# A Bayesian view of doubly robust causal inference

## TL;DR

This paper provides a Bayesian perspective on doubly robust causal inference, highlighting limitations of likelihood-based methods and proposing a posterior predictive approach that decouples outcome and treatment models for improved robustness.

## Contribution

It introduces a Bayesian posterior predictive method that incorporates inverse treatment probabilities as importance sampling weights, offering a new way to achieve doubly robust causal inference.

## Key findings

- Simulations demonstrate the proposed method's effectiveness.
- Likelihood-based Bayesian inferences cannot fully explain doubly robust properties.
- Decoupling models enhances robustness in causal effect estimation.

## Abstract

In causal inference confounding may be controlled either through regression adjustment in an outcome model, or through propensity score adjustment or inverse probability of treatment weighting, or both. The latter approaches, which are based on modelling of the treatment assignment mechanism and their doubly robust extensions have been difficult to motivate using formal Bayesian arguments, in principle, for likelihood-based inferences, the treatment assignment model can play no part in inferences concerning the expected outcomes if the models are assumed to be correctly specified. On the other hand, forcing dependency between the outcome and treatment assignment models by allowing the former to be misspecified results in loss of the balancing property of the propensity scores and the loss of any double robustness. In this paper, we explain in the framework of misspecified models why doubly robust inferences cannot arise from purely likelihood-based arguments, and demonstrate this through simulations. As an alternative to Bayesian propensity score analysis, we propose a Bayesian posterior predictive approach for constructing doubly robust estimation procedures. Our approach appropriately decouples the outcome and treatment assignment models by incorporating the inverse treatment assignment probabilities in Bayesian causal inferences as importance sampling weights in Monte Carlo integration.