# The posterior covariance matrix with missing data

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 $$X$$?

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 $$Σ | X_c$$.