Let $X\sim N(\mu,\sigma^2)$ be a normal distribution and $Y$ a logit-normal distribution $logit(Y)=X$ of parameters $\mu$ and $\sigma$.
I need to calculate the mean of $Y$ but it does not have an analytical expression. Therefore, I want to derive a formula that approximates the mean of $Y$ with respect to $\mu$ and $\sigma$. I guess it could be achieved through integral approximation and/or Taylor expansion but didn't manage to find a closed form approximation formula.
To visualise how the mean varies with respect to both $\mu$ and $\sigma$, I ran this simple code which approximates the mean through sampling:
d=expand.grid(mu=seq(-20,20,length.out=21),
sigma=seq(0,20,length.out=21)) %>%
dplyr::mutate(mean=NA)
for(i in 1:dim(d)[1]){
d$mean[i] = mean(inv_logit(rnorm(1000000,d[i,"mu"],d[i,"sigma"])))
print(i)
}
d %>%
ggplot(aes(x=mu,y=mean,col=factor(sigma)))+
geom_point()+
geom_line()
