Variational message passing for skew t regression

We extend recent work concerning variational approximations via message passing to accommodate approximate fitting and inference for skew t regression models. Derivation of variational message passing is challenging owing to the presence of non-standard exponential families and numerical integration being needed. Nevertheless, the factor graph fragment approach means that algorithm updates only need to be derived once for a particular response model, which can be integrated in an arbitrarily complex model. Another advantage of our work is that all skew t parameters are inferred, rather than being held fixed. Furthermore, we show that posterior dependence arising in an auxiliary variable representation of a skew t model may lead to poor performances in terms of variational message passing approximation when using simple auxiliary variable representations of the likelihood fragment and convenient factorizations of the approximating densities.