New Foundations for Ockham’s Razor in Statistical Inductive Inference, with Applications to Causal Discovery from Non-experimental Data

In machine learning, the standard explanation of Ockham's razor is to minimize predictive risk. But prediction is interpreted passively---one may not rely on predictions to change the probability distribution used for training. That limitation may be overcome by studying alternatively manipulated systems in randomized experimental trials, but experiments on multivariate systems or on human subjects are often infeasible or immoral. Happily, the past three decades have witnessed the development of a range of statistical techniques for discovering causal relations from non-experimental data. One characteristic of such methods is a strong Ockham bias toward simpler causal theories---i.e., theories with fewer causal connections among the variables of interest. Our question is what Ockham's razor has to do with finding true (rather than merely plausible) causal theories from non-experimental data. The traditional story of minimizing predictive risk does not apply, because uniform consistency is often infeasible in non-experimental causal discovery: without strong and implausible assumptions, the probability of erroneous causal orientation may be arbitrarily high at any sample size. The standard justification for causal discovery methods is point-wise consistency, or convergence in probability to the true causes. But Ockham's razor is not necessary for point-wise convergence: a Bayesian with a strong prior bias toward a complex model would also be point-wise consistent. Either way, the crucial Ockham bias remains disconnected from learning performance. A method reverses its opinion in probability when it probably says A at some sample size and probably says B incompatible with A at a higher sample size. A method cycles in probability when it probably says A, then probably says B incompatible with A, and then probably says A again. Uniform consistency allows for no reversals or cycles in probability. Point-wise consistency allows for arbitrarily many. Lying plausibly between those two extremes is straightest possible convergence to the truth, which allows for only as many cycles and reversals in probability as are necessary to solve the learning problem at hand. We show that Ockham's razor is necessary for cycle-minimal convergence and that patience, or waiting for nature to choose among simplest theories, is necessary for reversal-minimal convergence. The idea yields very tight constraints on inductive statistical methods, both classical and Bayesian, with causal discovery methods as an important special case. It also provides a valid interpretation of significance and power when tests are used to fish inductively for models. The talk is self-contained for a general scientific audience. Novel concepts are illustrated amply with figures and simulations.