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Let $\Sigma$ be a $p\times p$ positive definite matrix and $\nu>p-1$. I would like to simulate a realization of the Inverse-Wishart distribution $X\sim\mathcal W^{-1}(\Sigma, \nu)$ with the constraint that certain off-diagonal terms of $X$ should be 0. For example, the constraint could be that $X_{1,p}=X_{p,1}=0$.

In practice, this will be useful in the following Bayesian setting: $X$ will be the covariance matrix of a multivariate normal distribution for which I know that certain components are independent. This will be used in an MCMC, in which the parameters $\Sigma$ and $\nu$ change at every iteration, so I am looking for a reasonably efficient way of getting a single realization. The dimension $p$ will typically be between 3 and 15.

I would appreciate any pointers on how to handle this problem.

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