Suppose I have a random variable $\theta=(\theta\_1,\dotsc,\theta\_n)$; where the $\theta\_i$ might have pairwise correlations. I decompose it into $\theta=\hat\theta(\phi\_1,\dotsc,\phi\_k)$, where $\hat\theta$ is a deterministic function, and the $\phi\_i$ are random variables such that each $\phi\_i$ is independent from all others.


Does this operation have a name? "Independent decomposition" or something?



Is there a name for "splitting a probability distribution into independent components"?
I think this particular property is unlikely to have a generally accepted name. 


However, a more general concept is commonly called decoupling, when the joint distribution of a function of dependent random variables (r.v.'s) is represented as a mixture of distributions of this function of independent r.v.'s. See e.g. [Wikipedia](https://en.wikipedia.org/wiki/Decoupling_(probability)) and the reference there to the book *Decoupling: From Dependence to Independence* by de la Peña and Giné. 


