Covariance of log-transformed variable Consider a model with parameters $a,b$. We know the priors for the parameters are $b\sim\mathcal{N}(\mu_b,\nu_b^2)$ and $g=log(a)\sim\mathcal{N}(\mu_g,\nu_g^2)$.
Using observed data we produce maximum-likelihood estimates $\hat{b}, \hat{a}$ and their estimated covariance matrix $\hat{\Sigma}=\begin{bmatrix} \hat{\sigma}_a^2 & \hat{\sigma}_{ab} \\ \hat{\sigma}_{ab} &  \hat{\sigma}_b^2\\ \end{bmatrix}$.
What would be the covariance matrix for $b$ and $g$? How can I express it using the abovementioned terms?

If $g$ and $b$ are jointly normal with means $\mu_g, \mu_b$ and covariance matrix
\begin{pmatrix}
q & r \\
r & s \\
\end{pmatrix}
then $a=\exp(g)$ and $b$ have covariance matrix
$$
\begin{pmatrix}
(e^{2q}-e^q)\exp(2\mu_g) & r \exp(\mu_g+\frac{q}{2}) \\
r \exp(\mu_g+\frac{q}{2}) & s \\
\end{pmatrix}
=
\begin{pmatrix}
\hat{\sigma}_a^2 & \hat{\sigma}_{ab} \\
\hat{\sigma}_{ab} & \hat{\sigma}_b^2 \\
\end{pmatrix}
$$
We can solve this and get
\begin{align}
q &= \log\left(\frac12+\sqrt{\frac14+\hat{\sigma}_a^2\exp(-2\mu_g)}\right)\\
r &= \hat{\sigma}_{ab}/\exp(\mu_g+\frac{q}{2})\\
s &= \hat{\sigma}_b^2\
\end{align}
