Stacking interventions for equitable outcomes

Predictive risk scores estimating probabilities for a binary outcome on the basis of observed covariates are frequently developed with the intent of avoiding that outcome by intervening on covariates in response to estimated risks. Since risk scores are usually developed in complex systems, interventions often take the form of expert actors responding to estimated risks as they best see fit. In this case, interventions may be complex and their effects difficult to observe or infer, meaning that explicit specification of interventions in response to risk scores is impractical. The capacity to design the aggregate model-intervention scheme in a way which optimises objectives is hence limited. We propose an algorithm by which a model-intervention scheme can be developed by `stacking' possibly unknown intervention effects. By repeatedly observing and updating the intervention and model, this scheme leads to convergence or almost-convergence of eventual outcome risk to an equivocal value for any initial value of covariates, given reasonable assumptions. Roughly, our approach involves deploying a series of risk scores to expert actors, with instructions to act on them in succession. Our algorithm uses only observations of pre-intervention covariates and the eventual outcome as input. It is not necessary to know the action of the intervention, other than a general assumption that it is `well-intentioned'. This algorithm can also be used to safely update risk scores in the presence of unknown interventions, an important contemporary problem in machine learning. We demonstrate convergence of expectation of outcome in a range of settings, and give sufficient conditions for convergence in distribution of covariate values. Finally, we demonstrate a potential practical implementation by simulation to optimise population-level outcome frequency.