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We show that one can perform causal inference in a natural way forcontinuous-time scenarios using tools from stochastic analysis. This providesnew alternatives to the positivity condition for inverse probability weighting.The probability distribution that would govern the frequency of observations inthe counterfactual scenario can be characterized in terms of a so-calledmartingale problem. The counterfactual and factual probability distributionsmay be related through a likelihood ratio given by a stochastic differentialequation. We can perform inference for counterfactual scenarios based on theoriginal observations, re-weighted according to this likelihood ratio. This ispossible if the solution of the stochastic differential equation is uniformlyintegrable, a property that can be determined by comparing the correspondingfactual and counterfactual short-term predictions. Local independence graphsare directed, possibly cyclic, graphs that represent short-term predictionamong sufficiently autonomous stochastic processes. We show through an examplethat these graphs can be used to identify and provide consistent estimators forcounterfactual parameters in continuous time. This is analogous to how JudeaPearl uses graphical information to identify causal effects in finite stateBayesian networks.

Counterfactual analyses with graphical models based on local independence