Bayesian stats actually let you integrate stochasticity into deterministic models fairly easily, in a way you can't really do with frequentist stats. Bayesian methods are common for geophysical inverse modelling, for example. Probabilistic programming does this, for example. Exact Bayesian inference is impossible in the general case, but approximate inference often works well.

Subjective priors are one of the main advantages of Bayesian stats. Regularization used in ML corresponds to using subjective priors, for example. L2 regularization finds the MAP with a normal prior, favoring parsimonious solutions.