i will introduce some recent progress towards understanding the scalability of markov chain monte carlo (mcmc) methods and their comparative advantage with respect to variational inference. i will discuss an optimization perspective on the infinite dimensional probability space, where mcmc leverages stochastic sample paths while variational inference projects the probabilities onto a finite dimensional parameter space. three ingredients will be the focus of this discussion: non-convexity, acceleration, and stochasticity. this line of work is motivated by epidemic prediction, where we need uncertainty quantification for credible predictions and informed decision making with complex models and evolving data.