  Scientific fields such as genomics, ecology, and political science often
collect multivariate count data. In these fields, the data are often
sufficiently noisy such that inferences regarding the total size of the
measured systems have substantial uncertainty. This uncertainty can hinder
downstream analyses, such as differential analysis in case-control studies.
There have historically been two approaches to this problem: one considers the
data as compositional and the other as counts that can be normalized. In this
article, we use the framework of partially identified models to rigorously
study the types of scientific questions (estimands) that can be answered
(estimated) using these data. We prove that satisfying Frequentist inferential
criteria is impossible for many estimation problems. In contrast, we find that
the criteria for Bayesian inference can be satisfied, yet it requires a
particular type of model called a Bayesian partially identified model. We
introduce Scale Simulation Random Variables as a flexible and computationally
efficient form of Bayesian partially identified models for analyzing these
data. We use simulations and data analysis to validate our theory.
