# Find distribution of underlying feature

Let's say I have a dataset with values for the variable $W$. The distribution of $W$ doesn't follow any obvious known distribution.

I have a model that gives $W$ as a functions of $E$ and $K, W = f(E, K)$. Function $f$ is not trivial (not just sums and multiplications, and each variable appears more than once).

Know: from another source I derive the distribution of $E$.

So, I know the distribution of $W$ (by having the values) and the distribution of $E$. Can I find the distribution of $K$? And how?

If needed I can give more details.

I had tried the following but I think it's flawed. For each observed value of $W$ I sample 100 values of $E$ and calculate the necessary $K$ to explain that $W$. Then I join all the calculated $K$ and try to fit distributions to this data.

• Is function f known? And do you know the distribution of E as a mathematical expression with a few constants or just as an empirically derived PDF (like E = density(x)? And when you want to infer the distribution of K do you then mean which family (normal OR chi-square OR uniform OR...) or just the posterior distribution over a known distribution family? Dec 20 '14 at 19:47
• Thanks for your reply. f is known and E is a mixture of two normal distributions, for witch I have the parameters. I do not know the family distribution of K, but I suspect it to be a mixture of a discrete distribution (value 0) with a mixture of normal distributions. However, I might be wrong in this assumption. Dec 21 '14 at 0:10

I am not an expert on this, but I can give a vague answer with an overall strattegy. I too would take a sampling based approach. However, you are probably right that you actually have the possibility to do better than just isolating K for every observed W.
If bayesian inference sounds scary and you have a reasonable guess at the distribution-family for K you can compare alternative models using nonlinear frequentist approach. Look at R packages nls and nlm. Or nlmer if you want to do mixed model.