This question shows how to derive an analytical expression for the conditional distribution from a multivariate normal. I am curious how well this extends to when there's a Gaussian copula, but potentially different margins.

This would suggest a joint distribution $$f\left(X_{1},X_{2}\right)=|C|^{-0.5}exp\left\{ -\frac{1}{2}\left[\begin{array}{cc} Z_{1} & Z_{2}\end{array}\right]\left(C^{-1}-I\right)\left[\begin{array}{cc} Z_{1} & Z_{2}\end{array}\right]'\right\} \times\prod_{i=1}^{n_{1}}f_{i}\left(x_{i}\right)\times\prod_{j=1}^{n_{2}}f_{j}\left(x_{j}\right) $$ where $C$ is the correlation matrix, $n_{1}$ and $n_{2}$ are the number of respective variables of $X1$ and $X2$, and the $i^{th}$ element of $Z\equiv\left[\begin{array}{cc} Z_{1} & Z_{2}\end{array}\right]$ is given by $z_{i}\equiv\Phi^{-1}\left(u_{i}\right)$ where $u_{i}$ is the grade.

There is also the marginal distribution $$f\left(X_{2}\right)=|C_{22}|^{-0.5}exp\left\{ -\frac{1}{2}Z_{2}\left(C_{22}^{-1}-I\right)Z_{2}'\right\} \times\prod_{j=1}^{n_{2}}f_{j}\left(x_{j}\right)$$

Combining together would give the conditional distribution $$f\left(X_{1}|X_{2}\right) = \frac{f\left(X_{1},X_{2}\right)}{f\left(X_{2}\right)} = \left(\prod_{i=1}^{n_{1}}f_{i}\left(x_{i}\right)\right)\times\left(\frac{|C|}{|C_{22}|}\right)^{-0.5}\times exp\left\{ -\frac{1}{2}\left(\left[\begin{array}{cc} Z_{1} & Z_{2}\end{array}\right]\left(C^{-1}-I\right)\left[\begin{array}{cc} Z_{1} & Z_{2}\end{array}\right]'-Z_{2}\left(C_{22}^{-1}-I\right)Z_{2}'\right)\right\} = \left(\prod_{i=1}^{n_{1}}f_{i}\left(x_{i}\right)\right)\left(|C_{11}-C_{12}C_{22}^{-1}C_{21}|\right)^{-0.5} exp\left\{ -\frac{1}{2}Q\right\} $$ where $Q$ is the term in parentheses.

I have tried simplifying $Q$ so that if I were to assume Gaussian marginals, then the result would match that for the multivariate normal. I can't seem to work out the matrix algebra of it though. I get something similar to what is used for the conditional covariance matrix in the multivariate Gaussian case, which may not come out to a valid correlation matrix, and then a few extra terms that presumably correct the marginals.

  • 1
    $\begingroup$ In the bivariate case, yes, but I'm assuming that $X_{1}$ and $X_{2}$ are both multi-dimensional. $\endgroup$ – John Nov 29 '12 at 19:53
  • $\begingroup$ I had seen that from the other thread and was trying to match calculations to make it work. Perhaps my biggest worry is the effective conditional copula would not have a correlation matrix with ones on the diagonals. $\endgroup$ – John Nov 29 '12 at 21:23

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