Too Fast Causal Inference under Causal Insufficiency
  Causally insufficient structures (models with latent or hidden variables, or
with confounding etc.) of joint probability distributions have been subject of
intense study not only in statistics, but also in various AI systems. In AI,
belief networks, being representations of joint probability distribution with
an underlying directed acyclic graph structure, are paid special attention due
to the fact that efficient reasoning (uncertainty propagation) methods have
been developed for belief network structures. Algorithms have been therefore
developed to acquire the belief network structure from data. As artifacts due
to variable hiding negatively influence the performance of derived belief
networks, models with latent variables have been studied and several algorithms
for learning belief network structure under causal insufficiency have also been
developed.
  Regrettably, some of them are known already to be erroneous (e.g. IC
algorithm of [Pearl:Verma:91]. This paper is devoted to another algorithm, the
Fast Causal Inference (FCI) Algorithm of [Spirtes:93]. It is proven by a
specially constructed example that this algorithm, as it stands in
[Spirtes:93], is also erroneous. Fundamental reason for failure of this
algorithm is the temporary introduction of non-real links between nodes of the
network with the intention of later removal. While for trivial dependency
structures these non-real links may be actually removed, this may not be the
case for complex ones, e.g. for the case described in this paper. A remedy of
this failure is proposed.
