On the limits of probabilistic forecasting in nonlinear times series analysis

The ignorance score measures the quality of probabilistic forecasting. In this paper, we study its basic properties in the perfect model scenario, i.e., under the assumption that the system producing the data is perfectly known. Two further qualifications are added to this general setting. First, the system is a discrete-time, measure-preserving dynamical system. Moreover, randomness results from the quantization of the state space (i.e., from the finite precision of the observations), rather than being introduced via observational noise. In this “non-linear” perfect model scenario we derive, in particular, the admissible domain of the ignorance score and relate it with the ignorance score in imperfect models.