We present a novel method for approximate inference. Using some of the constructs from expectation propagation (EP), we derive a lower bound of the marginal likelihood in a similar fashion to variational Bayes (VB). The method combines some of the benefits of VB and EP: it can be used with light-tailed likelihoods (where traditional VB fails), and it provides a lower bound on the marginal likelihood. We apply the method to Gaussian process classification, a situation where the Kullback-Leibler divergence minimized in traditional VB can be infinite, and to robust Gaussian process regression, where the inference process is dramatically simplified in comparison to EP. Code to reproduce all the experiments can be found at github.com/SheffieldML/TVB.