The paper is concerned with unconstrained and constrained regularization of highly ill-posed problems in the form of Fredholm integral equations of the first kind. In the first part of the paper unconstrained method of Maximum likelihood is used to find the optimal value of the regularization parameter. In the second part constrained Maximum likelihood is used on the same test problems in order to compare the efficiency of the two methods. Comparison of the methods is established through Tables of results and computer diagrams. Highly improperly posed problems available in the literature are tested by the methods.