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The quantitative analysis of dynamic PET data to obtain kinetic constants in compartmental models involves the use of non-linear weighted least squares regression. Current estimation techniques often have poor mean square error estimation properties. Ridge regression is a technique for improving parameter estimation accuracy in ordinary linear regression. This technique has been found to have potential for improving mean square error when adapted to the nonlinear PET estimation problem. The effectiveness of ridge regression in this context, however, relies heavily on the correct selection of an unknown biasing parameter and the precise specification of a penalty function. In this study, an approach is explored for improving the effectiveness of ridge regression by incorporation of more rigorous Bayesian formulations for specification of the ridge penalty function. Using a variance component model, a prior covariance for the ridge penalty term is developed. Application of the resulting ridge estimation technique shows that the use of the Bayesian formulation for the penalty can reduce current ridge regression parameter loss by up to 19%. An adaptive approach to the selection of the biasing parameter is also developed.