Geographically weighted regression (GWR) models handle geographical dependence through a spatially varying coefficient model and have been widely used in applied science, but its general Bayesian extension is unclear because it involves a weighted log-likelihood which does not imply a probability distribution on data. We present a Bayesian GWR model and show that its essence is dealing with partial misspecification of the model. Current modularized Bayesian inference models accommodate partial misspecification from a single component of the model. We extend these models to handle partial misspecification in more than one component of the model, as required for our Bayesian GWR model. Information from the various spatial locations is manipulated via a geographically weighted kernel and the optimal manipulation is chosen according to a Kullback–Leibler (KL) divergence. We justify the model via an information risk minimization approach and show the consistency of the proposed estimator in terms of a geographically weighted KL divergence.
Yang Liu was supported by a Cambridge International Scholarship from the Cambridge Commonwealth, European and International Trust. Robert J. B. Goudie was funded by the UK Medical Research Council [programme code MC_UU_00002/2].
The data analysed and the code used in this study are freely available at https://github.com/MathBilibili/Bayesian-geographically-weighted-regression.
For the purpose of open access, the author has applied a Creative Commons Attribution (CC BY) licence to any Author Accepted Manuscript version arising.
"Generalized Geographically Weighted Regression Model within a Modularized Bayesian Framework." Bayesian Anal. Advance Publication 1 - 36, 2023. https://doi.org/10.1214/22-BA1357