Combining information from multiple distributions I have 13 classes. For each class, I have a different distribution: e.g.

For each distribution, the y-axis indicates the probability and the x-axis indicates a count value.
Given some input data, I want to map it to a count value. I have calculated the probabilities that the input falls into each class (i.e. a 1 x 13 vector). But now, how can I combine information from all 13 distributions to obtain a single count value? I thought about two ways:


*

*Compute the weighted sum of the modes for each distribution

*Compute the weighted sum of each distribution to get a new distribution, and find the mode of this new distribution


However, I could not find any academic justifications or formal math theories to support my intuitions. Can anyone point me in the right direction? Are my intuitions entirely wrong? I checked out mixture distributions and linear polling, but they do not quite seem to be what I am looking for...
 A: Let $x$ be the input data, $c \in \{1, \dots, 13\}$ be the class, and $n$ be the count. What you're  after is $p(n \mid x)$, the conditional distribution over count, given the input data (or some statistic that summarizes this distribution). Following the definition of marginal probability, this is obtained by summing $p(n,c \mid x)$ (the joint distribution of count and class, given the input data) over all possible classes:
$$p(n \mid x) = \sum_{c=1}^{13} p(n,c \mid x)$$
Following the definition of conditional probability, $p(n,c \mid x)$ can factorized as $p(n \mid c,x) p(c \mid x)$. If the count depends directly only on the class (i.e. count and input are conditionally independent, given class), then $p(n \mid c,x) = p(n \mid c)$ and therefore $p(n,c \mid x) = p(n \mid c) p(c \mid x)$. Plugging this in:
$$ p(n \mid x) = \sum_{c=1}^{13} p(n \mid c) p(c \mid x)$$
This can be seen as a sum of the count distributions for each class, weighted by the probabilities that the input is a member of each class. This corresponds to the second approach you mentioned. Keep in mind that this requires the conditional independence assumption I mentioned above (i.e. the input doesn't carry any additional information about the count, beyond that carried by its class). If you want, you can then find a statistic that summarizes this distribution (e.g. the mean or mode).
