Let $X$ be a categorical random variable with possible outcomes $o_1,...,o_n \subset [l, u]$ (real numbers with a known lower bound $l>0$ and known upper bound $u$) that occur with probability $p(X = o_i) = p_i$ for all $i$ and the probabilities sum to one, $\sum_{i=1}^n p_i = 1$.
This random variable cannot be observed. Instead, we observe $Y$ which is a sum of outcomes, $$ Y = \sum_{i=1}^{k} X. $$ For every sample from $Y$, we also known the number of elements in the sum $k$.
How can we estimate
- the set of outcomes $o_1,...,o_n$, and
- the corresponding probabilities $p_i, ..., p_n$,
from a sample of $Y$ denoted by $(y_1, k_1), ..., (y_m, k_m)$ where each $k_i$ is the number of elements in the sum? Also, given an estimate, how can we update it as soon as more data becomes available?
What do we know and what can we assume:
- the total number of outcomes $n$ is known
- the upper and lower bound of the outcomes $l$ and $u$ is known
- we can place a well-informed prior on each outcome $o_i$
- we have no a-priori knowledge about the probabilities $p_i$