Suppose we have $n$ probabilities $p_1, p_2, \ldots, p_n \in [0, 1]$, and let $A_1, A_2, \ldots, A_n \in \{0, 1\}$ be corresponding independent Bernoulli events, where $A_i = 1$ with probability $p_i$. For each subset $X \subseteq [n]$, the probability of $X$ is thus $\Pr[X] = (\prod_{x \in X} p_x) (\prod_{x' \in [n] \setminus X} (1 - p_x))$.
Given a positive integer $k$, we want to sample a $k$-subset of $[n]$. How can we sample $k$ elements $X = \{x_1, x_2, \ldots, x_k\} \in \binom{[n]}{k}$ with probability exactly $\frac{\Pr[X]}{\sum_{X' \in \binom{[n]}{k}} \Pr[X']}$? I know we can always do that by sampling a subset without caring about its cardinality, and re-trying until we obtain a $k$-subset. But this can be very inefficient. I also know that $\sum_{X' \in \binom{[n]}{k}} \Pr[X']$ can be represented as a Poisson binomial distribution. Are there better ways with theoretical justification?