  The widely recommended procedure of Bayesian model averaging is flawed in the
M-open setting in which the true data-generating process is not one of the
candidate models being fit. We take the idea of stacking from the point
estimation literature and generalize to the combination of predictive
distributions, extending the utility function to any proper scoring rule, using
Pareto smoothed importance sampling to efficiently compute the required
leave-one-out posterior distributions and regularization to get more stability.
We compare stacking of predictive distributions to several alternatives:
stacking of means, Bayesian model averaging (BMA), pseudo-BMA using AIC-type
weighting, and a variant of pseudo-BMA that is stabilized using the Bayesian
bootstrap. Based on simulations and real-data applications, we recommend
stacking of predictive distributions, with BB-pseudo-BMA as an approximate
alternative when computation cost is an issue.
