Calibrating Meta-Learned Feature Maps for Pooled Measurement Models

Amortized Bayesian experimental design and context-adaptation methods learn a shared model across many systems, then identify a per-system latent quantity from a handful of observations at deployment. Prior work in this lineage, to our knowledge, assumes each observation directly measures the quantity being modeled. We study a setting where this assumption fails by construction: each observation is a pooled measurement, formed by summing an unknown pairwise interaction law over an experimenter-chosen subset of nodes in a relational system, the same observation model used in quantitative group testing. We show that a shared feature map trained under this pooled model is subject to an identifiability failure that pooling amplifies relative to the pointwise case: training can converge to a reparametrization of the true interaction law that fits the data equally well but is misaligned with the fixed prior used for exact Bayesian inference, producing systematic recovery error even when the feature map's functional form is otherwise correct. We derive the exact condition under which a reparametrization is harmless, and show that the pooled training objective constrains the gauge only weakly: the evidence is invariant to orthogonal transformations and, under sum-pooling, a per-pair signal-to-interference argument suggests diminishing signal against non-orthogonal drift as pool size grows, so finite training leaves ill-conditioned reparametrizations in practice. We mitigate this with a calibration anchor: known ground truth on a small subset of training-system pairs, used to directly constrain the feature map during meta-training rather than the inference-time prior. An orthogonality-penalized variant of the calibration anchor further improves interaction-law recovery accuracy over the baseline at most acquisition budgets tested, on Kuramoto and Winfree coupling.