Adaptive Lower Predictive Bound (LPB)

• Lower Predictive Bound (LPB) is a one-sided inferential guarantee ensuring outcomes exceed a specified data-driven quantile with high confidence.
• Adaptive calibration methods, including conformal prediction and importance reweighting, enable LPBs to robustly manage complex censoring and heterogeneity.
• Empirical evaluations demonstrate that adaptive LPBs produce sharper, stable predictions, outperforming traditional models in survival and risk analysis.

A Lower Predictive Bound (LPB) is a rigorous, one-sided inferential guarantee that the realized value of a random variable of interest (e.g., survival time, risk threshold, or task completion time) for a particular unit, covariate profile, or model input will exceed a specified data-driven lower quantile, with at least a pre-specified (usually high) coverage probability. In modern predictive inference, especially in survival analysis and risk assessment, LPBs are constructed to provide distribution-free, marginally or conditionally valid lower limits for outcomes under complex data structures, often in the presence of right-censoring or other forms of partial observation. Recent work demonstrates methodologies that achieve sharp, minimally conservative, and valid lower predictive bounds by leveraging conformal calibration, adaptive data subsetting, and importance reweighting in high-dimensional and heterogeneous settings.

1. Definition and Interpretation of Lower Predictive Bound (LPB)

An LPB aims to quantify the minimal value a future or unobserved instance will likely exceed, providing a statistical guarantee of the form: $$\mathbb{P}_{(X, T) \sim P}(T \geq \hat{L}(X)) \geq 1 - \alpha$$ where $X$ represents observed covariates, $T$ the outcome variable (e.g., survival time), $\hat{L}(X)$ the lower bound constructed from data, $\alpha$ a small error probability (often 0.05 or 0.1), and $P$ denotes the data-generating distribution. The probability is over both the data used to derive $\hat{L}$ (including all sources of randomness) and a new realization $(X, T)$. For censored data, the definition and construction account for incomplete observations via robust design.

Notably, LPBs differ from two-sided prediction intervals by focusing exclusively on lower limits, which are often more actionable in applications where early events (failures, unsafe behaviors, churn) are critical.

2. Adaptive Cutoff Methodology for Efficient LPBs

Conformalized survival analysis with adaptive cutoffs introduces a procedure for constructing distribution-free LPBs that directly accommodates censoring and heterogeneity in the censoring process. Building on precursors that subset calibration data based on a global fixed threshold for the censoring time (i.e., discarding samples censored "too early" to inform about high survival quantiles), this method:

• Allows the censoring threshold (the cutoff) to be covariate-dependent and data-adaptive, reflecting variability in censoring across subpopulations.
• Subsets calibration data by including only those points where the observed censoring time exceeds an individualized (fit-dependent) cutoff.
• Applies importance reweighting to correct for covariate shift induced by subsetting.

This confers several advantages:

• Sharper bounds: Only data truly uninformative for a particular quantile/covariate are excluded, reducing unnecessary conservativeness.
• Heterogeneity adaptation: The procedure better captures complex, possibly non-proportional, censoring patterns.

3. Mathematical Framework and Calibration Guarantees

Let $\hat{q}_a(X)$ denote an estimate (from, e.g., quantile regression) of the $a$-th conditional quantile of the survival time, and let $\hat{w}_a(X)$ be an estimate of the inverse probability of not being censored before $\hat{q}_a(X)$: $$\hat{w}_a(x) \approx \frac{1}{P(C \geq \hat{q}_a(x) \mid X = x)}$$ For a calibration set $\mathcal{I}_2$, the empirical miscoverage for a candidate quantile level $a$ is estimated as: $\widehat{\alpha}(a) = \frac{ \sum_{i \in \mathcal{I}_2} \hat{w}_a(X_i) \cdot \mathbbm{1}\{T_i < \hat{q}_a(X_i) \leq C_i\} }{ \sum_{i \in \mathcal{I}_2} \hat{w}_a(X_i) \cdot \mathbbm{1}\{\hat{q}_a(X_i) \leq C_i\} }$ The largest quantile level $\hat{a}$ with $\widehat{\alpha}(a') \leq \alpha$ for all $a' \leq a$ is selected, and the final LPB is set as

$\hat{L}(X) = \hat{q}_{\hat{a}}(X)$

This construction provides marginal coverage in the sense that for new data, the resulting LPB is satisfied with at least $1 - \alpha$ probability. The method accommodates a general family of monotone candidate prediction functions $\{ \hat{f}_a(X) \}_{a \in [0,1]}$.

A key theoretical property of this approach is double robustness: LPB validity is ensured provided that either the model for the censoring distribution or for the survival quantile is well specified and estimated, with error control quantified as calibration sample size increases.

4. Assumptions, Scope, and Practical Conditions

The approach presupposes:

• Type I right-censoring with all censoring (and event) times observable.
• Conditional independence of censoring: $C \perp T \mid X$.
• The ability to reasonably estimate either the conditional quantile of survival time or the conditional probability of not being censored for each covariate profile.

It is especially effective in high-dimensional, data-rich, or heterogeneous settings where parametric assumptions, such as those underpinning Cox proportional hazards or other parametric survival models, may be difficult to justify or result in invalid inference. The methodology is not limited to a particular choice of regression or survival modeling; any technique capable of providing quantile or monotonic function estimation for $T \mid X$ can be employed within the adaptive calibration framework.

5. Empirical Evaluation and Benchmarking

Empirical experiments compare the adaptive LPB methodology with a range of approaches:

• Parametric: Cox proportional hazards model.
• Machine learning: Random survival forests.
• Distribution-free: Conformal prediction using a fixed cutoff ("DFT-fixed") and a method that only provides bounds on the censored time ("DFT-baseline").

Findings indicate:

• Random forests and Cox models can undercover (coverage below the nominal level) due to model misspecification or finite sample effects.
• DFT-baseline is usually overly conservative (producing LPBs that are too low).
• DFT-fixed is less conservative but may not stabilize coverage under heterogeneous censoring.
• The adaptive method ("DFT-adaptive") achieves or slightly exceeds nominal coverage, provides higher (less conservative) and more stable LPBs, and is particularly advantageous when censoring patterns are heterogeneous across the covariate space.

Empirical results thus support that the adaptive cutoff procedure “is able to avoid under- or over-coverage; it achieves essentially the target coverage rate and returns a higher (i.e., more precise) LPB,” with benefits most pronounced under non-uniform censoring mechanisms.

6. Real-World Use Case: Survival Prediction in Mobile App Analytics

The adaptive LPB construction was applied to user activity data from a mobile app. The task was to predict the time until a user reaches their 14th active day (in the presence of censoring by the paper window). Features included user metadata (gender, age, number of children) and activity logs.

Evaluation leveraged both available event/censoring times to report upper and lower empirical bounds on coverage. The adaptive method yielded coverage near the target (e.g., 90%), consistently outperformed Cox and random survival forests (which undercovered), was less conservative and more stable than other distribution-free baselines, and produced more informative (tighter) LPBs. This demonstrates immediate utility in applications such as targeted engagement or retention interventions.

7. Implications for Predictive Inference and Future Directions

The adaptive cutoff method for LPBs constitutes a broadly applicable, robust, and efficient framework for distribution-free predictive inference under censoring. Its salient properties, such as finite-sample marginal coverage, double robustness, and efficient use of data via individualized cutoff policy, make it particularly suited for high-dimensional or heterogeneous real-world survival analysis tasks.

Implications extend to improved uncertainty quantification in modern predictive pipelines where the cost of over-conservatism (loss of informativeness) and undercoverage (invalid guarantees) are substantial. The method’s flexibility allows for integration with state-of-the-art regression and machine learning models, subject to the underlying calibration and estimation principles. Future work may address conditional coverage beyond marginal guarantees, extension to more complex censoring schemes, and investigation of the bias-variance trade-off in selection of adaptive cutoffs for optimal inference.