MA-SBI: Computable Ceilings Decide Which Misspecification Corrector to Fit

Simulation-based inference degrades when the simulator misspecifies the process that produced the real observations, and the existing repairs ask for data the practitioner rarely has, such as ground-truth parameters paired with real observations, a functional form for the noise, or a credible alternative prior. What practitioners often do have is a written record of the conditions under which each observation was collected, such as the instruction text given to participants or the policy bulletin in force that week. We show that the share of the simulator's error this side information can remove has a ceiling that is computable in advance from held-out residuals, without evaluating any posterior, and that a second ceiling says whether the correction also needs to see the observation. We introduce misspecification-aware simulation-based inference (MA-SBI), a family of observation-space correctors that the two ceilings select between, fitted in front of a frozen amortised posterior. In the Gaussian-linear model the conditional mean provably attains both ceilings, and the fitted correction is identically zero when the simulator is well specified, however the prior behaves. A second result separates the estimators by sample complexity, since the structured one pays for the rank of the embedded design where an unstructured one pays for the width of the encoder. Across four benchmarks and three physical datasets the ceilings are tight, with every arm landing between 0.99 and 1.0 of its own, so the number tells a practitioner what an estimator will achieve before one is fitted. On a wind-tunnel benchmark a corrector that uses no parameter labels matches a calibration-based method that uses them, and we report the negative results too, including an epidemiological dataset whose apparent coverage gain does not survive our pre-registered test.