This article describes a full Bayesian treatment for simultaneous fixed-effect selection and parameter estimation in high-dimensional generalized linear mixed models. The approach consists of using a Bayesian adaptive Lasso penalty for signal-level adaptive shrinkage and a fast Variational Bayes scheme for estimating the posterior mode of the coefficients. The proposed approach offers several advantages over the existing methods, for example, the adaptive shrinkage parameters are automatically incorporated, no Laplace approximation step is required to integrate out the random effects. The performance of our approach is illustrated on several simulated and real data examples. The algorithm is implemented in the R package glmmvb and is made available online.

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Let X and Y be separable Banach spaces and E be a subset of X. By a random operator from X into Y we mean a rule Φ that assigns to each element x ∈ X a unique Yvalued random variable Φx. Taking into account many circumstances in which the inputs are also subject to the influence of a random environment there arises the need to define the action of Φ on some random inputs. In this paper, some procedures for extending random mappings will be proposed. Some conditions under which a random mapping can be extended to apply to all X-valued random variables will be presented.Key words. Random mappings; random operators; action on random inputs; extension of the domain of a random mapping.

Using density-matrix theory, an analytical expression of the self-Kerr nonlinear coefficient of a three-level lambda EIT medium for a weak probe light is derived. Influences of the coupling light and Doppler broadening on the self-Kerr coefficient are investigated and compared to experimental observation with a good agreement. The self-Kerr nonlinearity is basically modified and greatly enhanced in the spectral region corresponding to EIT transparent window. Furthermore, sign, slope, and magnitude of the self-Kerr coefficient can be controlled with frequency and intensity of the coupling light and temperature. Such controllable Kerr nonlinearity can find interesting applications in optoelectronic devices working with low-light intensity.

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