound estimator error.Source: §7.1 · pp. 23–24

7.2 (Functional) central limit theorems

• Uniform asymptotic linearity, under the stated positive-window-probability conditions, supports weak convergence of the sequential probability and threshold processes to Gaussian limits.The arguments use Donsker classes, the central limit theorem, and Slutsky’s lemma.Source: §7.2 · pp. 27, 29, 30

Appendix A. Degeneracy of the weak limit

• The scaled process has a degenerate weak limit in L2(R) as h → 0 and nh → ∞, despite non-degenerate Gaussian finite-dimensional limits.Integral means may nevertheless converge to non-degenerate limits because infinite-dimensional integrals have a different convergence rate.Source: Appendix A · p. 33
• The proof establishes degeneracy by showing that inner products of the process with each weighted Hermite polynomial converge to zero in probability.The weighted Hermite polynomials span L2(R).Source: Appendix A · p. 33
• The appendix concludes the argument by bounding the variances of the resulting terms.The excerpt supplies no further bound values.Source: Appendix A · p. 34

Appendix B. Additional proofs

• Given that exactly k observations fall in E, their indices form a uniformly distributed k-element subset of the sample.Conditioning on a particular selected subset preserves the conditional distribution of the observations in that subset.Source: Appendix B · p. 34
• Conditional on the selected indices, the indicators that those observations fall in A are independent Bernoulli variables with success probability P(X ∈ A | X ∈ E).Their sum therefore has the corresponding conditional binomial probabilities.Source: Appendix B · pp. 34–35
• A Taylor expansion of (1+x)^(1/n) is applied when u(1 − q_n)/q_n < 1.The passage states this condition for the expansion used in the derivation.Source: Appendix B · p. 35

• The network ensemble uses k = 1, 2, 3.Source: Appendix C · p. 36
• The input prepends a learned classification token to t1 and t2, and the model extracts the final CLS representation.Source: Appendix C · p. 36
• Each Transformer block uses two-head self-attention with head dimension 32 and residual connections; attention-weight dropout is 0.1, and all dropout is disabled at prediction time.Source: Appendix C · p. 36