I am looking at the machine learning paper, and I came across expected-value in the following notation:

$E_{s_t \sim \rho^B, \alpha \sim \beta, r_t \sim E}[(Q(s_t, a_t |\theta^Q)-y_t)^2]$

(eqn 4) on https://arxiv.org/pdf/1509.02971.pdf

My Question is, Does it mean that the expectation is taken over $s_t, \alpha,$ and $r_t$? what is that $_{s_t \sim \rho^B, \alpha \sim \beta, r_t \sim E}$ means? Does it mean that the expectation is taken over $s_t, \alpha,$ and $r_t$?

Thanks

Yes; it means that you take the expectation over those variables. The ~ sign is to show you what "space" they live in; so the variable $$s_t$$ comes from the space $$\rho^\beta$$. So, when you take an expectation over those, they will disappear, and the remaining quantity will be a function of theta. (note that $$y_t$$ is also a function of $$\theta$$ as they mention it in their paper; the other parameters it depends on also disappear after taking expectation)