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.