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Deconvolution is the process of estimating a system's input using measurements of a causally related output where the relationship between the input and output is known and linear. Regularization parameters are used to balance smoothness of the estimated input with accuracy of the measurement values. In this paper we present a maximum marginal likelihood method for estimating unknown regularization (and other) parameters used during deconvolution of dynamical systems. Our computational approach uses techniques that were developed for Kalman filters and smoothers. As an example application we consider estimating insulin secretion rate (ISR) following an intravenous glucose stimulus. This procedure is referred to in the medical literature as an intravenous glucose tolerance test (IVGTT). This estimation problem is difficult because ISR is a strongly non-stationary signal; it presents a fast peak in the first minutes of the experiment, followed by a smoother release. We use three regularization parameters to define a smooth model for ISR variance. This model takes into account the rapid variation of ISR during the beginning of an IVGTT and its slower variation as time progresses. Simulations are used to assess marginal likelihood estimation of these regularization parameters as well as of other parameters in the system. Simulations are also used to compare our model for ISR variance with other stochastic ISR models. In addition, we apply maximum marginal likelihood and our ISR variance model to real data that have previous ISR estimation results reported in the literature.

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