Interpretations of directed information in portfolio theory, data compression, and hypothesis testing

Haim H. Permuter; Young Han Kim; Tsachy Weissman

doi:10.1109/TIT.2011.2136270

Interpretations of directed information in portfolio theory, data compression, and hypothesis testing

Haim H. Permuter, Young Han Kim, Tsachy Weissman

Ben-Gurion University of the Negev

Research output: Contribution to journal › Article › peer-review

Abstract

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 to Xⁿ) as the amount of information that a random sequence Y ⁿ = (Y₁,Y₂, Y_n) causally provides about another random sequence Xⁿ = (X₁,X ₂,X_n). A new measure, directed lautum information, is also introduced and interpreted in portfolio theory, data compression, and hypothesis testing.

Original language	American English
Article number	5773045
Pages (from-to)	3248-3259
Number of pages	12
Journal	IEEE Transactions on Information Theory
Volume	57
Issue number	6
DOIs	https://doi.org/10.1109/TIT.2011.2136270
State	Published - 1 Jun 2011

Keywords

Causal conditioning
Kelly gambling
causal side information
directed information
hypothesis testing
instantaneous compression
lautum information
portfolio theory

All Science Journal Classification (ASJC) codes

Information Systems
Computer Science Applications
Library and Information Sciences

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10.1109/TIT.2011.2136270

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title = "Interpretations of directed information in portfolio theory, data compression, and hypothesis testing",

abstract = "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 to Xn) as the amount of information that a random sequence Y n = (Y1,Y2, Yn) causally provides about another random sequence Xn = (X1,X 2,Xn). A new measure, directed lautum information, is also introduced and interpreted in portfolio theory, data compression, and hypothesis testing.",

keywords = "Causal conditioning, Kelly gambling, causal side information, directed information, hypothesis testing, instantaneous compression, lautum information, portfolio theory",

author = "Permuter, {Haim H.} and Kim, {Young Han} and Tsachy Weissman",

note = "Funding Information: Manuscript received December 24, 2009; revised November 24, 2010; accepted November 27, 2010. Date of current version May 25, 2011. This work was supported in part by NSF Grant CCF-0729195 and in part by BSF Grant 2008402. H. H. Permuter was supported in part by the Marie Curie Reintegration fellowship.",

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N2 - 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 to Xn) as the amount of information that a random sequence Y n = (Y1,Y2, Yn) causally provides about another random sequence Xn = (X1,X 2,Xn). A new measure, directed lautum information, is also introduced and interpreted in portfolio theory, data compression, and hypothesis testing.

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Interpretations of directed information in portfolio theory, data compression, and hypothesis testing

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Keywords

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