  In the era of open data, Poisson and other count regression models are
increasingly important. Provided this, we develop a closed-form inference for
an over-dispersed Poisson regression, especially for (over-dispersed) Bayesian
Poisson wherein the exact inference is unobtainable. The approach is derived
via mode-based log-Gaussian approximation. Unlike closed-form alternatives, it
remains accurate even for zero-inflated count data. Besides, our approach has
no arbitrary parameter that must be determined a priori. Monte Carlo
experiments demonstrate that the estimation error of the proposed method is a
considerably smaller estimation error than the closed-form alternatives and as
small as the usual Poisson regressions. We obtained similar results in the case
of Poisson additive mixed modeling considering spatial or group effects. The
developed method was applied for analyzing COVID-19 data in Japan. This result
suggests that influences of pedestrian density, age, and other factors on the
number of cases change over periods.
