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I am using the twang package in R to balance two groups by creating propensity scores, which are then used as weights in the svyglm for a weighted regression of the two groups.

I would like however to use the weights in a bayesian glm, since this is the model employed also previously in the analysis. How could I implement this, or, is there even a package which allows for propensity-weighted regression in a bayesian context?

Edit: I have read that the weights parameter in stan is not equal to the parameter in svyglm, however, it seems to be that bmrs is allowing for survey-weighted regression in the same manner as svyglm does. Is that correct?

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According to the developer of brms:

brms takes the weights literally, which means that an observation with weight 2 receives 2 times more weight than an observation with weight 1. It also means that using a weight of 2 is equivalent to adding the corresponding observation twice to the data frame.

I think that is in line with the rationale of pseudo-populations in the context of IPTW as discussed in the causal inference literature.

Additionally, according to Zigler:

In essence, the use of Bayesian methods in either the propensity score modeling stage or the outcome modeling stage is straightforward, but combining these two stages into a single analysis is met with the issues discussed in Section 3.1.[The fundamental problem of joint Bayesian estimation of propensity scores and causal effects].

Because IPTW is deterministic, it does not generate design decision uncertainty, and therefore the only source of design uncertainty is the design estimation uncertainty, which needs to be propagated into the outcome model as discussed by Liao and Zigler

But I am not aware of an R package allowing the incorporation of the design estimation uncertainty into the outcome model in a unified way. A potential solution to that is discussed here, but that does not work with brms or another stan package, at least for now.

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For those interested, I found out in the meantime that a Bayesian weighted regression is not that straightforward - according to the Bayesian approach, propensity scores would have to be treated as random variables and accordingly possessing a distribution.

One possible implementation is discussed here:

Uncertainty in Propensity Score Estimation: Bayesian Methods for Variable Selection and Model Averaged Causal Effects

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