  Under-reporting of count data poses a major roadblock for prediction and
inference. In this paper, we focus on the Pogit model, which deconvolves the
generating Poisson process from the censuring process controlling
under-reporting using a generalized linear modeling framework. We highlight the
limitations of the Pogit model and address them by adding constraints to the
estimation framework. We also develop uncertainty quantification techniques
that are robust to model mis-specification. Our approach is evaluated using
synthetic data and applied to real healthcare datasets, where we treat
in-patient data as `reported' counts and use held-out total injuries to
validate the results. The methods make it possible to separate the Poisson
process from the under-reporting process, given sufficient expert information.
Codes to implement the approach are available via an open source Python
package.
