A robust Bayesian and game-theoretic framework for certifying AI-enabled safety-critical systems under structural misspecification

Artificial intelligence is entering safety-critical systems that regulators must certify before deployment. Rare modes may remain dormant, leaving failures partly unknown. We determine how much testing suffices and when certification is justified. We develop a robust Bayesian decision framework whose ambiguity set of inferential regimes spans up to fifty-four priors, likelihood learning rates and three reliability-growth model classes, embedded in a Stackelberg game between regulator and developer. On the real NTDS failure data the robust rule recommends release after 1,600 additional days beyond the 250-day campaign under an eighteen-member prior set, against 905 under the standard Bayesian rule and 1,065 under minimax regret; widening that layer to fifty-four members gives 1,910 days. We distinguish two quantities previously both called coverage: conservatism rate CR and credible-interval coverage CIC. In Monte Carlo simulations with a dormant mode the standard approach underestimates residual faults 8.5-fold, with CIC of 1.0% (SE 0.7%). Prior robustness leaves CR at 0.5% (SE 0.5%). A uniformly more optimistic model member changes nothing; a structurally richer change-point member raises CR to 70% (SE 4.6%). Likelihood tempering raises CR to 86% (SE 2.5%) at a learning-rate floor of 0.2, but rises monotonically to 100% as the floor falls to 0.1: the floor, not the method, is the operative dial. Across sixty campaigns the mandated robust rule reduces median societal loss by 94% under a threshold grid capped at 0.97, and by 76% when extended. Regulators gain three levers: the learning-rate floor, the certification threshold and audit intensity.