-- Certifying safety in dynamical systems is crucial, but barrier certificates -- widely used to verify that system trajectories remain within a safe region -- typically require explicit system models. When dynamics are unknown, data-driven methods can be used instead, yet obtaining a valid certificate requires rigorous uncertainty quantification. For this purpose, existing methods usually rely on full-state measurements, limiting their applicability. This paper proposes a novel approach for synthesizing barrier certificates for unknown systems with latent states and polynomial dynamics. A Bayesian framework is employed, where a prior in state-space representation is updated using input-output data via a targeted marginal Metropolis-Hastings sampler . The resulting samples are used to construct a candidate barrier certificate through a sum-of-squares program. It is shown that if the candidate satisfies the required conditions on a test set of additional samples, it is also valid for the true, unknown system with high probability. The approach and its probabilistic guarantees are illustrated through a numerical simulation. Ensuring the safety of dynamical systems is a critical concern in applications such as human-robot interaction, autonomous driving, and medical devices, where failures can lead to severe consequences. In such scenarios, safety constraints typically mandate that the system state remains within a predefined allowable region. Barrier certificates [1] provide a systematic framework for verifying safety by establishing mathematical conditions that guarantee that system trajectories remain within these regions.

As control engineering methods are applied to increasingly complex systems, data-driven approaches for system identification appear as a promising alternative to physics-based modeling. While the Bayesian approaches prevalent for safety-critical applications usually rely on the availability of state measurements, the states of a complex system are often not directly measurable. It may then be necessary to jointly estimate the dynamics and the latent state, making the quantification of uncertainties and the design of controllers with formal performance guarantees considerably more challenging. This paper proposes a novel method for the computation of an optimal input trajectory for unknown nonlinear systems with latent states based on a combination of particle Markov chain Monte Carlo methods and scenario theory. Probabilistic performance guarantees are derived for the resulting input trajectory, and an approach to validate the performance of arbitrary control laws is presented. The effectiveness of the proposed method is demonstrated in a numerical simulation.