Hyperparameters Tuning in Regularized System Identification with Nonzero Prior Means

We consider the problem of identifying linear time invariant systems using regularization schemes, and address the fact that generally the mean value of the corresponding parameter prior is set to zero. We thus consider the scenario where it is beneficial to use a prior with nonzero-mean, where this mean moreover depends on some hyperparameters. We show how to construct such priors and do hyperparameter tuning by marginal likelihood, and since a parameter dependent mean may slow down optimization, we also derive an efficient and stable way of treating them, leading to an overall scheme whose leading order numerical complexity is the same as in the case where the prior mean is zero. The proposed method thus allows including new types of external information in the prior, and we exemplify how this extension can improve the existing regularization techniques.