How would you know if a signal contains any predictable structure? Maybe you would look at its power spectrum. If the power spectrum is completely flat, you might conclude that the signal is purely random white noise.
Not so fast! Complex processes with high-order correlation can completely evade typical methods of detection. Check out my paper on fraudulent white noise to learn more.
So what to do? Is there any rigorous theory for how to incorporate all information from the past relevant for predicting the future? Indeed, this is the theory of computational mechanics, developed by Jim Crutchfield and many colleagues including myself, over many years.
Typical processes have both intrinsic randomness and predictable structure. We have learned that more memory is required to predict a process than to generate it: predators must be more sophisticated than their prey.
We also found that quantum states can be used to simulate classical stochastic processes with less memory than otherwise classically necessary.
Current research explores optimal prediction of quantum patterns, the theory of input—output processes, inference, and more.
