The goal of this paper is to integrate the notions of stochastic conditional independence and variation conditional independence under a more general notion of extended conditional independence. We show that under appropriate assumptions the calculus that applies for the two cases separately (axioms of a separoid) still applies for the extended case. These results provide a rigorous basis for a wide range of statistical concepts, including ancillarity and sufficiency, and, in particular, the Decision Theoretic framework for statistical causality, which uses the language and calculus of conditional independence in order to express causal properties and make causal inferences.

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Taking a rigorous formal approach, we consider sequential decision problems involving observable variables, unobservable variables, and action variables. We can typically assume the property of extended stability, which allows identification (by means of “-computation”) of the consequence of a specified treatment strategy if the “unobserved” variables are, in fact, observed—but not generally otherwise. However, under certain additional special conditions we can infer simple stability (or sequential ignorability), which supports -computation based on the observed variables alone. One such additional condition is sequential randomization, where the unobserved variables essentially behave as random noise in their effects on the actions. Another is sequential irrelevance, where the unobserved variables do not influence future observed variables. In the latter case, to deduce sequential ignorability in full generality requires additional positivity conditions. We show here that these positivity conditions are not required when all variables are discrete.

Despite intense active surveillance among thousands of hospital employees with >97% annual compliance, tuberculin conversion rates were low, and no cases of active TB were identified during 9 years of follow-up. There was no evidence of transmission of M tuberculosis from infected patients to employees during uncontrolled exposures. Rates of TB in the community were low. These data suggest that rigorous application of the Centers for Disease Control and Prevention guidelines and Occupation Safety and Health Administration regulations for preventing nosocomial TB in pediatric hospitals may be excessive and costly. Special provisions should be made for pediatric hospitals with a proven low risk of transmission of M tuberculosis.

Despite intense active surveillance among thousands of hospital employees with >97% annual compliance, tuberculin conversion rates were low, and no cases of active TB were identified during 9 years of follow-up. There was no evidence of transmission of M tuberculosis from infected patients to employees during uncontrolled exposures. Rates of TB in the community were low. These data suggest that rigorous application of the Centers for Disease Control and Prevention guidelines and Occupation Safety and Health Administration regulations for preventing nosocomial TB in pediatric hospitals may be excessive and costly. Special provisions should be made for pediatric hospitals with a proven low risk of transmission of M tuberculosis.