Information preserving sufficient summaries for dimension reduction

We discuss a type of confounder dimension reduction summary which retains all of the information in the covariates about both an outcome variable and an intervention or grouping variable. These sufficient dimension reduction summaries share much with sufficient statistics for parameters indexing a family of probability distributions and are directly related to the dimension reduction summaries considered in regression theory and propensity theory. These sufficient dimension reduction summaries yield conditional independence, or balance, of the covariates and intervention given the value of the summary. Further, in contrast to other widely used dimension reduction summaries, the regression function for the outcome given the intervention and the sufficient summary is the same as that given the intervention and the original set of confounders.