Retrodicting Chaotic Systems: An Algorithmic Information Theory Approach

Making accurate inferences about data is a key task in science and mathematics. Here, we study the problem of {retrodiction}, inferring past values of a series, in the context of chaotic dynamic systems. Specifically, we are interested in inferring the starting value x0 in the series x0, x1, x2,... xn given the value of xn and the associated function f, which determines the series as f(xi) = xi +1. Even in the deterministic case, this is a challenging problem due to mixing and the typical exponentially many candidate past values in the preimage of any given value xn (e.g., a current observation). We study this task from the perspective of algorithmic information theory, which motivates searching for the `simplest' value in the set of candidates. Different methods for estimating candidate (conditional) complexity are examined, including to look for values in the lowest-density regions. We test these methods numerically on the logistic map, Tent map, Bernoulli map, and Julia/Mandelbrot map, which are well-studied maps in chaos theory. The methods aid in retrodiction by assigning low ranks to the candidates that are more likely to be the true starting value. Our approach works well in some parameter and map cases, while several other retrodiction techniques perform poorly. Nonetheless, our approach is not effective in all cases, and several open problems remain including computational cost and sensitivity to noise. We also give an explicit description-length-weighted covariance construction supporting a Gaussian-process research direction, without identifying all candidate ranking methods as Gaussian processes.