Parental refusal and delay of childhood vaccination has increased in recent years in the United States. This phenomenon challenges maintenance of herd immunity and increases the risk of outbreaks of vaccine-preventable diseases. We examine US county-level vaccine refusal for patients under five years of age collected during the period 2012-2015 from an administrative healthcare dataset. We model these data with a Bayesian zero-inflated negative binomial regression model to capture social and political processes that are associated with vaccine refusal, as well as factors that affect our measurement of vaccine refusal. Our work highlights fine-scale socio-demographic characteristics associated with vaccine refusal nationally, finds that spatial clustering in refusal can be explained by such factors, and has the potential to aid in the development of targeted public health strategies for optimizing vaccine uptake.

Models with intractable normalizing functions have numerous applications ranging from network models to image analysis to spatial point processes. Because the normalizing constants are functions of the parameters of interest, standard Markov chain Monte Carlo cannot be used for Bayesian inference for these models. A number of algorithms have been developed for such models. Some have the posterior distribution as the asymptotic distribution. Other "asymptotically inexact" algorithms do not possess this property. There is limited guidance for evaluating approximations based on these algorithms, and hence it is very hard to tune them. We propose two new diagnostics that address these problems for intractable normalizing function models. Our first diagnostic, inspired by the second Bartlett identity, applies in principle to any asymptotically exact or inexact algorithm. We develop an approximate version of this new diagnostic that is applicable to intractable normalizing function problems. Our second diagnostic is a Monte Carlo approximation to a kernel Stein discrepancy-based diagnostic introduced by Gorham and Mackey (2017). We provide theoretical justification for our methods. We apply our diagnostics to several algorithms in the context of challenging simulated and real data examples, including an Ising model, an exponential random graph model, and a Markov point process.

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