Simultaneous causal inference for multiple treatments via sufficiency
  Some units from a population receive the same treatment that is different from treatments available for other reservoir populations. The minimal sufficient statistic $s$ for the pre-treatment $x$-covariates's distributions in the populations is the coarsest balancing score. $s$ is used to select matching units for simultaneous causal comparisons of multiple treatments.Necessary and sufficient conditions on the posterior distribution of the treatment variable (given $x$) determine whether a statistic is either sufficient or minimal sufficient for the x-covariates' distributions. Results in the literature are thus extended. Strong ignorability of treatment assignment given $s(x)$ is also established. Consequently, the expected treatments' differences given $s(x)$ are shown to be simultaneously unbiased for the average causal effects of all treatments' differences. The existing statistical theory for $s$ and its estimates support their use in causal inference. 
Yannis G. Yatracos
math.ST stat.TH