In object-tracking and state-estimation problems, ambiguous evidence such as imprecise measurements and the absence of detections can contain valuable information and thus be leveraged to further refine the probabilistic belief state. In particular, knowledge of a sensor’s bounded field of view (FoV) can be exploited to incorporate evidence of where an object was not observed. This paper presents a systematic approach for incorporating knowledge of the FoV geome-try, position, and object inclusion/exclusion evidence into object state densities and random finite set multiobject cardinality distributions. The resulting state estimation problem is nonlinear and solved using a new Gaussian mixture approximation based on recursive component splitting. Based on this approximation, a novel Gaussian mixture Bernoulli filter for imprecise measurements is derived and demonstrated in a tracking problem using only natural language statements as inputs. This paper also considers the relationship between bounded FoVs and cardinality distributions for a representative selection of multiobject distributions, which can be used for sensor planning, as is demonstrated through a problem involving a multi-Bernoulli process with up to 100 potential objects.