The estimation of spatial intensity is an important inference problem in spatial epidemiologic studies. A standard data assimilation component of these studies is the assignment of a geocode, that is, point-level spatial coordinates, to the address of each subject in the study population. Unfortunately, when geocoding is performed by the standard automated method of street-segment matching to a georeferenced road file and subsequent interpolation, it is rarely completely successful. Typically, 10-30% of the addresses in the study population, and even higher percentages in particular subgroups, fail to geocode, potentially leading to a selection bias, called geographic bias, and an inefficient analysis. Missing-data methods could be considered for analyzing such data; however, because there is almost always some geographic information coarser than a point (e.g., a Zip code) observed for the addresses that fail to geocode, a coarsened-data analysis is more appropriate. This article develops methodology for estimating spatial intensity from coarsened geocoded data. Both nonparametric (kernel smoothing) and likelihood-based estimation procedures are considered. Substantial improvements in the estimation quality of coarsened-data analyses relative to analyses of only the observations that geocode are demonstrated via simulation and an example from a rural health study in Iowa.