Continuous binomial rng

[simonbrauer]

@adamConnerSax

Where the paper confuses me though, is in what to use for the number of successes. What would make sense to me, is to compute the weighted proportion of successes and then multiply that ratio by the ESS. But that’s not what happens in the paper. There, the successes are set to the weighted proportion of successes multiplied by the sum of sample sizes (sss rather than ESS).

I agree. My sense is that the equations should read

\text{weighted proportion} = \bar{y}^*_j = \frac{\sum_{i \in j} w_i y_i}{\sum_{i \in j} w_i}\\ \text{adjusted sample size} = n^*_j = \frac{n_j}{\text{design effect}} \\ \text{adjusted number of successes} = y^*_j = \bar{y}^*_j n^*_j \\ y^*_j \sim Binomial(n^*_j, \theta_j)

I checked the published version of the paper and confirmed that the equations are the same as the working draft that you linked. So either it is a typo or there is something we are not understanding.

I’m sure there is a reasonable way to generate random draws from such a density. The problem is that you/Ghitza and Gelman are use a pseudo-posterior, which does not describe the actual data-generating process. So there isn’t a natural link between how the data were actually generated and how you’re modelling the data. This is, by my reading, one of the core challenges in Bayesian approaches to weighted survey data. See Survey weighted regression, Are complex surveys feasible in brms?, and (Pseudo) Bayesian Inference for Complex Survey Data.