AI 中文总结
本研究指出贝叶斯计算中常将边缘似然视为静态常数的疏漏，提出将其恢复为实时正则项，引入诊断指标与混合概率，为非平稳场景下的序贯估计与风险管理提供正则化机制。
AI 中文摘要
在现代贝叶斯计算与参数估计中，边缘似然作为贝叶斯定理中的分母P(D)，常通过未归一化的比例关系被绕过。即便在需显式计算该分母以求得贝叶斯因子的专门模型选择框架内，其也被纯粹当作静态常数处理。本研究分析了这种计算便利性所导致的一处微妙分析疏漏。通过对简化的序贯部分信息系统的分析，我们表明边缘概率具有关键的双重信息层：后验概率决定单次试验中信念更新的局部幅度，而边缘分母则决定该更新在历史时间范围内的物理、长期频率主义节奏。丢弃分母仅能计算观察者在特定数据集出现后应持有的信念，却抹去了决定该推断状态在自然中出现频率的数据生成现实。我们提出将边缘似然恢复为主动的实时正则项，引入三种诊断指标以调节递归在线估计增益，并构建了有效的混合概率，将先验与后验融合为受意外触发或保守的调节器。该配对概率框架或可为非平稳分布漂移下的序贯估计架构与定量风险管理场景提供稳健的正则化机制。
英文摘要
In modern Bayesian computation and parametric estimation, the marginal likelihood, serving as the denominator P(D) in Bayes' Theorem, is routinely bypassed via unnormalized proportionality relations. Even within specialized model-selection frameworks where it is explicitly evaluated to compute Bayes Factors, the denominator is treated purely as a static constant. This note evaluates a subtle analytical oversight resulting from this computational convenience. Through the analysis of a simplified, sequential partial-information system, we show that the marginal probability possesses a critical dual layer of information: while the posterior probability determines the local magnitude of a belief update upon a solitary trial, the marginal denominator governs the physical, long-run frequentist cadence of that update across a historical horizon. Discarding the denominator computes what an observer ought to believe once a specific dataset manifests, but erases the data-generating reality that governs how frequently that inferential state occurs in nature. We propose reclaiming the marginal likelihood as an active, real-time regularizer. We introduce three diagnostic measures to regulate recursive online estimation gains, and construct valid mixture probabilities that blend the prior and posterior to function as surprise-activated or conservative regulators. This paired probability framework may offer a robust regularizing mechanism for sequential estimation architectures and quantitative risk management scenarios under non-stationary distribution shifts.
