Optimal Causal Inference

We consider an information-theoretic objective function for statistical modeling of time series that embodies a parametrized trade-off between the predictive power of a model and the model’s complexity. We study two distinct cases of optimal causal inference, which we call optimal causal filtering (OCF) and optimal causal estimation (OCE). OCF corresponds to the ideal case of having infinite data. We show that OCF leads to the exact causal architecture of a stochastic process, in the limit in which the trade-off parameter tends to zero, thereby emphasizing prediction. Specifically, the filtering method reconstructs exactly the hidden, causal states. More generally, we establish that the method leads to a graded model-complexity hierarchy of approximations to the causal architecture. We show for nonideal cases with finite data (OCE) that the correct number of states can be found by adjusting for statistical fluctuations in probability estimates.