I have a beta-binomial model given as
\begin{align} Y|\theta &\sim Bin(n, \theta) \\ \theta &\sim Beta(\alpha, \beta). \end{align}
If I am observing $Y=y$, then my posterior is \begin{align} \theta|Y=y \sim Beta(\alpha + y,\beta +n -y). \end{align}
Now, what I like to find is an expression for the conditional expectation of $\theta$ given that I observe at least $Y=y$, i.e., how can I find
\begin{align} \mathbb{E}(\theta |Y \ge \underline{y}) \end{align} ? Any hint or help is highly appreciated! Thank you in advance.