论文简报
stat.ME 2605.26568v1 值得读

Target-Oriented Statistical Compression: Sufficiency, Reverse Martingales, and Sequential Monitoring

Yuan-chin Ivan Chang

发布日期：2026-05-26 05:37 相关性：0.7000 价值：0.7000 分类：stat.ME cs.IT math.ST
arXiv 摘要 PDF 原文 网页正文

摘要

Statistical procedures rarely retain all features of the observed data. A sufficient statistic removes information irrelevant to a parameter; a maximum likelihood estimate compresses an empirical objective into an optimizing point; and a hidden state in a sequential model compresses past observations into a learned representation. This article develops these practices under the unified notion of \emph{target-oriented statistical compression}: a useful summary preserves what matters for an inferential, predictive, or decision-relevant target, rather than every detail of the realized data path. The central object is the conditional target process \(M_n=\E(Z\given\G_n)\), where \(Z\) is the target and \(\G_n=σ(T_n)\) is the information retained by the compression map \(T_n\). When \((\G_n)\) is a decreasing filtration, \((M_n)\) is a reverse martingale with limit \(M_\infty=\E(Z\given\G_\infty)\). Exact sufficiency corresponds to lossless compression, while approximate summaries such as penalized estimators, principal components, and neural-network hidden states produce reverse quasi-martingale defects measuring coherence loss across compression levels. The diagnostic \(r_n=|M_n-M_{n-1}|\) is treated as an observable stability proxy, not as an unbiased estimator of the theoretical defect. Boundary degeneracy in sequential binary problems is developed as a central application. Practical boundary claims require joint assessment of boundary closeness, uncertainty control, and trajectory stability. The companion paper \citet{chang2025rm} develops the corresponding stopping procedures, finite-sample bounds, and numerical evidence; the present paper provides the broader theoretical infrastructure and extends the framework to Gaussian, Poisson, and quasi-martingale monitoring problems.

相关性判断

medium
相关方向
information_theory sequential_analysis statistical_inference
判断依据

The paper is mainly statistical theory, but it explicitly frames sufficiency as information compression and is tagged `cs.IT`; the reverse-martingale and sequential monitoring angle makes it adjacent to information-theoretic review.

价值判断

Clear theoretical framing connecting statistical sufficiency, compression, reverse martingales, and sequential monitoring. Structure analysis shows coherent technical machinery and explicit cs.IT adjacency, but the primary contribution appears statistical rather than core information theory. Main practical procedures, finite-sample bounds, and numerical evidence are deferred to a companion paper, reducing urgency for deep review of this paper alone.

核心问题与主要方法

核心问题

Characterize statistical summaries as target-oriented compression and use that structure to decide when a sequential binary process is credibly near a boundary.

场景：Sequential inference with a compression map T_n, retained sigma-fields G_n, and conditional target processes M_n under decreasing filtrations; includes binary boundary monitoring plus Gaussian, Poisson, logistic, and quasi-martingale examples.

主要方法

Define a compression map T_n, retained sigma-field G_n=sigma(T_n), and conditional target projection M_n=E(Z|G_n), separating the statistic from the martingale object. Use a decreasing filtration, including a tail sigma-field construction, so that conditional target projections form a reverse martingale and converge to M_infty. Model approximate summaries through a reverse quasi-martingale defect delta_n=E(M_n|G_{n+1})-M_{n+1}, with r_n=|M_n-M_{n-1}| serving as an empirical stability signal. Declare practical binary boundary behavior only when B_n<=epsilon, W_n<=w, and r_n<=eta hold together. Show that when summaries are exactly sufficient and the implemented stability diagnostic is linked to the zero defect, tau_RM reduces to the two-condition rule.

关键贡献与后续阅读

关键贡献

Introduces target-oriented statistical compression as a common language for sufficient statistics, MLEs, penalized estimators, risk scores, and learned hidden states. Identifies the conditional target process M_n=E(Z|G_n), rather than the statistic itself, as the object governed by reverse-martingale theory under decreasing retained information. Provides a framework for approximate sufficiency via reverse quasi-martingale defects that quantify loss of coherence across compression levels. Formulates a three-condition sequential boundary scorecard combining boundary closeness, uncertainty width, and trajectory stability. Connects exact sufficiency to lossless compression and states a structural reduction in which the stability screen imposes no additional delay when the defect vanishes under suitable diagnostic linkage.

研究启发

How much of the finite-sample error control and numerical evidence is actually proved or reproduced in this paper versus deferred to the companion paper? Is the decreasing-filtration tail construction operationally useful for online monitoring, or mainly a retrospective theoretical device? Can the relationship between the theoretical defect delta_n and practical diagnostics r_n be sharpened beyond proxy behavior for learned representations or penalized estimators? Are the reported simulation claims reproducible from the supplementary scripts, especially the comparisons with boundary-only, two-condition, SPRT, and CUSUM baselines?

限制与不确定性

Assessment relies on abstract and structure analysis only, not full-paper validation. Novelty may be overstated if the framework is mostly a unifying reinterpretation of known sufficiency and martingale tools. Empirical and procedural support appears to be outside this paper.

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