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.Comment: 12 pages, 4 figure