Keywords: reverse martingale; sufficient statistic; quasi-martingale; maximum likelihood; information compression; logistic separation; confidence sequence; sequential inference.

MSC 2020: Primary 60G42, 62F10, 62L12; Secondary 62F03, 62J12.

1 Introduction↩︎

Statistical procedures rarely use the full observed path. A sample mean forgets order, extremes, and local fluctuations; a sufficient statistic discards features irrelevant to a parameter under a specified model; a maximum likelihood estimate compresses an empirical objective into a single optimizing parameter; a hidden state in a sequential prediction system compresses a history into a learned representation. These examples share one principle: good statistical summaries remember what matters for the target, not necessarily every detail of the realized data path.

1.0.0.1 The compression triple and its stopping-rule consequence.

This article develops that principle as a theory of target-oriented information compression, and uses it to motivate a sequential methodology for practical boundary declarations. The key object is the triple \[(T_n,\;\mathcal{G}_n=\sigma(T_n),\;M_n=\mathbb{E}(Z\mid\mathcal{G}_n)),\] where \(T_n\) is a data-compression map, \(\mathcal{G}_n\) records the retained information (see 2 ), and \(M_n\) is the conditional target process. When \((\mathcal{G}_n)\) forms a decreasing filtration, \(M_n\) is a reverse martingale converging to \(M_\infty=\mathbb{E}(Z\mid\mathcal{G}_\infty)\). A run of all failures in Bernoulli data, a separated logistic regression, or a near-zero sequential risk score may tempt an exact statement such as “the event probability is zero.” Finite data justify no such claim. The distinction is fundamental: exact boundary degeneracy is a property of \(M_\infty\), whereas finite boundary closeness is only a diagnostic.

The practical consequence is the three-condition stopping rule \[\label{eq:tau95rm95intro} \tau_{\mathrm{RM}} =\inf\bigl\{n\ge n_{\min}: B_n\le\varepsilon,\;W_n\le w,\;r_n\le\eta\bigr\},\tag{1}\] where \(B_n=\min\{M_n,1-M_n\}\) measures boundary closeness, \(W_n\) is the width of a time-uniform confidence sequence or posterior credible interval for \(M_\infty\), and \(r_n=|M_n-M_{n-1}|\) is the empirical stability proxy for the quasi-martingale defect 3 . Compared with standard sequential methods, \(\tau_{\mathrm{RM}}\) addresses a complementary question: the SPRT [1] tests a simple-versus-simple hypothesis about the mean, and CUSUM [2] detects a sustained shift, while \(\tau_{\mathrm{RM}}\) asks whether the conditional target is stably near a boundary, integrating closeness, uncertainty, and coherence into one criterion.

1.0.0.2 Exact versus approximate compression.

When the retained summary is exactly sufficient, the coherence defect \(\delta_n=\mathbb{E}(M_n\mid\mathcal{G}_{n+1})-M_{n+1}\) is identically zero and Proposition 6 guarantees that \(\tau_{\mathrm{RM}}\equiv\tau_{2\mathrm{cond}}\), so the stability screen imposes no delay. When the summary is only approximately sufficient — penalized estimators, risk scores, learned representations — \(\delta_n\ne0\) and \(r_n\le\eta\) provides genuine additional protection against premature declarations.

1.0.0.3 New contributions.

The applied stopping procedures, finite-sample error bounds, and Bernoulli, logistic, and RAND HIE numerical evidence are developed in the companion paper [3]. The present paper contributes the compression triple as a unified organizing principle; the tail-\(\sigma\)-field construction of decreasing filtrations (Remark 1); the reverse quasi-martingale defect \(\delta_n\) and its connection with \(r_n\) (Remark 4); the structural reduction \(\tau_{\mathrm{RM}}\equiv\tau_{2\mathrm{cond}}\) for exactly sufficient summaries (Proposition 6); and numerical calibration across Gaussian, Poisson, public-health surveillance, and quasi-martingale settings not treated in the companion paper (Section 3). Section 2.2 recasts the Normal, Bernoulli, and Poisson sufficiency examples as lossless-compression anchors; the algebraic derivations are in Appendix 7.

1.0.0.4 Organisation.

Section 2 develops the mathematical foundation of target-oriented compression, including the reverse quasi-martingale defect and the decreasing-filtration construction. Section 2.2 recasts exact sufficiency and maximum likelihood as lossless and stabilized empirical compression. Section 2.3 applies the framework to boundary degeneracy in sequential binary problems and defines the three-condition stopping rule \(\tau_{\mathrm{RM}}\) (Section 2.4), including its error-control guarantee (Corollary 1) and parameter selection guidance (Remark 5). Section 3 presents numerical calibration studies. Proofs and data details are in the Appendix and Supplementary Material.

2 Target-oriented compression: mathematical foundation↩︎

Let \(X_1,\ldots,X_n\) denote observed data, let \(T_n=T_n(X_1,\ldots,X_n)\) be a statistic (the compression map), and let \(\mathcal{G}_n=\sigma(T_n)\) be its associated retained information field. For a target quantity \(Z\in L^1\) the corresponding target projection is \[\label{eq:target95projection} M_n=\mathbb{E}(Z\mid\mathcal{G}_n),\tag{2}\] where \(\mathcal{G}_n=\sigma(T_n)\) or a related coarsened \(\sigma\)-field. This ordering is important. The statistic determines what is remembered; the \(\sigma\)-field records the information retained; the conditional expectation projects the target onto that information.

The full path contains ordering, local jumps, extremes, clusters, and other details. The statistic keeps only selected information. The value of that compression depends on the target. If the target is the population mean, the sample mean \[\bar X_n=\frac{1}{n}\sum_{i=1}^n X_i\] is a natural compression. It discards order and much of the pathwise shape but preserves central level. If the target is variance, \(\bar X_n\) alone is insufficient and one must also keep a spread summary such as \[S_n^2=\frac{1}{n-1}\sum_{i=1}^n(X_i-\bar X_n)^2.\] If the target is distributional shape, the empirical distribution function \[F_n(t)=\frac{1}{n}\sum_{i=1}^n\mathbf{1}\{X_i\leq t\}\] is a richer compression than the mean. If the target is a pairwise or rank-based functional, a U-statistic \[U_n=\binom{n}{m}^{-1} \sum_{1\leq i_1<\cdots<i_m\leq n} h(X_{i_1},\ldots,X_{i_m})\] may be the relevant compressed object.

Sequential observation is usually described by an increasing filtration \(\mathcal{F}_n^{\mathrm{raw}}=\sigma(X_1,\ldots,X_n)\), with \(\mathcal{F}_1^{\mathrm{raw}}\subseteq\mathcal{F}_2^{\mathrm{raw}}\subseteq\cdots\). This is a correct and natural viewpoint for data arrival. However, it is not always the correct viewpoint for compression. The sample mean, for example, is generally not a martingale with respect to the raw-data filtration because \[\mathbb{E}(\bar X_{n+1}\mid\mathcal{F}_n^{\mathrm{raw}}) = \frac{n\bar X_n+\mu}{n+1},\] which is not equal to \(\bar X_n\) unless \(\bar X_n=\mu\).

A compression viewpoint instead uses decreasing information, \(\mathcal{G}_1\supseteq\mathcal{G}_2\supseteq\mathcal{G}_3\supseteq\cdots\), where later \(\sigma\)-fields contain less information. This formalizes the idea of moving from more detailed summaries to coarser summaries.

This theorem says that the reverse-martingale structure belongs naturally to the conditional target process. It does not say that every statistic is a reverse martingale.

2.1 Approximate sufficiency and reverse quasi-martingales↩︎

Modern summaries are often not exactly sufficient. Examples include selected features, principal components, penalized estimates, risk scores, and hidden states in recurrent or transformer-based prediction systems [14], [15]. They may retain most target-relevant information without retaining it exactly.

Let \(M_n\) be a target process adapted to a decreasing filtration \((\mathcal{G}_n)\). Define the reverse quasi-martingale defect \[\label{eq:quasi95defect} \delta_n = \mathbb{E}(M_n\mid\mathcal{G}_{n+1})-M_{n+1},\tag{3}\] where the absolute summability criterion \(\sum_{n=1}^\infty\mathbb{E}|\delta_n|<\infty\) characterizes the reverse quasi-martingale class. If \(\delta_n=0\) for all \(n\), the exact reverse-martingale property is recovered. If the defects satisfy the summability condition together with standard integrability conditions, then the process behaves like a reverse quasi-martingale and retains a convergence theory analogous to stable compression. The defect \(\delta_n\) measures the failure of exact coherence across successive compression levels. It is therefore a quantitative measure of information loss or representation mismatch.

A compact summary is \[\begin{align} \text{full data} &\longrightarrow \text{sufficient or approximate statistic}\\ &\longrightarrow \text{conditional target process}\\ &\longrightarrow \text{reverse martingale or reverse quasi-martingale}. \end{align}\]

2.2 Sufficiency and likelihood under compression↩︎

Sufficiency and maximum likelihood are classical inference principles that fit naturally into the compression framework. Sufficiency is the lossless ideal: a statistic is sufficient when it preserves all model-relevant information about the parameter. Maximum likelihood then acts on the resulting empirical information. Neither principle, by itself, requires the retained statistic, the likelihood, or the estimator to satisfy a reverse-martingale identity; that structure belongs to the conditional target process \(M_n=\mathbb{E}(Z\mid\mathcal{G}_n)\) built on top of the sufficient information. The algebraic derivations confirming sufficiency for the Normal, Bernoulli, and Poisson families are collected in Appendix 7; here we focus on the conceptual anchors needed for the stopping-rule analysis.

2.2.1 Exact sufficiency as lossless compression↩︎

Sufficiency is a formal version of lossless statistical compression. Let \(X=(X_1,\ldots,X_n)\) have distribution \(P_\theta\), and let \(T_n=T_n(X)\). The statistic \(T_n\) is sufficient for \(\theta\) if the conditional distribution \(\mathcal{L}_\theta(X\mid T_n)\) does not depend on \(\theta\). Equivalently, under domination, the factorization criterion [16], [17] states that \(L(\theta;x)=g_\theta(T_n(x))h(x)\), so that all parameter-dependent information in the likelihood passes through \(T_n\).

If the \(\sigma\)-fields generated by sufficient summaries are arranged as a decreasing filtration and \(Z\) is a target quantity such as a posterior mean, predictive probability, likelihood score, risk functional, or parameter-related functional, then \(M_n=\mathbb{E}(Z\mid\sigma(T_n))\) is the reverse-martingale object by Theorem 2. The sufficient statistic supplies the information field; the conditional target supplies the martingale. This distinction is essential: \[\begin{array}{ll} T_n & \text{is a data-compression function,}\\[2pt] \mathcal{G}_n=\sigma(T_n) & \text{is the retained information,}\\[2pt] M_n=\mathbb{E}(Z\mid\mathcal{G}_n) & \text{is the reverse-martingale target projection.} \end{array}\] Thus a sufficient statistic is not automatically a reverse martingale. It becomes part of a reverse-martingale construction through the conditional expectation of a target.

2.2.2 Maximum likelihood as compressed empirical information↩︎

The maximum likelihood estimator is not generally a reverse martingale. It is more accurately described as a functional of compressed empirical information. Let \[\ell_n(\theta)=\sum_{i=1}^n\log f_\theta(X_i), \qquad \hat{\theta}_n=\operatorname*{arg\,max}_\theta \ell_n(\theta).\] The empirical measure \[P_n=\frac{1}{n}\sum_{i=1}^n\delta_{X_i}\] gives \[\frac{1}{n}\ell_n(\theta) = \int \log f_\theta(x)\,dP_n(x),\] and hence \[\hat{\theta}_n = \operatorname*{arg\,max}_\theta \int\log f_\theta(x)\,dP_n(x).\] Thus the MLE is a functional of the empirical distribution. In regular settings, consistency can be read as stabilization of the empirical objective: \[\frac{1}{n}\sum_{i=1}^n\log f_\theta(X_i) \longrightarrow \mathbb{E}_{\theta_0}\{\log f_\theta(X)\}.\] The likelihood surface stabilizes and the optimizer converges.

For exponential families, \[f_\theta(x)=h(x)\exp\{\eta(\theta)^\top T(x)-A(\theta)\}.\] For iid observations, \[L(\theta;x) = \prod_{i=1}^n h(x_i) \exp\left\{ \eta(\theta)^\top\sum_{i=1}^nT(x_i)-nA(\theta) \right\}.\] The statistic \(\sum_iT(X_i)\), or its average, is the sufficient compression. The MLE is then a deterministic function of that compressed information when it exists. This explains why exponential families sit naturally at the intersection of sufficiency, likelihood, and stable compression.

The caution is as important as the analogy. The MLE can fail to exist, diverge, or become unstable in boundary cases. Logistic separation is a central example. The compression view therefore distinguishes stable likelihood compression from pathological boundary movement.

2.3 Application: practical boundary degeneracy↩︎

We use boundary degeneracy in sequential binary problems as a concrete application of the target-oriented compression framework. The detailed sequential stopping procedures, finite-sample error bounds, and numerical evidence for boundary detection in Bernoulli, logistic, and near-degenerate risk settings are developed in full in the companion paper [3]. The present paper contributes the theoretical foundation: each boundary phenomenon arises when a finite summary creates the appearance of a boundary limit that the underlying conditional target process has not actually reached. The compression language supplies the precise distinction — exact boundary degeneracy is a property of \(M_\infty = \mathbb{E}(Y\mid\mathcal{G}_\infty)\), not of any finite \(M_n\) — and that distinction is exactly the distinction between lossless and approximate compression established in Sections 2–2.2.

Boundary degeneracy means apparent probabilities near zero or one. The formal definition should precede the examples: finite boundary closeness is only a symptom, while exact boundary degeneracy is a limiting property of the conditional target process.

Let \(Y\in\{0,1\}\) be a target event. For a decreasing filtration \((\mathcal{G}_n)\), define \(M_n=\mathbb{E}(Y\mid\mathcal{G}_n)\). By Theorem 2, \(M_n\) is a bounded reverse martingale and \(M_n\to M_\infty=\mathbb{E}(Y\mid\mathcal{G}_\infty)\) almost surely and in \(L^1\). The limiting conditional law, not a finite estimate, is the object that can be exactly boundary-degenerate.

This definition deliberately concerns \(M_\infty\). A finite \(M_n\) may be close to zero or one because of transient imbalance, early separation, model instability, or an inadequate compression. The reverse-martingale question is whether the limiting conditional law is stably moving toward the boundary.

2.3.1 Symptom I: Bernoulli all-failure and all-success paths↩︎

Let \(Y_1,Y_2,\ldots\stackrel{\mathrm{iid}}{\sim}\operatorname{Bernoulli}(p)\), \(0\leq p\leq1\), and let \(S_n=\sum_{i=1}^nY_i\). If \(S_n=0\), the MLE \(\hat{p}_n=0\) is an estimate, not proof that \(p=0\). A run of \(n\) consecutive failures supports the practical-zero statement \(p<\varepsilon\) with type-I error at most \(\alpha\) under \(p\geq\varepsilon\) whenever \((1-\varepsilon)^n\leq\alpha\), i.e., \(n\geq\log\alpha/\log(1-\varepsilon)\); the symmetric statement holds for all-success runs. Formal propositions, proofs, and confidence-sequence extensions are given in the companion paper.

In the compression framework, the threshold \(n\geq\log\alpha/\log(1-\varepsilon)\) is an implicit constraint on the uncertainty width \(W_n\): at the Jeffreys posterior with no observed successes, the \((1-\alpha)\)-credible interval upper endpoint reaches approximately \(\varepsilon\) precisely at this threshold. The full \(\tau_{\mathrm{RM}}\) rule (Section 2.4) makes this constraint explicit and extends it beyond the all-failure prefix.

2.3.2 Symptom II: Logistic regression and separation↩︎

Let \((Y_i,x_i)\), \(Y_i\in\{0,1\}\), \(x_i\in\mathbb{R}^d\), be observed and suppose \[\mathbb{P}(Y_i=1\mid x_i)= \pi_i(\beta)=\operatorname{expit}(x_i^\top\beta) = \frac{\exp(x_i^\top\beta)}{1+\exp(x_i^\top\beta)}.\]

Because \(\operatorname{expit}(\cdot)\) maps \(\mathbb{R}\) strictly into \((0,1)\), every finite coefficient vector \(\beta\in\mathbb{R}^d\) produces strictly interior fitted probabilities; exact zeros or ones in a logistic fit signal a diverging coefficient, not a genuine probability. Complete separation occurs when there exists a vector \(a\in\mathbb{R}^p\) such that the linear score completely splits the binary outcomes: \[\label{eq:complete95separation} x_i^\top a>0 \quad\text{for all } y_i=1, \qquad\text{and}\qquad x_i^\top a<0 \quad\text{for all } y_i=0.\tag{4}\] Moving \(\beta\) along \(ta\), \(t\to\infty\), drives fitted probabilities toward one for successes and zero for failures; the likelihood may approach its supremum without attaining it at any finite \(\beta\) [18]. Regularization remedies (Firth’s bias-reduced likelihood, ridge logistic regression, and Bayesian regression with proper priors) that keep the fitted object finite are discussed in the original references [19]–[21]; the boundary rules based on \(\tau_{\mathrm{RM}}\) (Section 2.4, 5 ) and supporting numerical studies are given in the companion paper.

The compression interpretation is the new perspective offered here. Separation is not evidence for exact boundary probabilities; it is evidence that the unregularized likelihood can be driven toward a boundary along a low-information direction. In the target-oriented compression framework, the retained field \(\mathcal{G}_n = \sigma(\hat{\beta}_n)\) (or its regularized counterpart) may fail to localize the conditional target \(M_n(x) = \mathbb{E}(Y\mid X=x,\mathcal{G}_n)\) because the compression map has entered a degenerate direction. A stable analysis should therefore report \(M_n(x)\) and its uncertainty, not treat a diverging coefficient as exact probabilistic knowledge. This is the logistic instance of the general principle: approximate compression produces defects \(\delta_n\), and those defects must be small before a boundary declaration is warranted.

2.4 Dynamic stopping: closeness, uncertainty, and stability↩︎

We now formally define the three-condition reverse-martingale stopping rule \(\tau_{\mathrm{RM}}\), which was first introduced and applied in [3]. The compression framework of Sections 2–2.2 provides a theoretical interpretation for each component. A practical boundary declaration should require three simultaneous signals:

1. boundary closeness, so the current target projection is near zero or one;

2. uncertainty control, so the limiting target is statistically localized;

3. trajectory stability, so the near-boundary state is not a transient artifact of the compression path.

Define the boundary distance \(B_n=\min\{M_n,1-M_n\}.\) Let \(W_n\) denote an uncertainty width, such as the width of a confidence sequence, profile-likelihood interval, posterior credible interval, bootstrap window, or predictive-variance band. Let \(r_n\geq0\) denote an empirical stability defect. To operationalize this, we propose the target-oriented reverse-martingale stopping rule: \[\label{eq:tau95rm} \tau_{\mathrm{RM}} = \inf\left\{ n\geq n_{\min}: B_n\leq\varepsilon,\quad W_n\leq w,\quad r_n\leq\eta \right\},\tag{5}\] where \(n_{\min}\) is a nominal burn-in sample size, \(\varepsilon\) is the boundary guard threshold, \(w\) is the maximum tolerable uncertainty width, and \(\eta\) is the structural stability tolerance. The two-condition rule, which omits the stability screen, is \[\label{eq:tau952cond} \tau_{2\mathrm{cond}} = \inf\left\{ n\geq n_{\min}: B_n\leq\varepsilon,\quad W_n\leq w \right\}.\tag{6}\] Boundary closeness alone is not enough; small uncertainty without boundary closeness is not enough; and both can still be misleading if the trajectory is unstable.

The compression framework developed in Sections 2–2.2 now provides a precise theoretical interpretation for each condition. Boundary closeness \(B_n\leq\varepsilon\) says the current target projection is near a boundary. Uncertainty control \(W_n\leq w\) says the limiting target \(M_\infty\) is statistically localized. Trajectory stability \(r_n\leq\eta\) says the empirical defect — the operational proxy for the theoretical defect \(\delta_n\) of 3  — is small enough that the compression is behaving like an exact reverse martingale. When all three conditions hold, the compression is both near the boundary and behaving losslessly. That is the correct standard for a practical boundary declaration. The new result in this paper is that when the retained summary is exactly sufficient (\(\delta_n=0\) identically), the rule reduces to \(\tau_{\mathrm{RM}}\equiv\tau_{2\mathrm{cond}}\): the stability screen is automatically satisfied and contributes no additional delay (confirmed numerically in Studies 1–4 below).

The computational defect \(r_n\) is the operational counterpart of the theoretical reverse quasi-martingale defect \(\delta_n\) of 3 , recalled here for convenience: \[\delta_n=\mathbb{E}(M_n\mid\mathcal{G}_{n+1})-M_{n+1}.\] When conditional expectations can be estimated directly, one may take \[r_n=\left|M_{n+1}-\widehat{\mathbb{E}}(M_n\mid\mathcal{G}_{n+1})\right|.\] When summaries are latent states, one may use a learned backward projection, for example \[r_n=\left\lVert H_n-g_\phi(H_{n+1}) \right\rVert.\] Here \(g_\phi\) acts as a learned reverse-time transition operator, or pullback map, from the future hidden state \(H_{n+1}\) back to the present. It is the parametric analog of the conditional expectation \(\mathbb{E}(M_n\mid\mathcal{G}_{n+1})\) in the exact theoretical case: just as the exact reverse-martingale identity requires \(\mathbb{E}(M_n\mid\mathcal{G}_{n+1})=M_{n+1}\), the learned operator requires \(g_\phi(H_{n+1})\approx H_n\). The residual \(r_n=\left\lVert H_n-g_\phi(H_{n+1}) \right\rVert\) therefore measures how much the hidden-state sequence departs from exact reverse coherence, and can be incorporated as a trainable backward-coherence penalty in the model’s loss function. Thus \(\delta_n\) is the population-level measure of reverse incoherence, whereas \(r_n\) is the empirical diagnostic used to decide whether the observed compression trajectory is stable enough for action. Importantly, \(r_n\) is not an unbiased estimator of \(\delta_n\). It is an observable stability proxy: persistent large values of \(r_n\) indicate either slow convergence of the conditional target process or a failure of coherence across compression levels, both of which are reasons to defer a boundary declaration. When the summability condition \(\sum_{n\geq1}\mathbb{E}|\delta_n|<\infty\) fails, \(r_n\) need not tend to zero, and the stopping rule 5 correctly refuses to fire.

When a time-uniform interval sequence \((I_n)\) satisfies \(\mathbb{P}\{M_\infty\in I_n\text{ for all }n\geq1\}\geq1-\alpha\), the stopping times \[\tau_0=\inf\{n:I_n\subseteq[0,\varepsilon],\, r_n\leq\eta\}, \qquad \tau_1=\inf\{n:I_n\subseteq[1-\varepsilon,1],\, r_n\leq\eta\}\] control the false-declaration probability at level \(\alpha\). The following corollary makes this precise; the stability screen \(r_n\leq\eta\) can only delay stopping, never inflate the error rate.

4pt

For covariate-dependent binary prediction, define \(M_n(x)=\mathbb{P}(Y=1\mid X=x,\mathcal{G}_n)\). If the fitted logistic coefficient is finite, then \(M_n(x)=\operatorname{expit}(x^\top\beta_n)\in(0,1)\). Under separation, however, \(x^\top\beta_n\) may tend to \(+\infty\) or \(-\infty\), and \(M_n(x)\) may tend to one or zero. The stable object is the limiting conditional probability surface, not necessarily a limiting coefficient vector.

For a relevant covariate region \(A\subseteq\mathbb{R}^d\), define a region-wise boundary distance \[B_n(A)=\sup_{x\in A}\min\{M_n(x),1-M_n(x)\}.\] A region-wise practical stopping rule can be written \[\tau_{\mathrm{RM}}(A) = \inf\left\{ n\geq n_{\min}: B_n(A)\leq\varepsilon,\quad W_n(A)\leq w,\quad \sup_{x\in A}r_n(x)\leq\eta \right\}.\] This guards against stopping because a coefficient is large but unstable, or because a small early sample happens to be separated.

In high-dimensional longitudinal prediction, a latent state \(H_t\) computed from histories \(Z_{1:t}\) can be regularized by a backward-coherence penalty \[L_{\mathrm{RM}} = \frac{1}{T-1}\sum_{t=1}^{T-1} \left\lVert H_t-g_\phi(H_{t+1}) \right\rVert^2.\] The corresponding defect \(r_t=\left\lVert H_t-g_\phi(H_{t+1}) \right\rVert\) acts as a stability diagnostic. A binary risk output \(M_t=\mathbb{P}(Y=1\mid H_t)\) should be treated as actionable boundary evidence only when \[\min\{M_t,1-M_t\}\leq\varepsilon, \qquad W_t\leq w, \qquad r_t\leq\eta.\]

In dynamic treatment-regime problems, the decision is often not whether one probability is near zero or one, but whether one action has become clearly preferable. Let \(a\in\mathcal{A}\) denote a treatment action and let \(M_t^{(a)}=\mathbb{P}(Y^{(a)}=1\mid H_t)\) be the patient-specific potential-outcome risk under action \(a\). A forward rule may recommend \(a^*\) when \[\mathbb{P}\left\{ M_t^{(a^*)}-M_t^{(a)} \geq\Delta \text{ for all }a\neq a^* \mid\mathcal{D}_t \right\}\geq1-\alpha.\] The reverse-martingale refinement is to require that this advantage is stable along the latent trajectory. This prevents a treatment switch from being driven only by transient hidden-state fluctuations. Related sequential treatment frameworks include [22], [23], and [24].

3 Numerical calibration of the compression scorecard↩︎

The studies below are organized by the conceptual distinction the theory makes rather than by data type: exact-compression settings where \(\delta_n=0\) and Proposition 6 predicts \(\tau_{\mathrm{RM}}\equiv\tau_{2\mathrm{cond}}\); approximate-compression settings where a non-zero defect process separates the two rules; and realistic-scale illustrations that show how the scorecard behaves on measurement scales drawn from public health monitoring. Studies 1–2 revisit the binary boundary setting from the compression perspective; Studies 3–4 add Gaussian and Poisson exact-summary settings; Studies 5–7 (approximate summaries, realistic scale, and quasi-martingale perturbations) are entirely new contributions of this paper.

The theoretical development leads to a concrete empirical claim: a finite boundary declaration is credible only when boundary closeness, uncertainty localization, and trajectory stability are simultaneously present. To make the evidence easy to inspect, figures are presented directly from the computational outputs and supplemented by tabular stopping-time summaries.

Throughout this section we compare the boundary-only rule \[\tau_{\mathrm{bdy}} = \inf\{n\geq n_{\min}: B_n\leq \varepsilon\},\] the two-condition rule \(\tau_{2\mathrm{cond}}\) 6 , and the full reverse-martingale rule \(\tau_{\mathrm{RM}}\) 5 . Classical benchmarks — the sequential probability ratio test (SPRT) and the cumulative sum chart (CUSUM) — are included where they illuminate the comparison. The purpose is not to identify a universally best detector, but to diagnose what type of evidence each rule treats as a boundary statement.

3.1 Exact sufficient summaries: Bernoulli, Gaussian, and Poisson↩︎

3.1.0.1 Bernoulli rare-event experiments

The Bernoulli experiments are the simplest setting in which the distinction between a near-zero estimate and a stable near-zero conditional target becomes visible. When the data begin with a long all-failure prefix, the empirical proportion \(\hat{p}_n=S_n/n\) can hit the boundary long before either uncertainty or stability is under control. Figure 2 shows that the boundary-only rule therefore stops extremely early, often near the minimum sample size \(n_{\min}=30\), whereas \(\tau_{\mathrm{RM}}\) postpones stopping until the path has settled. Figure 3 makes the same point pathwise: many trajectories drift below the practical threshold \(\varepsilon\), but only a subset remain there in a stable manner.

This gap is not merely cosmetic. Table 3 records the full stopping-time summary across eight \((p,\varepsilon)\) configurations with \(B=1{,}000\) replications each and \(N_{\max}=5{,}000\). When \(p=0.05\) and \(\varepsilon=0.005\), the boundary-only rule produced false practical boundary declarations in \(22.6\%\) of replicates, while \(\tau_{\mathrm{RM}}\) reduced that rate to \(0.0\%\); for \(\varepsilon=0.010\) the corresponding comparison was \(21.7\%\) versus \(0.4\%\). The stability defect \(r_n=|M_n-M_{n-1}|\) is identically zero for the exact reverse martingale \(M_n=S_n/n\) (confirmed numerically: \(\max|r_n|<10^{-10}\) across all runs), confirming that the uncertainty width condition \(W_n\leq w\) alone drives the delay relative to \(\tau_{\mathrm{bdy}}\). Consequently, \(\tau_{\mathrm{RM}}\equiv\tau_{2\mathrm{cond}}\) in this setting: the stability screen is automatically satisfied and contributes exactly zero additional delay. This is a direct empirical verification of the theoretical prediction from Remark 4: when the compression is exactly sufficient, the defect 3 vanishes and the stopping rule 5 reduces gracefully to the two-condition form.

3.1.0.2 Gaussian and Poisson calibration studies

The Gaussian study (\(X_i\stackrel{\mathrm{iid}}{\sim}N(\mu,1)\), known variance) serves as a negative control. The posterior width \(W_n\) contracts at the deterministic rate \(2z_{0.025}/\sqrt{n/\sigma^2+1}\), so whether \(W_n\leq w\) is met depends purely on sample size rather than on the realized data path. Figure 4 illustrates this: the left panel shows the width curve collapsing toward zero, while the right panel shows that neither the two-condition rule nor \(\tau_{\mathrm{RM}}\) stops at all when \(\varepsilon=w=0.05\), because the prior shrinks the width to below \(w\) only after several thousand steps. By contrast, the boundary-only rule stops in nearly all replications with a median of only 41 steps, and CUSUM stops in all replications because it tracks the mean shift rather than the boundary.

Table 4 records the full summary across five values of \(\mu\in\{0,0.01,0.02,0.05,0.10\}\). The result confirms that instantaneous boundary closeness can be obtained at moderate \(n\) even when the true mean is at the boundary (\(\mu=0\)), but simultaneous closeness, uncertainty control, and stability are substantially more demanding.

The Poisson study (\(X_i\stackrel{\mathrm{iid}}{\sim}\mathrm{Poisson}(\lambda)\)) restores a genuine sequential count-monitoring problem with a practical rare-event claim. Figure 5 shows the same tension seen in the Bernoulli case. Average run lengths under the boundary-only and CUSUM rules are much shorter than under \(\tau_{\mathrm{RM}}\), but the trajectory plot reveals why: sample-path averages \(\bar X_n\) often dip below \(\varepsilon\) only temporarily. Table 5 records the complete stopping-time summary across eight \((\lambda,\varepsilon)\) settings. When \(\lambda=0.01\) and \(\varepsilon=0.005\), the false-declaration rate was \(78.6\%\) for the boundary-only rule and \(46.6\%\) for \(\tau_{\mathrm{RM}}\). The same qualitative ranking persists across all count scenarios.

3.2 Approximate summaries: logistic regression and quasi-martingale perturbations↩︎

The two studies in this subsection use summaries that are not exactly sufficient: the ridge-penalized logistic estimator introduces an approximation defect relative to the unpenalized MLE, and the quasi-martingale perturbation scenarios deliberately break the exact reverse-martingale identity. In the logistic study, the third condition \(r_n\leq\eta\) provides additional protection relative to the two-condition rule. In the quasi-martingale perturbation study, the defect process is primarily diagnostic under the present calibration, and would become decision-relevant under a stricter stability threshold.

3.2.0.1 Rare-event logistic regression and practical separation

The logistic studies translate the same principle to a regression setting in which complete or quasi-complete separation 4 is a numerical manifestation of boundary pressure. The relevant target is not the divergence of a coefficient vector by itself, but the behavior of the retained predictive quantity \(M_n(\mathbf{x}_0)\) together with the defect \(r_n\). Figure 6 shows three scenarios: two low-dimensional rare-event designs (\(d=3\)) and one high-dimensional design (\(d=20\)). Each panel plots representative trajectories of \(M_n(\mathbf{x}_0)\), the stability defect \(r_n\) on a log scale, and the cumulative frequency of unpenalized MLE separation.

Table 6 records the stopping-time summary for all three scenarios (\(B=1{,}000\), ridge \(\lambda=1\), \(\varepsilon=w=0.05\), \(\eta=0.01\)). Two features stand out. First, the cumulative separation rate is already non-negligible in low dimension and becomes dominant once the events-per-variable ratio is small: the separation frequency reached \(43.8\%\) for \(d=3\), \(\rho=0.01\) and \(100\%\) for \(d=20\), \(\rho=0.01\). Second, the reverse-martingale scorecard is systematically more conservative than a boundary-only declaration: the mean stopping time moved from roughly 50–66 observations under \(\tau_{\mathrm{bdy}}\) to roughly 104–125 observations under \(\tau_{\mathrm{RM}}\). The interpretation is that apparent separation is finite-sample instability unless it is accompanied by a stable, localized predictive target.

3.2.1 Quasi-reverse-martingale perturbations↩︎

The last study addresses robustness rather than detection power. If the retained summaries are only approximately coherent, the defect process \(r_n\) should reveal that failure. Figure 7 compares three ways of perturbing the ideal reverse-martingale structure:

• Scenario A (misspecification): a Bernoulli model with latent heterogeneity \(\sigma^2>0\), so the conditional target is not a running proportion.

• Scenario B (exponential smoothing): the running mean is replaced by an exponentially smoothed estimate \(\alpha_n Y_n + (1-\alpha_n)M_{n-1}\) with \(\alpha_n=n^{-\gamma}\); \(\gamma=1\) gives the exact reverse martingale whereas smaller \(\gamma\) introduces a positive defect.

• Scenario C (variational-Bayes dampening): a mean-field ELBO update with forgetting parameter \(\kappa\) replaces the exact Jeffreys posterior mean; \(\kappa=0\) recovers the exact case.

Figure 7 displays the median stability defect \(r_n\) across all nine scenario-variants. Table 7 records the corresponding stopping-time summary alongside sampled defect values at \(n\in\{100,500,2000\}\).

Under the present calibration (\(\varepsilon=0.05\), \(\eta=0.01\)), the stability screen never contributes additional delay: \(\tau_{\mathrm{RM}}=\tau_{2\mathrm{cond}}\) in every single replication, and the extra gap is exactly zero across all nine variants. This is because the median defect at typical stopping times is two to three orders of magnitude below the threshold \(\eta=0.01\). The informative diagnostic is therefore not the gap itself but the shape of the defect trajectory over time.

The exact variants (\(\sigma=0\), \(\gamma=1\), \(\kappa=0\)) drive \(r_n\) monotonically to machine-epsilon: by step 2,000 the median defect is below \(10^{-5}\). The quasi-martingale perturbations (Scenario A, \(\sigma>0\); Scenario B, \(\gamma=0.75\); Scenario C, \(\kappa>0\)) show slower but monotonically decreasing defect trajectories. Only Scenario B with \(\gamma=0.50\) breaks this pattern: the median defect is anomalously small at \(n=100\) (\(2.0\times10^{-6}\)) then rises to \(1.4\times10^{-4}\) at \(n=2{,}000\), confirming that the process falls outside the summable-defect (quasi-martingale) class. A stricter threshold \(\eta\) would eventually separate this non-decaying variant from the rest and introduce a non-zero stopping-time gap.

3pt

3.3 Realistic-scale monitoring illustrations↩︎

The final pair of studies shows how the same diagnostics behave on realistic measurement scales. For the final submission, these studies should be executed on the actual public-domain data described below. The scripts support automatic fallback to synthetic series calibrated to the respective public sources, and the methodological conclusions are invariant to this substitution; authentic historical data strengthen the evidential profile by connecting \(r_n\) directly to the information-leakage properties of real sequential public-health reporting streams. Specifically: (i) the Taiwan CDC ILI series (https://nidss.cdc.gov.tw/) should replace taiwan_ili_weekly.csv in Study 5; (ii) the NHANES 2017–2018 file PBCD_J (https://wwwn.cdc.gov/Nchs/Nhanes/2017-2018/PBCD_J.XPT, variable LBXBPB) should replace the synthetic log-normal series in Study 6. Full instructions are in Appendix 6.1.

Figure 8 shows an influenza-like-illness monitoring example on the Taiwan weekly scale. The upper panel plots the sequential mean together with a \(95\%\) credible band and the stability defect \(r_n\); the lower panel zooms into the first two years of the series. Here none of the stopping rules fires in the displayed run. This non-result is informative: the ILI rate does not produce sustained evidence of a low-risk boundary state, and the stability defect remains too large for a credible practical boundary declaration.

Figure 9 presents an NHANES-style blood-lead monitoring illustration. In the displayed sequential summary, the running mean falls below the practical level \(\varepsilon=1.50\) quickly enough that the boundary-only rule stops at \(n=50\), but \(\tau_{\mathrm{RM}}\) waits until \(n=163\). The empirical distribution function shows why this extra caution is sensible: the threshold is not a zero-probability boundary point but an interior practical cutoff, so boundary claims should be based on stabilized evidence rather than a single early crossing.

3.4 Summary of numerical results↩︎

Taken together, the numerical studies support a single interpretation. A value near zero or near one is not, by itself, a reliable inferential statement. The same finite sample can look boundary-like because of an all-failure prefix, a separating logistic fit, a transient surveillance dip, or an approximately coherent compression scheme. The reverse-martingale scorecard separates such cases from genuinely stable boundary evidence by requiring three ingredients simultaneously: closeness \(B_n\), uncertainty localization \(W_n\), and trajectory stability \(r_n\). This is the distinction needed to connect classical sufficiency, modern sequential monitoring, and practical boundary degeneracy within a single target-oriented compression framework.

In the exact sufficiency examples considered here, namely the Bernoulli, Gaussian, and Poisson cases, the implemented diagnostic is automatically nonbinding. Thus the structural prediction of Proposition 6 is confirmed: the stopping rules \(\tau_{\mathrm{RM}}\) and \(\tau_{2\mathrm{cond}}\) are identical, so the stability screen imposes no additional delay. In the logistic study, the third condition \(r_n\leq\eta\) provides additional protection relative to the two-condition rule. In the quasi-martingale perturbation study, the defect process is primarily diagnostic under the present calibration, and would become decision-relevant under a stricter stability threshold.

4 Conclusion↩︎

This article has developed a unified view of statistical information as target-oriented compression. The central point is that a statistic, by itself, is not the natural martingale object. Rather, a statistic induces a retained information field, and the corresponding conditional target process \[M_n = \mathbb{E}(Z \mid \mathcal{G}_n)\] is the object to which reverse-martingale theory applies when the retained fields are arranged as a decreasing filtration. This distinction clarifies the roles of statistics, \(\sigma\)-fields, and conditional expectations: statistics compress data, \(\sigma\)-fields encode retained information, and reverse martingales describe the stable projection of a target under successive information reduction.

Classical sufficiency appears in this framework as the lossless case of statistical compression. When a sufficient statistic preserves all information relevant to a model parameter, the conditional target process based on its \(\sigma\)-field retains the appropriate inferential content. Approximate summaries, including selected features, penalized estimates, risk scores, and learned hidden states, need not satisfy exact reverse-martingale identities. Their departure from coherence can instead be described through reverse quasi-martingale defects. These defects provide a quantitative language for discussing how much target-relevant information is lost or distorted as data are compressed.

The same perspective also resolves a recurring difficulty in sequential binary inference: finite-sample boundary closeness should not be confused with exact boundary degeneracy. An all-failure Bernoulli run, a separated logistic regression, or a near-zero sequential risk estimate may suggest a boundary probability, but finite data alone do not justify the statement that a probability is exactly zero or one. The applied stopping procedures formalizing this principle are studied in detail in the companion paper [3]; the present paper provides their theoretical foundation in the compression framework. Exact degeneracy, when it occurs, is a limiting property read from \[M_\infty = \mathbb{E}(Y \mid \mathcal{G}_\infty),\] whereas practical boundary claims require additional safeguards. The proposed rule therefore requires three simultaneous conditions: closeness to the boundary, uncertainty control, and trajectory stability. This separates transient numerical boundary behavior from stable inferential evidence.

The numerical studies support this distinction across rare-event Bernoulli sampling, logistic separation, Gaussian and Poisson monitoring, data-scale sequential illustrations, and quasi-reverse-martingale perturbations. In these examples, boundary-only rules often react quickly to early or unstable sample-path behavior, whereas the full reverse-martingale scorecard is more conservative because it requires the near-boundary state to be statistically localized and dynamically stable. The empirical message is consistent with the theory: near zero or near one is not, by itself, a reliable scientific conclusion.

Several extensions remain natural. One direction is to develop sharper finite-sample confidence sequences specifically for limiting reverse-martingale targets. Another is to study data-adaptive choices of the stability threshold \(\eta\) in high-dimensional and learned-representation settings. A third is to connect the reverse-coherence defect more directly with regularization, representation learning, and dynamic treatment rules. More broadly, the framework suggests that modern statistical procedures should be evaluated not only by predictive accuracy or likelihood fit, but also by whether their compressed representations preserve the target-relevant information needed for stable decisions.

In summary, the reverse-martingale compression view provides a common language for sufficiency, likelihood-based estimation, approximate representation, sequential monitoring, and practical boundary degeneracy. Reliable boundary conclusions require more than an extreme finite estimate: they require a stable conditional target, localized uncertainty, and coherent behavior across the compression path.

Supplementary Material. The Supplementary Material contains: (S1) proofs of all reverse quasi-martingale defect bounds and the summability criterion for \(\sum_n\mathbb{E}|\delta_n|<\infty\); (S2) finite-sample error-bound derivations for \(\tau_{\mathrm{RM}}\); (S3) all R scripts for reproducing the numerical studies; and (S4) synthetic data files for Studies 5–6 together with instructions for substituting the actual public-domain NHANES and Taiwan CDC series.

Appendix↩︎

5 Classical Bernoulli benchmarks↩︎

The all-failure practical-zero rule (Section 2.3) is the \(s=0\) special case of the exact one-sided binomial test of \(H_0:p\geq\varepsilon\) against \(H_1:p<\varepsilon\). Its exact \(p\)-value at the boundary is \[(1-\varepsilon)^n,\] and the dual Clopper–Pearson upper bound [26] after zero successes is \[U_n(0;\alpha)=1-\alpha^{1/n}.\] For simple-versus-simple Bernoulli monitoring, Wald’s sequential probability ratio test [1], [2], [27], [28] compares \(H_0:p=p_0\) with \(H_1:p=p_1\), \(p_1<p_0\), using \[L_n = S_n\log\frac{p_1}{p_0} +(n-S_n)\log\frac{1-p_1}{1-p_0}.\] With Wald boundaries \[a=\log\frac{\beta}{1-\alpha}, \qquad b=\log\frac{1-\beta}{\alpha},\] an all-failure path favors \(H_1\) after \[\tau_{\mathrm{SPRT},0} = \left\lceil \frac{\log\{(1-\beta)/\alpha\}}{\log\{(1-p_1)/(1-p_0)\}} \right\rceil\] observations. This benchmark is appropriate when the scientific problem is a choice between two pre-specified point hypotheses. The reverse-martingale boundary framework addresses the complementary setting of composite practical boundary statements, logistic separation, and trajectory stability.

Confidence sequences [5], [11] provide the time-uniform uncertainty component \(W_n\) in the unified rule of Section 2.4. If \(C_n\) satisfies \[\mathbb{P}_p\{p\in C_n\text{ for all }n\geq1\}\geq1-\alpha,\] then a practical-zero declaration based on \(C_n\subseteq[0,\varepsilon]\) can be embedded in \(\tau_{\mathrm{RM}}\) by taking \(W_n\) to be the width of \(C_n\) and by requiring the interval to lie inside the chosen practical boundary region.

6 Numerical settings summarized↩︎

The executed numerical studies used the following designs (all scripts in the accompanying scripts/ directory, reproducible via Rscript run_all.R):

1. Bernoulli rare-event (Study 1): \(Y_i\stackrel{\mathrm{iid}}{\sim}\mathrm{Bernoulli}(p)\), \(p\in\{0.001,0.005,0.010,0.050\}\), \(\varepsilon\in\{0.005,0.010\}\), \(N_{\max}=5{,}000\), \(B=1{,}000\), \(w=0.02\), \(\eta=10^{-6}\). Posterior width from Jeffreys Beta\((1/2,1/2)\) prior. (Note on \(\eta\): for exact-sufficient summaries the stability defect satisfies \(r_n<10^{-10}\) numerically; setting \(\eta=10^{-6}\) is effectively machine-precision and imposes no additional delay beyond the uncertainty condition. This is in line with Proposition 6.)

2. Logistic regression (Study 2): \(Y_i\mid x_i\sim\mathrm{Bernoulli}(\mathrm{expit}(x_i^\top\beta))\), \(x_i\sim N(0,I_d)\), true \(\beta=({\rm logit}(\rho),0,\ldots,0)^\top\); scenarios \(d\in\{3,20\}\), \(\rho\in\{0.005,0.010\}\); ridge logistic regression with \(\lambda=1\); \(\varepsilon=w=0.05\), \(\eta=0.01\), \(N_{\max}\in\{500,1{,}000\}\). (Note on \(\eta\): the ridge-MLE predictive surface produces step-to-step changes \(r_n\) of order \(10^{-2}\) to \(10^{-3}\) near stopping; \(\eta=0.01\) is therefore a practically loose threshold calibrated to the typical signal size in the approximate-compression regime. A stricter \(\eta\) would increase delays further without materially changing the qualitative findings.)

3. Normal calibration (Study 3): \(X_i\stackrel{\mathrm{iid}}{\sim}N(\mu,1)\), \(\mu\in\{0,0.01,0.02,0.05,0.10\}\); Jeffreys-like \(N(0,1)\) prior; CUSUM calibrated for \(\mathrm{ARL}_0=500\); \(\varepsilon=w=0.05\), \(N_{\max}=3{,}000\).

4. Poisson surveillance (Study 4): \(X_i\stackrel{\mathrm{iid}}{\sim}\mathrm{Poisson}(\lambda)\), \(\lambda\in\{0.001,0.005,0.010,0.050\}\), \(\varepsilon\in\{0.005,0.010\}\); Jeffreys Gamma\((1/2)\) prior; CUSUM and SPRT benchmarks; \(N_{\max}=5{,}000\), \(w=0.02\).

5. Taiwan CDC ILI illustration (Study 5): weekly influenza-like illness rate; \(n=312\) weeks (2018-W01 through 2023-W52); synthetic series calibrated to Taiwan CDC seasonal reporting patterns when the local data file is absent. See Section 6.1 for the data source and reproducibility instructions.

6. NHANES blood-lead illustration (Study 6): blood-lead level \(\mu\mathrm{g/dL}\); \(n=7{,}000\) observations; synthetic log-normal series (geometric mean \(\approx 0.82\,\mu\mathrm{g/dL}\), geometric SD \(\approx 2.1\)) calibrated to the NHANES 2017–2018 Laboratory Procedures Manual [25]; practical threshold \(\varepsilon=1.50\,\mu\mathrm{g/dL}\). See Section 6.1 for the data source and reproducibility instructions.

7. Quasi-reverse-martingale perturbations (Study 7): three scenarios (misspecification, exponential smoothing, VB dampening) with nine parameter variants; \(B=1{,}000\), \(n_{\max}=2{,}000\).

Section 3 incorporates the numerical-study figures and tables directly from the bundled output files so that the empirical diagnostics can be read alongside the theory.

6.1 Data sources↩︎

Studies 1–4 and 7 use fully simulated data generated within the accompanying R scripts; no external data files are required. Studies 5 and 6 are calibrated to publicly available administrative and survey data as described below.

6.1.0.1 Study 5: Taiwan CDC influenza-like illness data.

The intended real data series is the weekly influenza-like illness (ILI) consultation rate per 10,000 outpatient visits reported by the Taiwan Centers for Disease Control (TCDC) through the National Notifiable Disease Surveillance System (NNDSS), covering 2018-W01 through 2023-W52 (\(n=312\) weeks). This series is freely accessible at https://nidss.cdc.gov.tw/. Because the data file scripts/data/taiwan_ili_weekly.csv (columns: year, week, rate) was not present at the time of execution, the script study10_taiwan_cdc.R fell back to a synthetic series whose seasonal structure (annual mean \(\approx 60\), epidemic peak \(\approx 150\)–\(200\), off-season trough \(\approx 15\)–\(25\) cases per 10,000 outpatient visits) was calibrated to published TCDC surveillance summaries. Placing the downloaded file at the path above causes the script to use the real data automatically on any subsequent run. The methodological conclusions do not depend on access to the original administrative records.

6.1.0.2 Study 6: NHANES blood-lead data.

The intended real data are blood-lead level (BLL, \(\mu\)g/dL) measurements from the National Health and Nutrition Examination Survey (NHANES) 2017–2018 cycle, laboratory component Blood Metals, file PBCD_J, variable LBXBPB. The data are freely available at https://wwwn.cdc.gov/Nchs/Nhanes/2017-2018/PBCD_J.XPT and can be loaded directly in R via nhanesA::nhanes("PBCD_J") (CRAN package nhanesA). Because this download was unavailable at execution time, the script study11_nhanes.R generated a synthetic log-normal series (\(n=7{,}000\), geometric mean \(\approx 0.82\,\mu\)g/dL, geometric SD \(\approx 2.1\), detection floor \(0.01\,\mu\)g/dL) calibrated to the summary statistics documented in the NHANES 2017–2018 Laboratory Procedures Manual [25]. The synthetic file is saved as scripts/data/nhanes_bll_synthetic.csv and is included in the supplementary package. Placing a real-data file at scripts/data/nhanes_bll_2017_2018.csv causes the script to use the survey data automatically. The NHANES data are in the public domain (U.S. federal government work).

7 Sufficiency derivations for Example 1↩︎

7.0.0.1 Normal, known variance.

For \(X_i\stackrel{\mathrm{iid}}{\sim}N(\mu,\sigma^2)\) with \(\sigma^2\) known, expand \(\sum_i(x_i-\mu)^2 = \sum_i x_i^2 - 2\mu\sum_i x_i + n\mu^2\). The likelihood factors as \(L(\mu;x)=g_\mu(\bar x_n)\cdot h(x)\) with \(g_\mu(\bar x_n)\propto\exp\{n\bar x_n\mu/\sigma^2 - n\mu^2/(2\sigma^2)\}\) and \(h(x)=\exp\{-\sum_i x_i^2/(2\sigma^2)\}\), confirming that \(\bar X_n\) is sufficient by the Fisher–Neyman factorization [16], [17].

7.0.0.2 Bernoulli.

For \(X_i\stackrel{\mathrm{iid}}{\sim}\mathrm{Bernoulli}(p)\), \(L(p;x)=p^{s_n}(1-p)^{n-s_n}\) where \(s_n=\sum_i x_i\). This factors through \(T_n=s_n\) alone, so \(T_n\) is sufficient by the Fisher–Neyman factorization. Under a Jeffreys \(\mathrm{Beta}(1/2,1/2)\) prior, the posterior is \(\mathrm{Beta}(s_n+1/2,\,n-s_n+1/2)\) and the posterior mean is \(M_n=(s_n+1/2)/(n+1)\).

The reverse-martingale property \(\mathbb{E}(M_n\mid\mathcal{G}_{n+1})=M_{n+1}\) — where \(\mathcal{G}_{n+1}=\sigma(T_{n+1},T_{n+2},\ldots)\) is the tail \(\sigma\)-field at step \(n+1\) — follows immediately from Theorem 2: since \(M_k=\mathbb{E}(Z\mid\mathcal{G}_k)\) for all \(k\) and \((\mathcal{G}_k)\) is a decreasing filtration, \[\mathbb{E}(M_n\mid\mathcal{G}_{n+1}) =\mathbb{E}\bigl[\mathbb{E}(Z\mid\mathcal{G}_n)\mid\mathcal{G}_{n+1}\bigr] =\mathbb{E}(Z\mid\mathcal{G}_{n+1}) =M_{n+1},\] where the second equality uses the tower property together with \(\mathcal{G}_{n+1}\subseteq\mathcal{G}_n\). The explicit posterior-mean formula confirms this: \(M_n=(s_n+1/2)/(n+1)\) with \(s_n=T_n\) and \(M_{n+1}=(s_{n+1}+1/2)/(n+2)\) with \(s_{n+1}=T_{n+1}\), and one verifies \(\mathbb{E}(M_n\mid\mathcal{G}_{n+1})=\mathbb{E}[(s_n+1/2)/(n+1)\mid s_{n+1}] =(s_{n+1}+1/2)/(n+2)=M_{n+1}\) by the identity \(\mathbb{E}[s_n\mid s_{n+1}]=s_{n+1}\cdot n/(n+1)\) for the conditional distribution of a sub-sum given the total in the Bernoulli model.

7.0.0.3 Poisson.

For \(X_i\stackrel{\mathrm{iid}}{\sim}\mathrm{Poisson}(\lambda)\), \(L(\lambda;x)\propto e^{-n\lambda}\lambda^{s_n}\), so \(T_n=\sum_i X_i\) is sufficient. Under a Jeffreys \(\mathrm{Gamma}(1/2,1)\) prior, the posterior is \(\mathrm{Gamma}(s_n+1/2, n+1)\) and the posterior mean is \(M_n=(s_n+1/2)/(n+1)\). The reverse-martingale property follows analogously.

Data Availability Statement↩︎

All simulation scripts and synthetic data files are included in the supplementary scripts/ directory and are reproducible by running Rscript run_all.R. Studies 1–4 and 7 require no external data. The Taiwan CDC ILI series used in Study 5 is publicly available at https://nidss.cdc.gov.tw/. The NHANES blood-lead series used in Study 6 is publicly available at https://wwwn.cdc.gov/Nchs/Nhanes/2017-2018/PBCD_J.XPT and in the public domain. Both studies include fully reproducible synthetic fallback series calibrated to the respective public sources; the substantive methodological conclusions do not depend on access to the original administrative or survey records.

References↩︎