# data-averaged posterior

Hi, I was wondering how can the two following procedures with different orders of applying g and gathering samples be reflected in the mathematical expression of data-averaged posterior.

```
1. 1 time computation with y samples gathered from S datasets
2. S times computation and gathering theta samples
```

S is the number of samples from the simulation prior, g is posterior sampling method, f is the likelihood, \pi is the prior.

What procedure does

\int_{\tilde{\theta} \in \Theta} \int_{\tilde{y} \in \mathcal{Y}} g(\theta \mid \tilde{y}, f, \pi) f(\tilde{y} \mid \tilde{\theta}) \pi(\tilde{\theta}) d \tilde{y}d \tilde{\theta}

represent and what would be the other procedureâ€™s expression? Thanks!