that's a really good question. some aspects of VI vs MCMC are areas of active research. so it's tough to respond succinctly, but i'll try. the key disadvantage

that's a really good question. some aspects of VI vs MCMC are areas of active research. so it's tough to respond succinctly, but i'll try. the key disadvantages of VI (particularly ADVI) are: 1. mean-field variational inference cannot model posterior correlations. so if you expect your model + dataset to give a "skewed" posterior, then mean-field variational inference will have a difficult time describing such a posterior. (it will under estimate marginal variances.) 2. full-rank variational inference can model posterior correlations. but it can become too expensive for big models. there is a lot of great research coming up in this vein, such as [1,2]. 3. in either case, the version of variational inference we have in Stan (ADVI) uses a normal approximation in a transformed parameter space. thus, there is an additional mismatch of the shape of the variational posterior to the full MCMC posterior. in terms of advantages: 1. variational inference is (in general) a non-convex optimization problem. so it's easy to know when we've converged to a local optimum. convergence in MCMC is a bit more tricky to assess. 2. if your model has a multi-modal posterior, then variational inference will focus on just one of the modes. this is sometimes desirable as MCMC techniques might end up jumping around all of the modes and producing poor samples. this is just the tip of the iceberg. but i hope it helps! [1] http://arxiv.org/pdf/1506.03159.pdf http://arxiv.org/pdf/1506.03159.pdf [2] http://arxiv.org/pdf/1502.07685.pdf http://arxiv.org/pdf/1502.07685.pdf