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Interpreting measurements requires a physical theory, but the theory's accuracy may vary across the experimental domain. To optimize experimental design, and so to ensure that the substantial resources necessary for modern experiments are focused on acquiring the most valuable data, both the theory uncertainty and the expected pattern of experimental errors must be considered. We develop a Bayesian approach to this problem, and apply it to the example of proton Compton scattering. Chiral Effective Field Theory ($χ$EFT) predicts the functional form of the scattering amplitude for this reaction, so that the electromagnetic polarizabilities of the nucleon can be inferred from data. With increasing photon energy, both experimental rates and sensitivities to polarizabilities increase, but the accuracy of $χ$EFT decreases. Our physics-based model of $χ$EFT truncation errors is combined with present knowledge of the polarizabilities and reasonable assumptions about experimental capabilities at HI$γ$S and MAMI to assess the information gain from measuring specific observables at specific kinematics, \emph{i.e.}, to determine the relative amount by which new data are apt to shrink uncertainties. The strongest gains would likely come from new data on the spin observables $Σ_{2x}$ and $Σ_{2x^\prime}$ at $ω\simeq140$ to $200$ MeV and $40^\circ$ to $120^\circ$. These would tightly constrain $γ_{E1E1}-γ_{E1M2}$. New data on the differential cross section between $100$ and $200$\,MeV and over a wide angle range will substantially improve constraints on $α_{E1}-β_{M1}$, $γ_π$ and $γ_{M1M1}-γ_{M1E2}$. Good signals also exist around $160$ MeV for $Σ_3$ and $Σ_{2z^\prime}$. Such data will be pivotal in the continuing quest to pin down the scalar polarizabilities and refine understanding of the spin polarizabilities.