  This paper proposes new formulas for the probabilities of causation difined
by Pearl (2000). Tian and Pearl (2000a, 2000b) showed how to bound the
quantities of the probabilities of causation from experimental and
observational data, under the minimal assumptions about the data-generating
process. We derive narrower bounds than Tian-Pearl bounds by making use of the
covariate information measured in experimental and observational studies. In
addition, we provide identifiable case under no-prevention assumption and
discuss the covariate selection problem from the viewpoint of estimation
accuracy. These results are helpful in providing more evidence for public
policy assessment and dicision making problems.
