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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

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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)

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  • $\begingroup$ how do you take expectation over multiple variables $\endgroup$ – kou Apr 3 '18 at 6:29
  • $\begingroup$ Have you taken a class called multivariable calculus? $\endgroup$ – E-A Apr 3 '18 at 6:32
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    $\begingroup$ I tried to edit "perimeters" to say "parameters", but Stack wouldn't let me change less than 6 characters. $\endgroup$ – Joe May 19 '20 at 15:36
  • $\begingroup$ Yup; thanks! Appreciated. $\endgroup$ – E-A May 19 '20 at 23:53

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