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The Probability Estimation Framework involves direct estimation of theprobability of occurrences of outcomes conditioned on measurement settings andside information. It is a powerful tool for certifying randomness in quantumnon-locality experiments. In this paper, we present a self-contained proof ofthe asymptotic optimality of the method. Our approach refines earlier resultsto allow a better characterisation of optimal adversarial attacks on theprotocol. We apply these results to the (2,2,2) Bell scenario, obtaining ananalytic characterisation of the optimal adversarial attacks bound byno-signalling principles, while also demonstrating the asymptotic robustness ofthe PEF method to deviations from expected experimental behaviour. We alsostudy extensions of the analysis to quantum-limited adversaries in the (2,2,2)Bell scenario and no-signalling adversaries in higher $(n,m,k)$ Bell scenarios.

Asymptotically Optimal Adversarial Strategies for the Probability Estimation Framework