Title: Accelerating Monte Carlo Bayesian Inference via Approximating Predictive Uncertainty over the Simplex. Abstract: Estimating the predictive uncertainty of a Bayesian learning model is critical in various decision-making problems, e.g., reinforcement learning, detecting adversarial attack, self-driving car. As the model posterior is almost always intractable, most efforts were made on finding an accurate approximation the true posterior. Even though a decent estimation of the model posterior is obtained, another approximation is required to compute the predictive distribution over the desired output. A common accurate solution is to use Monte Carlo (MC) integration. However, it needs to maintain a large number of samples, evaluate the model repeatedly and average multiple model outputs. In many real-world cases, this is computationally prohibitive. In this work, assuming that the exact posterior or a decent approximation is obtained, we propose a generic framework to approximate the output probability distribution induced by model posterior with a parameterized model and in an amortized fashion. The aim is to approximate the true uncertainty of a specific Bayesian model, meanwhile alleviating the heavy workload of MC integration at testing time. The proposed method is universally applicable to Bayesian classification models that allow for posterior sampling. Theoretically, we show that the idea of amortization incurs no additional costs on approximation performance. Empirical results validate the strong practical performance of our approach.