Uncertainty of the second order

We study the stochastic dynamics of the quasispecies model with inverse Bayesian inference under environmental uncertainty. Inverse Bayesian inference is introduced through the correspondence between Bayesian inference and the replicator equation. We consider environmental uncertainty that is not modeled as the stochastic fitness called uncertainty of the second order. This is in contrast to uncertainty of the first order that can be subsumed by the stochastic fitness. The difference between these two kinds of uncertainty is discussed in the framework of categorical Bayesian probability theory. We analytically show that if the time scale of inverse Bayesian inference is sufficiently larger than that of Bayesian inference, then the quasispecies model exhibits a noise-induced transition. The theoretical result is verified by a numerical simulation.