Assume $\theta_{i} \sim i.i.d. Bernoulli(p)$ and $S(\theta_{i})$ is a continuous function.
Can we say that, $\theta_{i}|S(\theta_{i}) \leq \gamma$ is also $i.i.d. Bernoulli(p)$. Here, $\gamma$ is some fixed value.
In other words, can conditioning impair the independence feature?
