Let $f$ be a (deterministic) function and $W$ a Brownian motion. Let $\theta$ be a random variable that is independent of $W$. Let $\mathcal{F}$ denote the natural filtration of $W$. Then I am wondering how to compute $$\mathbb{E}[f(W_\theta)|\mathcal{F}_t],$$ where $t\geq 0$.
I imagine that the conditional expectation w.r.t. $\mathcal{F}_t$ equals the conditional expectation w.r.t. $\sigma(W_t)$ by Markovianity, but I am not sure if the resulting value is a function of $W_r$ or of $\theta$. I am not sure if the operator $\mathbb{E}$ affects only $W$ (and then we evaluate $\theta$) or if it averages out "all the omegas".
Any thought on this? Thanks!