  We consider the identification of large-scale linear and stable dynamic
systems whose outputs may be the result of many correlated inputs. Hence,
severe ill-conditioning may affect the estimation problem. This is a scenario
often arising when modeling complex physical systems given by the
interconnection of many sub-units where feedback and algebraic loops can be
encountered. We develop a strategy based on Bayesian regularization where any
impulse response is modeled as the realization of a zero-mean Gaussian process.
The stable spline covariance is used to include information on smooth
exponential decay of the impulse responses. We then design a new Markov chain
Monte Carlo scheme that deals with collinearity and is able to efficiently
reconstruct the posterior of the impulse responses. It is based on a variation
of Gibbs sampling which updates possibly overlapping blocks of the parameter
space on the basis of the level of collinearity affecting the different inputs.
Numerical experiments are included to test the goodness of the approach where
hundreds of impulse responses form the system and inputs correlation may be
very high.
