Confidence Regions for Simulations with Learned Probabilistic Models

Due to the growing amount of data and processing capabilities, machine learning techniques are increasingly applied for the identification of dynamical systems. Especially probabilistic methods are promising for learning models, which in turn are frequently used for simulations. Although confidence regions around predicted trajectories are of crucial importance in many control approaches, few rigorous mathematical analysis methods are available for learned probabilistic models. Therefore, we propose a novel method to estimate confidence regions for predicted trajectories, and assign them a confidence level based on Monte Carlo random trajectory sampling. Since the confidence level has a strongly nonlinear dependence on the number of Monte Carlo samples, we derive a lower bound on the number of samples that ensures a desired minimum confidence level. The efficiency and flexibility of the proposed method is demonstrated in simulations of a Bayesian hidden Markov model and a Gaussian process state space model.