Suppose I am uncertain about the covariance of a vector-valued random variable $X$, but the variance of some linear combination is known. How does this affect the distribution of $X$?
Specifically $X$ has Gaussian-Wishart distribution, $X\vert\Lambda\sim\mathcal N_p(0,\Lambda^{-1})$ and $\Lambda\sim\mathcal W_p(\mathbf W,n)$ - the conjugate prior for Gaussian of unknown covariance. What is the distribution of $X\vert var(\mathbb w'X)$, i.e. of $X\vert\mathbf w'\Lambda^{-1}\mathbf w$? I am hopeful it is another Gaussian-Wishart distribution...