Amortized Structural Inference for Media Mix Models: A Gated Architecture for Calibrated, Zero-Configuration Measurement

Media mix models (MMMs) require substantial expert configuration: adstock priors, saturation parameterizations, pooling strengths, and model-family choices are set by practitioners, and these choices materially affect the budget decisions the models inform. We ask whether that configuration can be amortized—learned once from simulation and applied in a forward pass—and, more importantly, whether the resulting inferences can be trusted when the simulator that trained the network is misspecified relative to reality. We propose a gated architecture in which the amortized posterior is never treated as a verdict but as an importance-sampling proposal audited against the exact model likelihood. A single scalar, the effective sample size (ESS) of the importance weights, routes each dataset to one of three outcomes: acceptance of reweighted draws (asymptotically exact inference at amortized cost), cheap refinement by short MCMC seeded at the proposal, or escalation to full inference with the dataset logged as a simulator gap. The same weights yield unbiased marginal-likelihood estimates, so discrete model-family selection inherits the identical safety property at no additional sampling cost. We validate the architecture on a synthetic MMM with known ground truth. Across proposal regimes spanning three orders of magnitude in ESS ratio (0.71 to 7×10⁻⁴), posterior means are statistically indistinguishable: proposal quality affects computational cost, not correctness. A confidently wrong model-family head (90% mass on the wrong adstock family) is overridden by a log Bayes factor of +20 computed from the same weights. A neural posterior estimator trained on 10,000 simulated panels attains near-nominal coverage of 80% credible intervals (0.799–0.842 across four structural parameters), recovers the saturation–scale identifiability correlation (correlation of +0.93) matching a long-run MCMC reference without supervision on that quantity, and earns acceptance on a held-out panel (ESS/n = 0.32) with no human configuration. All validation is on synthetic data whose generative process is known and shared with the training simulator; evaluation on real marketing panels with experimental ground truth remains future work. Code and prototypes are publicly available.