Empirical Risk Minimization (ERM) only utilizes the loss function defined for the task and is completely agnostic about sampling distributions. Thus it only covers half of the story. Furthermore, ERM is equivalent to Bayesian decision theory with a particular choice of prior.
The connection between frequentist methods and the Dirichlet process is explored further in the paper Bayesian nonparametric predictive inference and bootstrap techniques by P. Muliere and P. Secchi. It shows that classical bootstrap techniques are equivalent to assuming a Dirichlet process prior.