Almost all fields of science rely upon statistical inference to estimate unknown parameters in theoretical and computational models. While the performance of modern computer hardware continues to grow, the computational requirements for the simulation of models are growing even faster. This is largely due to the increase in model complexity, often including stochastic dynamics, that is necessary to describe and characterize phenomena observed using modern, high resolution, experimental techniques. Such models are rarely analytically tractable, meaning that extremely large numbers of stochastic simulations are required for parameter inference. In such cases, parameter inference can be practically impossible. In this work, we present new computational Bayesian techniques that accelerate inference for expensive stochastic models by using computationally inexpensive approximations to inform feasible regions in parameter space, and through learning transforms that adjust the biased approximate inferences to closer represent the correct inferences under the expensive stochastic model. Using topical examples from ecology and cell biology, we demonstrate a speed improvement of an order of magnitude without any loss in accuracy. This represents a substantial improvement over current state-of-the-art methods for Bayesian computations when appropriate model approximations are available.