Does the expectation of the log of a gaussian cumulative distribution function with respect to to a gaussian distribution have an analytical solution?

Question

I would like to know if the following has an analytical solution. If no such solution exists then I would like to know if there is an approximation which can be used instead.

$$\mathbb{E}_{N(a;\ m,\ \sigma ^2)}\left [\ln (\Phi(a)) \right ]$$

For extra info what I am trying to do is obtain gradients of this expression with respect to the parameters of the gaussian $m$ and $\sigma ^2$ so I want a symbolic expression which I can take derivatives of.

Answer 1

If I've understood you correctly, you want the mean of $\ln F$ where $F$ is the cdf. In other words, $$\int_\mathbb{R}F'\ln Fdx=\int_0^1\ln FdF=[F\ln F-F]_0^1=-1.$$