<p>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.</p>

<p>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 $x_k$. </p>

<p>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?</p>

CC BY-SA 3.0
2018-02-05T01:07:19.663Z
Interesting question. In general it definitely seems too hard -- what if one of the variables is itself the sum of independent variables?
usul
CC BY-SA 3.0
2018-02-09T03:46:30.413Z
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. http://www.cs.columbia.edu/~rocco/papers/focs13.html https://arxiv.org/abs/1505.00662 https://arxiv.org/abs/1107.2702
Clement C.
<approximation-algorithms><machine-learning><randomized-algorithms>
Learning hidden variable distribution
karmanaut
# Learning hidden variable distribution

(Asked by: username_0 on 2018-02-04)

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?

**Tags:** approximation-algorithms, machine-learning, randomized-algorithms

> **Comments:**
> **username_1**: Interesting question. In general it definitely seems too hard -- what if one of the variables is itself the sum of independent variables?  
> **username_2**: 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. http://www.cs.columbia.edu/~rocco/papers/focs13.html https://arxiv.org/abs/1505.00662 https://arxiv.org/abs/1107.2702  

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## 0 answers

