发表机构
Hebrew University of Jerusalem; Tel Aviv University（耶路撒冷希伯来大学; 特拉维夫大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
研究有限马尔可夫链演化状态预测的信息价值，证明平稳初始分布下最优终端价值随观测次数非递减且指数收敛，引入预测学习指数，非平稳初始分布下价值或随时间范围递减。
AI 中文摘要
我们研究有限马尔可夫链演化状态预测中的信息价值。在每一个阶段，决策者选择一个状态，仅观察马尔可夫链的当前状态是否与所选状态匹配；所获得的信息用于对问题最终阶段的状态进行预测。我们证明，当马尔可夫链从平稳分布（invariant distribution）出发时，最优终端价值随观测次数非递减，并以均匀指数速率收敛。我们引入预测学习指数（predictive learning index），该指数用于衡量在有限次观测后是否已提取所有可获得的信息价值，且证明该指数可存在有限值或无限值。相比之下，对于非平稳初始分布，该价值可能随时间范围严格递减。
英文摘要
We study the value of information in predicting the evolving state of a finite Markov chain. At each stage, a decision maker chooses a state and observes only whether the current state of the chain matches her choice; the resulting information is used to make a prediction on the state at the final stage of the problem. We show that, when the chain starts from an invariant distribution, the optimal terminal value is non-decreasing with the number of observations and converges at a uniform exponential rate. We introduce the predictive learning index, which measures whether all attainable informational value is extracted after finitely many observations, and show that both finite and infinite indices may occur. In contrast, for nonstationary initial distributions, the value may strictly decrease with the horizon.
