Bayesian analysis of treatment effects in an ordered potential outcomes model

We describe a new Bayesian estimation algorithm for fitting a binary treatment, ordered outcome selection model in a potential outcomes framework. We show how recent advances in simulation methods, namely data augmentation, the Gibbs sampler and the Metropolis-Hastings algorithm can be used to fit this model efficiently, and also introduce a reparameterization to help accelerate the convergence of our posterior simulator. Conventional "treatment effects" such as the Average Treatment Effect (ATE), the effect of treatment on the treated (TT) and the Local Average Treatment Effect (LATE) are adapted for this specific model, and Bayesian strategies for calculating these treatment effects are introduced. Finally, we review how one can potentially learn (or at least bound) the non-identified cross-regime correlation parameter and use this learning to calculate (or bound) parameters of interest beyond mean treatment effects.