I am kind of confused by the following problem:
Suppose there are two possible states of the world: $\omega\in \{A,B\}$. At beginning, an observer believes that $\mu_0\equiv \mathbb P(\omega=A)\in (0,1)$ (the prior). There is another random variable $s\in \Re^S$ ($S$ being the set of all possible values of $s$) whose distribution in state $\omega$ is $F(s\,|\,\omega)$. Let $s_t$ ($t=0,1,2,\dots$) be the $t$th realization of $s$ in a sequence of (conditionally independent) random draws. Then we obtain a sequence of random variables $s_0, s_1, \dots$.
Question: Are $s_0, s_1, \dots$ i.i.d.?
If we just compute the unconditional distribution for each $s_t$, then obviously they are identical. However, if we compute the distribution of, say, $s_1$, after observing $s_0$, then the observer's posterior about $\omega$ will change, and hence the conditional distribution of $s_1$ will be different from that of $s_0$. Are we required to just consider the unconditional distributions when determining independence?