Mean of the log and variance of the log I am struggling to derive the following identities:
$$
\mathbb{E}[\log Z]=2\log(\mathbb{E}[Z])-\frac12\log(\mathbb{E}[Z^2])
$$
$$
\mathrm{Var}[\log Z]=\log(\mathbb{E}[Z^2])-2\log(\mathbb{E}[Z])
$$
where $Z$ is a random variable. These are equations (B6) and (B7) in this paper.

Those apply to a log-normal distribution.  The paper says "The evidence is in
practice approximately log-normally distributed."
If it has parameters $\mu=\mathbb{E}[\log Z]$ and $\sigma^2=\mathrm{Var}[\log Z]$ then:

*

*$\mathbb{E}[ Z] = \exp\left(\mu + \frac{\sigma^2}{2}\right)$

*$\mathrm{Var}[Z]=(\exp(\sigma^2)-1)\exp(2\mu+\sigma^2)$

*$\mathbb{E}[ Z^2] =\exp(2\mu+2\sigma^2)$
which leads to  the desired

*

*$2\log(\mathbb{E}[Z])-\frac12\log(\mathbb{E}[Z^2]) = 2\mu+\sigma^2  - \mu-\sigma^2=\mu=\mathbb{E}[\log Z]$

*$\log(\mathbb{E}[Z^2])-2\log(\mathbb{E}[Z]) = 2\mu +2\sigma^2-2\mu-\sigma^2 = \sigma^2=\mathrm{Var}[\log Z]$
