Interpretations of Directed Information in Portfolio Theory, Data Compression, and Hypothesis Testing

We investigate the role of 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 lead to a natural interpretation of directed information
I
(
Yn
→
Xn
) as the amount of information that a random sequence
Yn
= (
Y
₁
,
Y
₂
,...,
Yn
) causally provides about another random sequence
Xn
= (
X
₁
,
X
₂
,...,
Xn
). A new measure, directed lautum information, is also introduced and interpreted in portfolio theory, data compression, and hypothesis testing.