  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.
