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

TL;DR

This paper explores the role of directed information in causality-constrained problems across portfolio optimization, data compression, and hypothesis testing, providing bounds, characterizations, and new measures.

Contribution

It establishes directed information as a key measure for causality in various information-theoretic and statistical contexts, introducing directed lautum information.

Findings

Directed information bounds portfolio growth with causal side info.

Characterizes causal side information in data compression.

Quantifies causal influence in hypothesis testing.

Abstract

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.