  Spatial count data models are used to explain and predict the frequency of
phenomena such as traffic accidents in geographically distinct entities such as
census tracts or road segments. These models are typically estimated using
Bayesian Markov chain Monte Carlo (MCMC) simulation methods, which, however,
are computationally expensive and do not scale well to large datasets.
Variational Bayes (VB), a method from machine learning, addresses the
shortcomings of MCMC by casting Bayesian estimation as an optimisation problem
instead of a simulation problem. Considering all these advantages of VB, a VB
method is derived for posterior inference in negative binomial models with
unobserved parameter heterogeneity and spatial dependence. P\'olya-Gamma
augmentation is used to deal with the non-conjugacy of the negative binomial
likelihood and an integrated non-factorised specification of the variational
distribution is adopted to capture posterior dependencies. The benefits of the
proposed approach are demonstrated in a Monte Carlo study and an empirical
application on estimating youth pedestrian injury counts in census tracts of
New York City. The VB approach is around 45 to 50 times faster than MCMC on a
regular eight-core processor in a simulation and an empirical study, while
offering similar estimation and predictive accuracy. Conditional on the
availability of computational resources, the embarrassingly parallel
architecture of the proposed VB method can be exploited to further accelerate
its estimation by up to 20 times.
