Meta-Uncertainty in Bayesian Model Comparison
This work addresses the issue of uncertain model rankings in BMC for researchers and practitioners, representing an incremental improvement by integrating existing techniques into a novel framework.
The paper tackles the problem of uncertainty in Bayesian model comparison (BMC) by conceptualizing meta-uncertainty in posterior model probabilities (PMPs) and developing a probabilistic framework to quantify it, enhancing BMC workflows across various inference methods.
Bayesian model comparison (BMC) offers a principled probabilistic approach to study and rank competing models. In standard BMC, we construct a discrete probability distribution over the set of possible models, conditional on the observed data of interest. These posterior model probabilities (PMPs) are measures of uncertainty, but -- when derived from a finite number of observations -- are also uncertain themselves. In this paper, we conceptualize distinct levels of uncertainty which arise in BMC. We explore a fully probabilistic framework for quantifying meta-uncertainty, resulting in an applied method to enhance any BMC workflow. Drawing on both Bayesian and frequentist techniques, we represent the uncertainty over the uncertain PMPs via meta-models which combine simulated and observed data into a predictive distribution for PMPs on new data. We demonstrate the utility of the proposed method in the context of conjugate Bayesian regression, likelihood-based inference with Markov chain Monte Carlo, and simulation-based inference with neural networks.