记忆预测超额：随机过程中预测增益与记忆长度的概率量
Memory Prediction Excess: A Probabilistic Quantity for Predictive Gain and Memory Length in Stochastic Processes
查看机构详情
- Southwest Jiaotong University（西南交通大学）
机构由 AI 辅助整理，请以论文原文为准。
浏览论文内容
中文总结 AI 辅助
本文提出记忆预测超额（MPE）这一概率量，定量衡量利用完整历史相对于静态分布带来的预测精度提升，并定义归一化版本及有限历史变体，证明最小记忆长度受马尔可夫阶数约束。
中文摘要 AI 辅助
在随机过程预测中，一个核心问题是过去信息能在多大程度上提高正确预测下一状态的概率。我们引入记忆预测超额（MPE）来定量回答这一问题。MPE衡量在离散时间、有限状态过程中，使用完整观测历史相对于仅使用静态边际分布所获得的平均预测精度提升。其定义为期望最优条件预测精度与最优静态预测精度之差。我们考察了其基本性质：MPE始终非负；它存在一个依赖于静态精度的上界，当且仅当未来几乎必然地是过去的确定性函数时达到该上界；并且刻画了MPE为零的退化情形。我们引入了一个归一化版本，取值在单位区间内，作为预测效率的无量纲度量。通过比较基于不同长度历史的预测，推导出一个下界，表明期望最优预测精度关于历史长度是单调的。该框架被扩展到有限长度历史，其中有限历史MPE（FH-MPE）衡量仅保留最近观测时可获得的预测增益。这引出了最小记忆长度的概念，即达到与完整历史相同预测性能所需的最小记忆长度。对于有限阶马尔可夫链，该最小记忆长度被证明受马尔可夫阶数限制。MPE及其变体以条件概率和预测精度形式表述，为记忆的预测效用提供了概率视角，补充了经典信息论方法。
英文摘要
A central question in the prediction of stochastic processes is the extent to which past information can improve the probability of correctly predicting the next state. We introduce the Memory Prediction Excess (MPE) to address this question quantitatively. The MPE measures the average improvement in prediction accuracy obtained by using the entire observed history relative to using only the static marginal distribution, in discrete-time finite-state processes. It is defined as the difference between the expected optimal conditional prediction accuracy and the optimal static prediction accuracy. Its basic properties are examined: the MPE is always non-negative; it admits an upper bound depending on the static accuracy, attained if and only if the future is almost surely a deterministic function of the past; and degenerate cases in which the MPE vanishes are characterized. A normalized version, taking values in the unit interval, is introduced as a dimensionless measure of predictive efficiency. A lower bound is derived by comparing predictions based on histories of different lengths, showing that the expected optimal prediction accuracy is monotone with respect to the history length. The framework is extended to finite-length histories, where the finite-history MPE (FH-MPE) measures the predictive gain attainable when only the most recent observations are retained. This leads to the notion of a minimal memory length required to achieve the same predictive performance as the full history. For finite-order Markov chains, this minimal memory length is shown to be bounded by the Markov order. The MPE and its variants are formulated in terms of conditional probabilities and prediction accuracies, offering a probabilistic perspective on the predictive utility of memory that is complementary to classical information-theoretic approaches.
