Semiparametric posterior limits

We review the Bayesian theory of semiparametric inference following Bickel and Kleijn (2012) and Kleijn and Knapik (2013). After an overview of efficiency in parametric and semiparametric estimation problems, we consider the Bernstein-von Mises theorem (see, e.g., Le Cam and Yang (1990)) and generalize it to (LAN) regular and (LAE) irregular semiparametric estimation problems. We formulate a version of the semiparametric Bernstein-von Mises theorem that does not depend on least-favourable submodels, thus bypassing the most restrictive condition in the presentation of Bickel and Kleijn (2012). The results are applied to the (regular) estimation of the linear coefficient in partial linear regression (with a Gaussian nuisance prior) and of the kernel bandwidth in a model of normal location mixtures (with a Dirichlet nuisance prior), as well as the (irregular) estimation of the boundary of the support of a monotone family of densities (with a Gaussian nuisance prior).Comment: 47 pp., 1 figure, submitted for publication. arXiv admin note: substantial text overlap with arXiv:1007.017

Convergence of the Exponentiated Gradient Method with Armijo Line Search

Consider the problem of minimizing a convex differentiable function on the probability simplex, spectrahedron, or set of quantum density matrices. We prove that the exponentiated gradient method with Armjo line search always converges to the optimum, if the sequence of the iterates possesses a strictly positive limit point (element-wise for the vector case, and with respect to the Lowner partial ordering for the matrix case). To the best our knowledge, this is the first convergence result for a mirror descent-type method that only requires differentiability. The proof exploits self-concordant likeness of the log-partition function, which is of independent interest.Comment: 18 page