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When working with multimodal Bayesian posterior distributions, Markov chainMonte Carlo (MCMC) algorithms have difficulty moving between modes, and defaultvariational or mode-based approximate inferences will understate posterioruncertainty. And, even if the most important modes can be found, it isdifficult to evaluate their relative weights in the posterior. Here we proposean approach using parallel runs of MCMC, variational, or mode-based inferenceto hit as many modes or separated regions as possible and then combine theseusing Bayesian stacking, a scalable method for constructing a weighted averageof distributions. The result from stacking efficiently samples from multimodalposterior distribution, minimizes cross validation prediction error, andrepresents the posterior uncertainty better than variational inference, but itis not necessarily equivalent, even asymptotically, to fully Bayesianinference. We present theoretical consistency with an example where the stackedinference approximates the true data generating process from the misspecifiedmodel and a non-mixing sampler, from which the predictive performance is betterthan full Bayesian inference, hence the multimodality can be considered ablessing rather than a curse under model misspecification. We demonstratepractical implementation in several model families: latent Dirichletallocation, Gaussian process regression, hierarchical regression, horseshoevariable selection, and neural networks.

Stacking for Non-mixing Bayesian Computations: The Curse and Blessing of Multimodal Posteriors