  The kernel-based method has been successfully applied in linear system
identification using stable kernel designs. From a Gaussian process
perspective, it automatically provides probabilistic error bounds for the
identified models from the posterior covariance, which are useful in robust and
stochastic control. However, the error bounds require knowledge of the true
hyperparameters in the kernel design and are demonstrated to be inaccurate with
estimated hyperparameters for lightly damped systems or in the presence of high
noise. In this work, we provide reliable quantification of the estimation error
when the hyperparameters are unknown. The bounds are obtained by first
constructing a high-probability set for the true hyperparameters from the
marginal likelihood function and then finding the worst-case posterior
covariance within the set. The proposed bound is proven to contain the true
model with a high probability and its validity is verified in numerical
simulation.
