  Evaluation of counterfactual queries (e.g., "If A were true, would C have
been true?") is important to fault diagnosis, planning, and determination of
liability. In this paper we present methods for computing the probabilities of
such queries using the formulation proposed in [Balke and Pearl, 1994], where
the antecedent of the query is interpreted as an external action that forces
the proposition A to be true. When a prior probability is available on the
causal mechanisms governing the domain, counterfactual probabilities can be
evaluated precisely. However, when causal knowledge is specified as conditional
probabilities on the observables, only bounds can computed. This paper develops
techniques for evaluating these bounds, and demonstrates their use in two
applications: (1) the determination of treatment efficacy from studies in which
subjects may choose their own treatment, and (2) the determination of liability
in product-safety litigation.
