投资组合理论、数据压缩和假设检验中有向信息的解释
Interpretations of Directed Information in Portfolio Theory, Data Compression, and Hypothesis Testing
摘要：
我们研究了梅西的有向信息在投资组合理论、数据压缩和因果约束统计中的作用。特别是，我们表明，有向信息是在股票市场中的最优投资组合的增长率由于{因果}边信息的增量上界。这个上限对于赛马赌博来说是紧的，这是股票市场的一个极端情况。有向信息还表征了瞬时压缩中{因果}边信息的值，并量化了两个随机过程联合压缩中{因果}推理的好处。在假设检验中，有向信息评估用于检验随机过程$Y$ {causally}是否影响另一过程$X$的最佳误差指数。这些结果给出了有向信息$I(Y^n \to X^n)$作为随机序列$Y^n = (Y_1,Y_2,..., Y_n)$ {causally}提供的关于另一随机序列$X^n = (X_1,X_2,...,X_n)$的信息量的自然解释。一个新的措施，{ \em定向lutum信息}，也被引入和解释投资组合理论，数据压缩，和假设检验。
英文摘要：
We investigate the role of Massey's directed information in portfolio theory, data compression, and statistics with causality constraints. In particular, we show that directed information is an upper bound on the increment in growth rates of optimal portfolios in a stock market due to {causal} side information. This upper bound is tight for gambling in a horse race, which is an extreme case of stock markets. Directed information also characterizes the value of {causal} side information in instantaneous compression and quantifies the benefit of {causal} inference in joint compression of two stochastic processes. In hypothesis testing, directed information evaluates the best error exponent for testing whether a random process $Y$ {causally} influences another process $X$ or not. These results give a natural interpretation of directed information $I(Y^n \to X^n)$ as the amount of information that a random sequence $Y^n = (Y_1,Y_2,..., Y_n)$ {causally} provides about another random sequence $X^n = (X_1,X_2,...,X_n)$. A new measure, {\em directed lautum information}, is also introduced and interpreted in portfolio theory, data compression, and hypothesis testing.
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