Pathfinder
Stan supports the Pathfinder algorithm Zhang et al. (2022). Pathfinder is a variational method for approximately sampling from differentiable log densities. Starting from a random initialization, Pathfinder locates normal approximations to the target density along a quasi-Newton optimization path, with local covariance estimated using the negative inverse Hessian estimates produced by the LBFGS optimizer. Pathfinder returns draws from the Gaussian approximation with the lowest estimated Kullback-Leibler (KL) divergence to the true posterior.
Stan provides two versions of the Pathfinder algorithm: single-path Pathfinder and multi-path Pathfinder. Single-path Pathfinder generates a set of approximate draws from one run of the basic Pathfinder algorithm. Multi-path Pathfinder uses importance resampling over the draws from multiple runs of Pathfinder. This better matches non-normal target densities and also mitigates the problem of L-BFGS getting stuck at local optima or in saddle points on plateaus. Compared to ADVI and short dynamic HMC runs, Pathfinder requires one to two orders of magnitude fewer log density and gradient evaluations, with greater reductions for more challenging posteriors. While the evaluations in Zhang et al. (2022) found that single-path and multi-path Pathfinder outperform ADVI for most of the models in the PosteriorDB evaluation set, we recognize the need for further experiments on a wider range of models.