Suppose a vector ${\bf x} = (m,1-m) \times s$ where $(m,1-m)$ is distributed according to a beta distribution beta$(a,b)$ with density $f(m,1-m) = \frac{m^{a-1}(1-m)^{b-1}} {B (a,b)}$, $s$ is distributed according to a Gamma distribution gamma$(\alpha, \beta)$ with density $g(s) = {\frac {\beta ^{\alpha }}{\Gamma (\alpha )}}s^{\alpha \,-\,1}e^{-\beta s}$. All parameters $a,b,\alpha,\beta$ are known. Also note $m,s$ are independent.
Then the question is: is there anyway to calculate the expectation of log beta function: $\log B ({\bf x})$, which is $\Bbb E [\log \Gamma (sm) + \log \Gamma (s(1-m)) - \log \Gamma (s) ]$ with both $m$ and $s$ as random variables?
See also related question.
My attempt:
I am not sure, but maybe we can use Stirling's approximation
That is
$\ln \Gamma (x) \approx (x - \frac{1}{2})\ln x - x + \frac{1}{{12(x + 1)}} + \frac{1}{2}\ln 2\pi + O(x)$
and my application scenario allows dropping the constant addends and the $O(x)$ term. Then,
$\ln \Gamma (sm) + \ln \Gamma (s(1 - m)) - \log \Gamma (s) = sm\ln sm - \frac{1}{2}\ln sm + \frac{1}{{12(sm + 1)}} + s(1 - m)\ln s(1 - m) - \frac{1}{2}\ln s(1 - m) + \frac{1}{{12(s(1 - m) + 1)}} - s\ln s - \frac{1}{2}\ln s + \frac{1}{{12(s + 1)}}$
Note $s,m$ are independent random variables, and $\Bbb E[s]$, $\Bbb E[m]$, $\Bbb E[\ln s]$, $\Bbb E[\ln m]$ have known form. The remaining problems are to calculate $\Bbb E[\frac {1}{s+1}]$, $\Bbb E[\frac {1}{sm+1}]$, $\Bbb E[s \ln s]$ and $\Bbb E[m \ln m]$, etc. See the related question.