Probability, Causality and Stochastic Formulations of Economic Theory

Abstract

The current paper discusses approximating a correct theory of cause and effect by minimizing distance to its associated probability measure in a space of measures in which each element is associated with a stochastic representation of a candidate theory. The discussion encourages researchers to use flexible dynamical models to model and discover the true quantitative relationships that may be hidden in interrelated stochastic data. The argument is based on the use of a decision criterion that scales to a metric that measures distance between any given measure. When this is the case, a metric space can be considered in which equivalences can be established by partitioning into classes of zero-distance points. Equivalence to the true measure, that is associated with the true frequencies in Markov chains of iterated causes and effects, is established by reaching zero distance in that space. When the hypothesis space is incorrectly constructed, equivalence is established with respect to a pseudo-true measure that by definition is closest to the correct hypothesis across all considered hypotheses. The specific case of Maximum Likelihood is further discussed. In particular, squared Hellinger distance marks a lower bound of Kullback-Leibler divergence. This implies that maximizing complexity penalized likelihood minimizes distance toward the true probability measure. As such, it is an objective that approximates the correct causal structure from interrelated stochastic data that are observed and modeled sequentially over time.