On variational approximations for frequentist and bayesian inference

Variational approximations are approximate inference techniques for complex statisticalmodels providing fast, deterministic alternatives to conventional methods that,however accurate, take much longer to run. We extend recent work concerning variationalapproximations developing and assessing some variational tools for likelihoodbased and Bayesian inference. In particular, the first part of this thesis employs a Gaussian variational approximation strategy to handle frequentist generalized linear mixedmodels with general design random effects matrices such as those including spline basisfunctions. This method involves approximation to the distributions of random effectsvectors, conditional on the responses, via a Gaussian density. The second thread isconcerned with a particular class of variational approximations, known as mean fieldvariational Bayes, which is based upon a nonparametric product density restriction on the approximating density. Algorithms for inference and fitting for models with elaborateresponses and structures are developed adopting the variational message passingperspective. The modularity of variational message passing is such that extensions tomodels with more involved likelihood structures and scalability to big datasets are relatively simple. We also derive algorithms for models containing higher level randomeffects and non-normal responses, which are streamlined in support of computationalefficiency. Numerical studies and illustrations are provided, including comparisons witha Markov chain Monte Carlo benchmark.