725

**Uniform
Consistency In Causal Inference**

**James M. Robins, Richard Scheines,
Peter Spirtes and Larry Wasserman**

### Abstract:

There is a long tradition of representing causal relationships by
directed acyclic graphs (Wright 1934). Spirtes (1992) and
Spirtes, Glymour and Scheines (1993) using some ideas in
Pearl and Verma (1991)
describe procedures for inferring the presence or absence
of causal arrows in the graph even if there might be unobserved confounding
variables, and/or an unknown time order, and
that under weak conditions, for certain combinations of DAGs and probability
distributions, are asymptotically (in sample size) consistent.
These results are surprising since they seem to contradict
the standard statistical wisdom that consistent estimates of causal effects
do not exist for non-randomized studies
if there are potentially unobserved confounding variables.

We resolve the apparent incompatibility of these views
by closely examining the asymptotic
properties of these causal inference procedures.
We show that the asymptotically
consistent procedures are ``pointwise consistent''
but ``uniformly consistent" tests do not exist.
Thus, no finite sample size can ever be guaranteed to
approximate the asymptotic results.
We also show the non-existence of
valid, consistent confidence intervals
for causal effects
and the non-existence of uniformly consistent point estimates.
Our results make no assumptions about the form of the tests
or estimates. In particular, the tests could be classical
independence tests, they could be Bayes tests or they could be
tests based on scoring methods such as BIC or AIC.
The implications of our results
for observational studies
are controversial and
are discussed briefly in the last section of the paper.

The results hinge on the following fact:
it is possible to find,
for each sample size *n*, distributions *P* and *Q*
such that:
*P* and *Q* are empirically indistinguishable
and yet *P* and *Q*
correspond to different causal effects.

*Heidi Sestrich*

*9/11/2000*
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