The posterior covariance matrix with missing data

Question

Let $X \sim N(0, \Sigma)$ be a multivariate normal vector, and let our prior for $\Sigma$ be inverse-Wishart: $\Sigma\sim IW(v,V)$. The posterior for $\Sigma$ given $X$ is also inverse-Wishart: $$\Sigma | X = x \sim IW(1 + v, V + x'x).$$

I'm interested in the case in which we observe only some elements of $X$. I.e., we observe $X_c \sim N(0, \Sigma_c)$, where $\Sigma_c$ is a submatrix of $\Sigma$. The posterior density for the full covariance matrix $\Sigma$ given an observation $X_c$ will be $$ p(\Sigma| X_c = x_c) \propto |\Sigma_c|^{-\frac{1}{2}} \exp\left(-\frac{1}{2} x_c' \Sigma_c^{-1} x_c\right)f(\Sigma), $$ where $f(\Sigma)$ is the prior density of $\Sigma$.

Is there a prior on $Σ$ such that this posterior is conjugate for any set of observed elements of $X$?

Answer 1

While I haven't been able to find a conjugate prior, I believe the following MCMC algorithm will suffice:

In each iteration $r$:
($r.1$) draw the missing elements ${X_{-c}}^r$ given $X_c, Σ^{r-1}$,
($r.2$) draw $Σ^r$ given $X_c, {X_{-c}}^r$.

Step ($r.1$) just requires drawing from a Normal distribution while step ($r.2$) can draw from an Inverse-Wishart as described in the original question.

I believe that $Σ^r$ should converge in distribution to $Σ | X_c$.