Strong Faithfulness and Uniform Consistency in Causal Inference

A fundamental question in causal inference is whether it is possible to reliably infer ma­ nipulation effects from observational data. There are a variety of senses of asymptotic reliability in the statistical literature, among which the most commonly discussed frequen­ tist notions are pointwise consistency and uniform consistency (see, e.g. Bickel, Dok­ sum [200 1]). Uniform consistency is in gen­ eral preferred to pointwise consistency be­ cause the former allows us to control the worst case error bounds with a finite sample size. In the sense of pointwise consistency, several reliable causal inference algorithms have been constructed under the Markov and Faithfulness assumptions [Pearl 2000, Spirtes et a!. 200 1]. In the sense of uniform con­ sistency, however, reliable causal inference is impossible under the two assumptions when time order is unknown and/or latent con­ founders are present [Robins et a!. 2000 ] . In this paper we present two natural generaliza­ tions of the Faithfulness assumption in the context of structural equation models, under which we show that the typical algorithms in the literature (in some cases with modifi­ cations) are uniformly consistent even when the time order is unknown. We also discuss the situation where latent confounders may be present and the sense in which the Faith­ fulness assumption is a limiting case of the stronger assumptions.