Improved log-Gaussian approximation for over-dispersed Poisson regression: application to spatial analysis of COVID-19

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.