A Bayesian framework for regularization

Regularization looks for an interpolating function which is close to the data and also "smooth". This function is obtained by minimizing an error functional which is the weighted sum of a "fidelity term" and a "smoothness term". However, using only one set of weights does not guarantee that this function will be the MAP estimate. One has to consider all possible weights in order to find the MAP function. Also, using only one combination of weights makes the algorithm very sensitive to the data. The solution suggested here is through the Bayesian approach: a probability distribution over all weights is constructed and all weights are considered when reconstructing the function or computing the expectation of a linear functional on the function space.