# Learning hidden variable distribution

Consider a set of $k$ continuous variables. Each variable $x_k$ is associated with a hidden distribution from which its value is sampled independently of other variables. I am given a set of observations of the sum of $k$ variables, i.e, $\sum_{i = 1}^{k} x_i$. The challenge is to learn the hidden distribution from the given observations.

This is the general variant of the problem I am trying to study. To simplify the problem, I can make the assumption that the hidden distribution is uniform and of the form $[0, u_k]$. Then, the challenge is to find $u_k$ for each variable $x_k$.

I am not very familiar with learning style algorithms but I am sure variants of this problem have been studied in great detail. Are there any references that I can look at to find out more about this problem?

• Interesting question. In general it definitely seems too hard -- what if one of the variables is itself the sum of independent variables? – usul Feb 5 '18 at 1:07
• Relevant: for the discrete case, you may want to look at the literature on learning Poisson Binomial Distributions, learning $k$-SIIRVs, and learning the parameters of Poisson Binomial Distributions. (Technically, the first two don't learn the hidden distribution, but learn an approximation of the distribution of the sum). See e.g. cs.columbia.edu/~rocco/papers/focs13.html arxiv.org/abs/1505.00662 arxiv.org/abs/1107.2702 – Clement C. Feb 9 '18 at 3:46