We study mixed convex exponential families obtained via a mixed parametrization with convex restrictions on the component corresponding to the mean value parameter in combination with convex constraints on the canonical parameter component. Exploiting the variation independence of the parameter components, we introduce a dual mixed estimator (DME) determined by the solution of two associated convex potimization problems. We show that the DME has the same asymptotic distribution as the maximum likelihood estimator.

Examples of mixed convex exponential families include locally associated Gaussian graphical models, coloured graphical models with patterned restrictions on the covariances, and variants of classical models for contingency tables with positivity restrictions. The lecture is based on joint work with Piotr Zwiernik. See also arXiv:2008.04688.