  Scientists use imaging to identify objects of interest and infer properties
of these objects. The locations of these objects are often measured with error,
which when ignored leads to biased parameter estimates and inflated variance.
Current measurement error methods require an estimate or knowledge of the
measurement error variance to correct these estimates, which may not be
available. Instead, we create a spatial Bayesian hierarchical model that treats
the locations as parameters, it using the image itself to incorporate
positional uncertainty. We lower the computational burden by approximating the
likelihood using a non-contiguous block design around the object locations. We
apply this model in a materials science setting to study the relationship
between the chemistry and displacement of hundreds of atom columns in crystal
structures directly imaged via scanning transmission electron microscopy.
Greater knowledge of this relationship can lead to engineering materials with
improved properties of interest. We find strong evidence of a negative
relationship between atom column displacement and the intensity of neighboring
atom columns, which is related to the local chemistry. A simulation study shows
our method corrects the bias in the parameter of interest and drastically
improves coverage in high noise scenarios compared to non-measurement error
models.
