Suppose that we have a sequence of $k$ probabilities $p_1, p_2, \ldots, p_k \in [0, 1]$, corresponding to $k$ independent events $A_1, A_2, \ldots, A_k$ and a Poisson-Binomial variable $X$ counting the number of successful event. Also, suppose that we have its PMF $\Pr[X = i]$ for each $0 \leq i \leq k$.
Now we want to compute the incremental change of its PMF when a single $p_i$ is revealed and "collapses" to $0$ or $1$. This is equivalent to computing the conditional PMF. WLOG, we assume that $p_k$ collapses, what would be the new PMF $\Pr[X = i | A_k]$ and $\Pr[X = i | \neg A_k]$? Can we have some techniques without computing from scratch?
What I have tried
We have
- $\Pr[X = i] = p_k \Pr[X = i | A_k] + (1-p_k) \Pr[X = i | \neg A_k]$
- $\Pr[X = i | A_k] = \Pr[X = i - 1 | \neg A_k]$
- $\Pr[X = 0 | A_k] = 0$
- $\Pr[X = 0 | \neg A_k] = \prod_{i \neq k} (1 - p_i)$
- $\Pr[X = k | \neg A_k] = 0$