This paper is part of an ongoing investigation of "pragmatic information", defined in Weinberger (2002) as "the amount of information actually used in making a decision". Because a study of information rates led to the Noiseless and Noisy Coding Theorems, two of the most important results of Shannon's theory, we begin the paper by defining a pragmatic information rate, showing that all of the relevant limits make sense, and interpreting them as the improvement in compression obtained from using the correct distribution of transmitted symbols. The first of two applications of the theory extends the information theoretic analysis of the Kelly Criterion, and its generalization, the horse race, to a series of races where the stochastic process of winning horses, payoffs, and strategies depend on some stationary process, including, but not limited to the history of previous races. If the bettor is receiving messages (side information) about the probability distribution of winners, the doubling rate of the bettor's winnings is bounded by the pragmatic information of the messages. A second application is to the question of market efficiency. An efficient market is, by definition, a market in which the pragmatic information of the "tradable past" with respect to current prices is zero. Under this definition, markets whose returns are characterized by a GARCH(1,1) process cannot be efficient. Finally, a pragmatic informational analogue to Shannon's Noisy Coding Theorem suggests that a cause of market inefficiency is that the underlying fundamentals are changing so fast that the price discovery mechanism simply cannot keep up. This may happen most readily in the run-up to a financial bubble, where investors' willful ignorance degrade the information processing capabilities of the market.

翻译：本文是对"实用信息"持续研究的一部分，该概念由Weinberger（2002）定义为"实际用于决策的信息量"。由于对信息率的研究催生了无噪与有噪编码定理——香农理论中两个最重要的成果，本文首先定义了实用信息率，证明所有相关极限均具有明确意义，并将其解释为使用正确传输符号分布所获得的压缩改进。该理论的第一个应用将凯利准则及其推广形式（赛马模型）的信息论分析，扩展到连续赛马场景中，其中获胜马匹的随机过程、赔付率与策略均依赖于某个平稳过程（包括但不限于历史赛事记录）。若投注者接收关于获胜概率分布的消息（边信息），其收益倍增率将受限于消息的实用信息量。第二个应用针对市场效率问题。根据定义，有效市场是指"可交易历史"相对于当前价格的实用信息为零的市场。在此定义下，收益率服从GARCH(1,1)过程的市场不可能是有效的。最后，基于香农有噪编码定理构建的实用信息类比表明：市场无效性的一个根本原因是底层基本面变化过快，以致价格发现机制无法及时响应。这种现象在金融泡沫形成期尤为显著，此时投资者的蓄意漠视会削弱市场的信息处理能力。