  A common goal in observational research is to estimate marginal causal
effects in the presence of confounding variables. One solution to this problem
is to use the covariate distribution to weight the outcomes such that the data
appear randomized. The propensity score is a natural quantity that arises in
this setting. Propensity score weights have desirable asymptotic properties,
but they often fail to adequately balance covariate data in finite samples.
Empirical covariate balancing methods pose as an appealing alternative by
exactly balancing the sample moments of the covariate distribution. With this
objective in mind, we propose a framework for estimating balancing weights by
solving a constrained convex program where the criterion function to be
optimized is a Bregman distance. We then show that the different distances in
this class render identical weights to those of other covariate balancing
methods. A series of numerical studies are presented to demonstrate these
similarities.
