Title: Pushing the bounds of dropout. Abstract: We push on the boundaries of our knowledge about dropout by showing theoretically that dropout training can be understood as performing MAP estimation concurrently for an entire family of conditional models whose objectives are themselves lower bounded by the original dropout objective. This discovery allows us to pick any model from this family after training, which leads to a substantial improvement on regularisation-heavy language modelling. The family includes models that compute a power mean over the sampled dropout masks, and their less stochastic subvariants with tighter and higher lower bounds than the fully stochastic dropout objective. The deterministic subvariant's bound is equal to its objective, and the highest amongst these models. It also exhibits the best model fit in our experiments. Together, these results suggest that the predominant view of deterministic dropout as a good approximation to MC averaging is misleading. Rather, deterministic dropout is the best available approximation to the true objective.