Sampling from a multivariate gaussian via affine transformation of uniform random samples? I saw a proof some number of months ago and seem to have forgotten to bookmark it. Essentially, the proof showed that with just a few elements, the {mean vector, cov vector} of the target gaussian and a scalar randomly sampled from the uniform distribution with range [0,1], the target distribution could be sampled by an "affine transformation."
Could someone link and/or how this works? There's a decent chance that I'm confused and conflating ideas...
 A: I doubt that your memory is correct.
An affine transformation of a multivariate uniform density will leave you with a uniform density over some hyper-parallelogram. Meanwhile a multivariate normal (of dimension $p$, say) has support over the whole of $\mathbb{R}^p$. This gap appears to be unbridgeable without some form of nonlinear transformation.
There are a variety of ways of generating independent standard Gaussian random variates from independent $U(0,1)$ random variates*, the simplest of which is perhaps the Box-Muller transformation, which takes a pair of independent standard uniform values to a pair of independent standard Gaussians.
If you have a source of independent standard Gaussians, then you can obtain a multivariate normal distribution from them with specified mean-vector $\mathbf{\mu}$ and variance-covariance matrix $\mathbf{\Sigma}$. Many questions on site discuss how to do this. Briefly: Let $\mathbf{z}$ be a vector of independent standard Gaussians. Let $A$ be a matrix such that $AA^\top=\mathbf{\Sigma}$. Then $\mathbf{x}=A\mathbf{z}+\mathbf{\mu}$ would have the required distribution.
As we see, the "affine" part would relate to getting correlated Gaussians with given mean from independent standard Gaussians.
A proof would require a few facts about means, variances and normal distributions:
- That $E(AX+\mathbf{\mu}) = AE(X)+\mathbf{\mu}$. (A basic property of expectations.)
- That $\text{Var}(AX+\mathbf{\mu}) = A\text{Var}(X)A^\top$. (A basic property of variances.)
- That linear combinations of multivariate normals have multivariate normal distributions.
Given the requisite facts the result follows immediately.
* if you have a source of such quantities
