Practical Boundary Degeneracy and Reverse-Martingale Limits in Sequential Binary Models

Technical Summary: Practical Boundary Degeneracy and Reverse-Martingale Limits in Sequential Binary Models

Problem Statement
The paper addresses a recurring statistical phenomenon in sequential binary data where finite samples create the appearance of "boundary degeneracy." This occurs in three distinct settings:

1. Bernoulli Trials: A long run of all failures (or all successes) leads an analyst to estimate a probability of exactly zero (or one).
2. Logistic Regression: Complete or quasi-complete separation causes fitted coefficients to diverge, driving predicted probabilities to the boundaries (0 or 1).
3. Sequential Risk Trajectories: A risk score approaches a boundary before the underlying state has stabilized.

In all three cases, there is a temptation to declare exact boundary probabilities ($p=0$ or $p=1$) based on finite data. The paper argues that while finite sequential data may justify practical boundary statements (e.g., $p \le \varepsilon$), they do not provide evidence for exact boundary probabilities. The core confusion lies in conflating a finite near-boundary estimate with the limiting conditional law to which the process converges.

Methodology and Framework
The paper unifies these three settings under a reverse-martingale framework, utilizing the classical reverse-martingale convergence theorem (Doob, 1953; Durrett, 2019).

• Reverse-Martingale Formulation: Instead of the standard forward filtration ($\mathcal{F}_1 \subseteq \mathcal{F}_2 \subseteq \dots$), the paper considers a decreasing filtration ($\mathcal{G}_1 \supseteq \mathcal{G}_2 \supseteq \dots$). For a binary outcome $Y \in \{0, 1\}$, the process $M_n = E(Y | \mathcal{G}_n)$ is a reverse martingale that converges almost surely to a limit $M_\infty = E(Y | \mathcal{G}_\infty)$.
• Definition of Degeneracy: Boundary degeneracy is redefined as a property of the limiting law $M_\infty$, not a finite-sample event. A process is practically boundary-degenerate if $M_\infty \le \varepsilon$ or $M_\infty \ge 1-\varepsilon$.
• The Three-Condition Stopping Rule ($\tau_{RM}$): To distinguish transient apparent certainty from genuine limiting degeneracy, the paper proposes a stopping rule requiring three simultaneous conditions:
  1. Boundary Closeness ($B_n \le \varepsilon$): The current estimate is near the boundary.
  2. Uncertainty Width ($W_n \le w$): The confidence interval or uncertainty measure is sufficiently narrow.
  3. Trajectory Stability ($r_n \le \eta$): The trajectory is stable, measured by a stability defect (e.g., the difference between the current estimate and a backward-looking average or a latent state projection).

This approach complements, rather than replaces, classical tools like one-sided binomial tests, Clopper-Pearson bounds, confidence sequences, and Wald's Sequential Probability Ratio Test (SPRT). While SPRT is optimal for simple-versus-simple hypothesis testing, it does not address composite one-sided statements, logistic separation, or trajectory stability.

Key Contributions

1. Unification: The paper provides a single theoretical structure (reverse martingales) to interpret Bernoulli all-failure runs, logistic separation, and unstable risk trajectories as manifestations of the same confusion between finite estimates and limiting laws.
2. Operational Remedy: It derives the $\tau_{RM}$ stopping rule, which prevents premature declarations of boundary probabilities by requiring stability and uncertainty control in addition to boundary closeness.
3. Regularization Insight: In the context of logistic regression, the paper advocates for regularized estimators (e.g., ridge, Firth, or Bayesian priors) to ensure finite fitted probabilities, interpreting divergent coefficients as numerical artifacts of separation rather than evidence of exact boundary probabilities.

Numerical Results
The paper presents five numerical studies to validate the framework:

• Bernoulli Rare-Event Experiments: Demonstrates that while an ordinary MLE ($\hat{p}_n = 0$) occurs frequently in rare-event samples, a practical-zero stopping rule (based on a pre-specified $\varepsilon$) controls Type-I error. The MLE equality to zero is shown to be a finite-sample artifact, not proof of $p=0$.
• Low- and High-Dimensional Logistic Regression: Shows that even with finite true coefficients, rare events combined with covariates (especially in high dimensions) lead to "separation-like" behavior, including one-class samples, non-convergence, and extreme fitted logits. Regularization (ridge) stabilizes these estimates, converting divergent coefficients into finite predictive probabilities.
• Stability vs. Closeness: In a controlled simulation of conditional-risk trajectories, a "boundary-only" rule stopped with probability 0.999 on a transient trajectory that later returned to the interior. In contrast, the three-condition rule ($\tau_{RM}$) reduced this false positive rate to 0.107 while maintaining 100% stopping power on genuinely stable boundary-convergent trajectories.
• Real-Data Illustration (RAND Health Insurance Experiment): Using a real dataset with a rare outcome (1.5% prevalence), the study confirms that small subsamples often produce unstable, extreme fitted probabilities. Adding flexible model terms (quadratic design) exacerbates this, reinforcing the need for stability diagnostics.

Significance and Claims
The paper claims that the proposed framework offers a more reliable way to interpret boundary phenomena in finite samples. Its primary significance is the distinction between finite near-boundary estimates and limiting boundary laws.

The authors state that acting on boundary closeness without evidence of stability is premature. The reverse-martingale view recasts boundary degeneracy as a property of the limiting conditional law ($M_\infty$). Consequently, the paper argues that finite samples justify controlled practical boundary statements (e.g., "we are confident $p \le \varepsilon$") but never exact boundary probabilities. The operational payoff is the ability to separate transient apparent certainty from genuine limiting degeneracy, preventing erroneous conclusions in sequential monitoring, adaptive treatment, and rare-event modeling.