Assume the symmetric and pd. matrix $\Sigma$ partioned with $$\Sigma = \begin{bmatrix} \boldsymbol\Sigma_{22} & \boldsymbol\Sigma_{12} \\ \boldsymbol\Sigma_{21} & \boldsymbol\Sigma_{22} \end{bmatrix} \text{ with sizes }\begin{bmatrix} q \times q & q \times (N-q) \\ (N-q) \times q & (N-q) \times (N-q) \end{bmatrix}.$$

I want to transform the parameters $\text{vech}(\Sigma) \rightarrow \theta :=[\text{vec}(B), \text{vech}(\Sigma_{t})],$ defined as $$B = \Sigma_{22}^{-1}\Sigma_{21} \\ \Sigma_{t} = \Sigma_{11}-\Sigma_{12}\Sigma_{22}^{-1}\Sigma_{21} .$$

Is there a way to compute the determinant of the Jacobian of this transformation?

Background: It is well-known that for $N$-dimensional $x$ multivariate gaussian distributed with parameters $\mu$ and $\Sigma$, partioned with $$x = \begin{bmatrix} \mathbf{x}_1 \\ \mathbf{x}_2 \end{bmatrix} \text{ with sizes }\begin{bmatrix} q \times 1 \\ (N-q) \times 1 \end{bmatrix}$$ and $μ$ and $Σ$ with

$$\mu=\begin{bmatrix} \mu_1 \\ \mu_2 \end{bmatrix} \text{ with sizes }\begin{bmatrix} q \times 1 \\ (N-q) \times 1 \end{bmatrix}$$

the conditional distribution of $x_1$ conditional on $x_2 = a$ is multivariate normal with parameters $$\bar{\mu}= \boldsymbol\mu_1 + \boldsymbol\Sigma_{12} \boldsymbol\Sigma_{22}^{-1} \left( \mathbf{a} - \boldsymbol\mu_2 \right) $$ and covariance matrix $$ \overline{\Sigma} = \boldsymbol\Sigma_{11} - \boldsymbol\Sigma_{12} \boldsymbol\Sigma_{22}^{-1} \boldsymbol\Sigma_{21}. $$ I want to assign priors on $\Sigma$ and perform Bayesian analysis on $\mu$ and $\Sigma$ working with the transformed parameters. In order to see how the transformation affects my priors I need the Jacobian.

  • $\begingroup$ Are you sure it's not $\Sigma_{t} = \Sigma_{11}-\Sigma_{21}\Sigma_{22}^{-1}\Sigma_{12}$ ? (with the middle term inverted) ? $\endgroup$ – Jean Marie Sep 25 '17 at 12:18
  • $\begingroup$ Thanks @JeanMarie for hinting at this point, I edited the typo. $\endgroup$ – muffin1974 Sep 25 '17 at 12:24

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