# How to estimate the weight matrix in distribution $X = VWV^T$?

Suppose the 1 x N vector $V\in \{0,1\}^N$ comes from the pdf $f(V) = VWV^T$, where $W$ is a N x N positive definite matrix.

If the weight matrix is given, I can use gibbs sampling to generate a random sample of those vectors. But now I want to do the reverse process. If I have a sample of vectors $V$, how can I estimate the weight matrix $W$?

My attempt is to try Metropolis-Hastings and use Wishart distribution as my prior and proposal. Suppose we start with matrix $W_0$ ~ Wishart$(n,I)$, then the proposal is $W_1$ ~ Wishart$(n,W_0/n)$, so the proposal has mean $W_0$.

I find that this algorithm can work on 2x2 matrices, though with a low acceptance rate(about 5%). But it seems unable to find solution for matrices of higher dimensions, mainly because the acceptance rate drops to below 1%.

Is there any better proposal than Wishart distribution with lower variance? Or should I do it in a different way?

• Posted a similar question on Mathexchange and got an answer. I can either scale Wishart or scale the parameters to control the variance so that acceptance rate is good. – J.H Nov 27 '17 at 9:16