Active Prediction in Dynamical Systems

Using a hidden Markov model (HMM) that describes the position of a damped stochastic harmonic oscillator as a stimulus input to a data processing system, we consider the optimal response of the system when it is targeted to predict the coming stimulus at a time shift later. We quantify the predictive behavior of the system by calculating the mutual information (MI) between the response and the stimulus of the system. For a passive sensor, the MI typically peaks at a negative time shift considering the processing delay of the system. Using an iterative approach of maximum likelihood for the predictive response, we show that the MI can peak at a positive time shift, which signifies the functional behavior of active prediction. We find the phenomena of active prediction in bullfrog retinas capable of producing omitted stimulus response under periodic pulse stimuli, by subjecting the retina to the same HMM signals encoded in the pulse interval. We confirm that active prediction requires some hidden information to be recovered and utilized from the observation of past stimulus by replacing the HMM with a Ornstein–Uhlenbeck process, which is strictly Markovian, and showing that no active prediction can be observed.