The Value of Information for Populations in Varying Environments (1010.5092v1)

Abstract: The notion of information pervades informal descriptions of biological systems, but formal treatments face the problem of defining a quantitative measure of information rooted in a concept of fitness, which is itself an elusive notion. Here, we present a model of population dynamics where this problem is amenable to a mathematical analysis. In the limit where any information about future environmental variations is common to the members of the population, our model is equivalent to known models of financial investment. In this case, the population can be interpreted as a portfolio of financial assets and previous analyses have shown that a key quantity of Shannon's communication theory, the mutual information, sets a fundamental limit on the value of information. We show that this bound can be violated when accounting for features that are irrelevant in finance but inherent to biological systems, such as the stochasticity present at the individual level. This leads us to generalize the measures of uncertainty and information usually encountered in information theory.

Summary of "The Value of Information for Populations in Varying Environments"

The paper "The Value of Information for Populations in Varying Environments" authored by Olivier Rivoire and Stanislas Leibler presents a rigorous analysis of population dynamics through the lens of information theory. Within this framework, the authors address the challenge of quantifying the value of information in biological systems—a topic often only qualitatively discussed due to the elusive nature of concepts like fitness.

Theoretical Framework

Rivoire and Leibler develop a model that likens population dynamics to investment strategies in financial markets. In this model, the population (analogous to a financial portfolio) faces environmental uncertainty akin to market fluctuations. Unlike its financial counterpart, biological systems exhibit inherent stochasticity at the individual level, which can potentially lead to a breach of the mutual information bounds traditionally seen as fundamental in financial contexts.

Mathematical Approach

Central to the paper is a mathematical approach that focuses on deriving a meaningful measure of information through population growth rates under varying environmental conditions. The authors explore the concept of mutual information from Shannon's communication theory, proposing that the mutual information sets a fundamental limit on the utility of information. They derive conditions where these bounds are violated due to biological factors like stochastic variation in individual organism responses.

Numerical Results and Key Findings

The paper particularly emphasizes the role of stochasticity, information inheritance, and acquisition from the environment on evolutionary strategies. The authors demonstrate that the stochastic nature and distributed information processing in biological populations significantly alter the dynamics compared to centralized financial systems. Numerical results indicate that violations of the standard entropy and mutual information bounds can occur, suggesting that populations can gain information beyond what individual members perceive.

Implications and Future Directions

Practically, these insights imply that biological organisms might evolve to exploit forms of informational advantage not easily quantifiable by classical measures. Theoretically, the paper lays the groundwork for extending information-theoretic concepts to other types of dynamical systems and raises profound questions about information processing in distributed, stochastic systems. As such, it proposes a more nuanced understanding of "fitness" in fluctuating environments, where classical entropy bounds do not always hold.

By bridging concepts from biology, information theory, and finance, the research opens new avenues for future explorations into the informational basis of life, suggesting that further refining these models could elucidate evolution and control mechanisms in both natural and synthetic biological systems.