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The quantitative analysis of dynamic positron emission tomography (PET) data to obtain kinetic constants in compartmental models involves the use of nonlinear weighted least squares regression. Current estimation techniques often have poor mean square error estimation properties. Ridge regression is a technique that 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. An adaptive approach to the selection of the biasing parameter is also evaluated. The adaptive selection of the biasing parameter was not shown to improve estimation over more standard ridge estimation techniques. Ridge regression with the Bayesian formulation for the penalty, however, reduces current ridge regression parameter loss by up to 16% when the penalty closely reflects the true kinetic parameter covariance structure and performs comparably to the current method when the penalty does not.