Is summing posterior probabilities valid for classification problems? A classification for two mutually exclusive problem can be formulated by having a decision hinge on whether $P_0(x) > P_1(x)$ or $P_0(x) < P_1(x)$  where $P_0(x)$ and $P_1(x)$ are posterior probabilities. However, is is valid to simply sum all the posterior probabilities $P_0(x)$ and $P_1(x)$ a to obtain total counts for both groups? 
 A: Yes, if the events are exclusive, then you can sum their probabilities to obtain probability of observing any of the events.
If $E_1, E_2,\dots,E_k$ are mutually exclusive events, then by definition $P(\cup_{i=1}^k E_i) = \sum_{i=1}^k P(E_i)$. So if you have estimated probabilities for $E_1$ and $E_2$, then $\hat P(E_1 \cup E_2) = \hat P(E_1) + \hat P (E_2)$. Same with merging the levels of Dirichlet distribution.
If $\mathbf{X}$ follows multinomial distribution $\mathbf{X} \sim \mathcal{M}(n, \mathbf{p})$, then marginally $X_i$'s are binomial distributed $X_i \sim \mathcal{B}(n, p_i)$ and for $i\ne j$, the sum of $X_i$ and $X_j$ is also binomial $X_i + X_j \sim \mathcal{B}(n, p_i+p_j)$.
A: $\newcommand\E{\mathbb{E}}$It's valid in the following way:
Suppose that the label of each class is distributed according to $\mathrm{Bernoulli}(P_1(x))$. Then the expected number of instances of class one is the sum of the posterior probabilities.
This is a fairly reasonable model for class variables, though I wouldn't take the values it gives out too literally unless you have much better classification probability estimates than are typical.
