Measurement-Induced Decision Loss: A feasible-set theory of microbiome intervention under observation loss

Microbiome measurements are commonly judged by how faithfully they reconstruct an underlying biological profile, using taxonomic similarity, prediction accuracy, or compositional distance. Yet faithfulness and usefulness are not the same: a distorted measurement may still support the correct intervention, while a small error can be decisive when it falls near the boundary between two treatment choices. We therefore evaluate measurement quality by whether information loss changes the decisions and outcomes that remain attainable. We treat the intervention-relevant state of the gut community as a hidden quantity seen only through an imperfect, information losing measurement, mapped by a decision rule to a graded intervention. Each decision rule is evaluated by net host benefit and ecological margin, generating a feasible set of attainable outcome pairs. We define the measurement-induced decision loss (MIDL) as the maximum net host benefit lost under imperfect measurement relative to perfect observation. For scalar affine information loss, increasing degradation is a Blackwell garbling; under post-processing closure, more informative measurement weakly expands the feasible set and cannot reduce optimal host benefit. In the binary-state model, MIDL increases linearly while an observation-dependent decision rule remains optimal and plateaus once the optimal policy becomes signal-independent. We extend the framework to finite states, state-specific loss, multi-segment stool aggregation, and ecological-margin constraints. The framework distinguishes measurement faithfulness from decision usefulness and provides a principled basis for comparing stool and richer measurements by the interventions and trade-offs they support.