# Using stacking to average Bayesian predictive distributions

## TL;DR

This paper extends stacking methods to combine Bayesian predictive distributions, improving model averaging in complex settings using proper scoring rules, Pareto smoothed importance sampling, and regularization, with empirical validation showing its advantages.

## Contribution

It generalizes stacking to predictive distributions with proper scoring rules, offering a more robust alternative to traditional Bayesian model averaging in the M-open setting.

## Key findings

- Stacking of predictive distributions outperforms BMA and AIC-based methods.
- Pareto smoothed importance sampling enables efficient leave-one-out computations.
- BB-pseudo-BMA offers a computationally cheaper alternative with comparable performance.

## Abstract

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.