Developing an information criterion for spatial data analysis through Bayesian generalized fused lasso

In the field of spatial data analysis, spatially varying coefficients (SVC) models, which allow regression coefficients to vary by region and flexibly capture spatial heterogeneity, have continued to be developed in various directions. Moreover, the Bayesian generalized fused lasso is often used as a method that efficiently provides estimation under the natural assumption that regression coefficients of adjacent regions tend to take the same value. In most Bayesian methods, the selection of prior distribution is an essential issue, and in the setting of SVC model with the Bayesian generalized fused lasso, determining the complexity of the class of prior distributions is also a challenging aspect, further amplifying the difficulty of the problem. For example, the widely applicable information criterion (WAIC), which has become standard in Bayesian model selection, does not target determining the complexity. Therefore, in this study, we adapted a criterion called the prior intensified information criterion (PIIC) to this setting. Specifically, under an asymptotic setting that retains the influence of the prior distribution, that is, under an asymptotic setting that deliberately does not provide selection consistency, we derived the asymptotic properties of our generalized fused lasso estimator. Then, based on these properties, we constructed an information criterion as an asymptotically bias-corrected estimator of predictive risk. In numerical experiments, we confirmed that PIIC outperforms WAIC in the sense of reducing the predictive risk, and in a real data analysis, we observed that the two criteria give rise to substantially different results.