Constrained Polynomial Likelihood

Starting from a distribution $z$, we develop a non-negative polynomial minimum-norm likelihood ratio $\xi$ such that $dp=\xi dz$ satisfies a certain type of shape restrictions. The coefficients of the polynomial are the unique solution of a mixed conic semi-definite program. The approach is widely applicable. For example, it can be used to incorporate expert opinion into a model, or as an objective function in machine learning algorithms.