23.2 Efficiency for probabilistic models and algorithms
Stan programs express models which are intrinsically statistical in nature. The algorithms applied to these models may or may not themselves be probabilistic. For example, given an initial value for parameters (which may itself be given deterministically or generated randomly), Stan’s optimization algorithm (L-BFGS) for penalized maximum likelihood estimation is purely deterministic. Stan’s sampling algorithms are based on Markov chain Monte Carlo algorithms, which are probabilistic by nature at every step. Stan’s variational inference algorithm (ADVI) is probabilistic despite being an optimization algorithm; the randomization lies in a nested Monte Carlo calculation for an expected gradient.
With probabilistic algorithms, there will be variation in run times (and maybe memory usage) based on the randomization involved. For example, by starting too far out in the tail, iterative algorithms underneath the hood, such as the solvers for ordinary differential equations, may take different numbers of steps. Ideally this variation will be limited; when there is a lot of variation it can be a sign that there is a problem with the model’s parameterization in a Stan program or with initialization.
A well-behaved Stan program will have low variance between runs with different random initializations and differently seeded random number generators. But sometimes an algorithm can get stuck in one part of the posterior, typically due to high curvature. Such sticking almost always indicates the need to reparameterize the model. Just throwing away Markov chains with apparently poor behavior (slow, or stuck) can lead to bias in posterior estimates. This problem with getting stuck can often be overcome by lowering the initial step size to avoid getting stuck during adaptation and increasing the target acceptance rate in order to target a lower step size. This is because smaller step sizes allow Stan’s gradient-based algorithms to better follow the curvature in the density or penalized maximum likelihood being fit.