  Models of discrete-valued outcomes are easily misspecified if the data
exhibit zero-inflation, overdispersion or contamination. Without additional
knowledge about the existence and nature of this misspecification, model
inference and prediction are adversely affected. Here, we introduce a robust
discrepancy-based Bayesian approach using the Total Variation Distance (TVD).
In the process, we address and resolve two challenges: First, we study
convergence and robustness properties of a computationally efficient estimator
for the TVD between a parametric model and the data-generating mechanism.
Second, we provide an efficient inference method adapted from Lyddon et al.
(2019) which corresponds to formulating an uninformative nonparametric prior
directly over the data-generating mechanism. Lastly, we empirically demonstrate
that our approach is robust and significantly improves predictive performance
on a range of simulated and real world data.
