There has been significant interest in developing new forms of acoustic model, in particular models which allow additional dependencies to be represented than those contained within a standard hidden Markov model (HMM). This paper discusses one such class of models, augmented statistical models. Here, a local exponential approximation is made about some point on a base model. This allows additional dependencies within the data to be modelled than are represented in the base distribution. Augmented models based on Gaussian mixture models (GMMs) and HMMs are briefly described. These augmented models are then related to generative kernels, one approach used for allowing support vector machines (SVMs) to be applied to variable length data. The training of augmented statistical models within an SVM, generative kernel, framework is then discussed. This may be viewed as using maximum margin training to estimate statistical models. Augmented Gaussian mixture models are then evaluated using rescoring on a large vocabulary speech recognition task.