can be used for both for readjusting learned policies to circumvent environmental changes and for controlling disparities between nonrepresentative samples and a target population. 3 It can also be used in the context of reinforcement learning to evaluate policies that invoke new actions, beyond those used in training. 35

Tool 6. Recovering from missing data. Problems due to missing data plague every branch of experimental science. Respondents do not answer every item on a questionnaire, sensors malfunction as weather conditions worsen, and patients often drop from a clinical study for unknown reasons. The rich literature on this problem is wedded to a model-free paradigm of associational analysis and, accordingly, is severely limited to situations where “missingness” occurs at random; that is, independent of values taken by other variables in the model. 6 Using causal models of the missingness process we can now formalize the conditions under which causal and probabilistic relationships can be recovered from incomplete data and, whenever the conditions are satisfied, produce a consistent estimate of the desired relationship. 12, 13

Tool 7. Causal discovery. The d -separation criterion described earlier enables machines to detect and enumerate the testable implications of a given causal model. This opens the possibility of inferring, with mild assumptions, the set of models that are compatible with the data and to represent this set compactly. Systematic searches have been developed that, in certain circumstances, can prune the set of compatible models significantly to the point where causal queries can be estimated directly from that set. 9, 18, 24, 31

Alternatively, Shimizu et al. 29 proposed a method for discovering causal directionality based on functional decomposition. 24 The idea is that in a linear model X → Y with non-Gaussian noise, P(y) is a convolution of two non-Gaussian distributions and would be, figuratively speaking, “more Gaussian” than P(x). The relation of “more Gaussian than” can be given precise numerical measure and used to infer directionality of certain arrows.

Tian and Pearl32 developed yet
another method of causal discovery
probability of the sentence is estima-
ble from experimental or observational
studies, or a combination thereof. 1, 18, 30

Of special interest in causal discourse are counterfactual questions concerning “causes of effects,” as opposed to “effects of causes.” For example, how likely it is that Joe’s swimming exercise was a necessary (or sufficient) cause of Joe’s death. 7, 20

Tool 4. Mediation analysis and the assessment of direct and indirect effects. Mediation analysis concerns the mechanisms that transmit changes from a cause to its effects. The identification of such an intermediate mechanism is essential for generating explanations, and counterfactual analysis must be invoked to facilitate this identification. The logic of counterfactuals and their graphical representation have spawned algorithms for estimating direct and indirect effects from data or experiments. 19, 27, 34 A typical query computable through these algorithms is: What fraction of the effect of X on Y is mediated by variable Z?

Tool 5. Adaptability, external validity, and sample selection bias. The validity of every experimental study is challenged by disparities between the experimental and the intended imple-mentational setups. A machine trained in one environment cannot be expected to perform well when environmental conditions change, unless the changes are localized and identified. This problem, and its various manifestations, are well-recognized by AI researchers, and enterprises (such as “domain adaptation,” “transfer learning,” “life-long learning,” and “ explainable AI”) 4 are just some of the subtasks identified by researchers and funding agencies in an attempt to alleviate the general problem of robustness. Unfortunately, the problem of robustness, in its broadest form, requires a causal model of the environment and cannot be properly addressed at the level of Association. Associations alone cannot identify the mechanisms responsible for the changes that occurred, 22 the reason being that surface changes in observed associations do not uniquely identify the underlying mechanism responsible for the change. The do- calculus discussed earlier now offers a complete methodology for overcoming bias due to environmental changes. It

Unlike the rules
of geometry,
mechanics, optics,
or probabilities,
the rules of cause
and effect
have been denied
the benefits
of mathematical
analysis.