The Bayesian Origins of Growth Rates in Stochastic Environments

Abstract: Stochastic multiplicative dynamics characterize many complex natural phenomena such as selection and mutation in evolving populations, and the generation and distribution of wealth within social systems. Population heterogeneity in stochastic growth rates has been shown to be the critical driver of diversity dynamics and of the emergence of wealth inequality over long time scales. However, we still lack a general statistical framework that systematically explains the origins of these heterogeneities from the adaptation of agents to their environment. In this paper, we derive population growth parameters resulting from the interaction between agents and their knowable environment, conditional on subjective signals each agent receives. We show that average growth rates converge, under specific conditions, to their maximal value as the mutual information between the agent's signal and the environment, and that sequential Bayesian learning is the optimal strategy for reaching this maximum. It follows that when all agents access the same environment using the same inference model, the learning process dynamically attenuates growth rate disparities, reversing the long-term effects of heterogeneity on inequality. Our approach lays the foundation for a unified general quantitative modeling of social and biological phenomena such as the dynamical effects of cooperation, and the effects of education on life history choices.