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In words, how would I express this notation?

"Given a random variable $f:Z\rightarrow \mathbb R$, its expected value is denoted by $\mathbb{E}_{z \sim \mathcal{D}}[f(z)]$"

This is from the first chapter of "Understanding Machine Learning" by Shwartz:

https://i.stack.imgur.com/skD5a.png https://i.stack.imgur.com/LtG3i.png

Xander Henderson
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Clay
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1 Answers1

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In general, the notation means that $f$ is a function whose domain is $Z$ and whose range is (a subset of ) $R.$ In this context, I would guess $Z$ is the sample space and $R$ is the real numbers based on the mathematical definition of a random variable.

spaceisdarkgreen
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  • Thank you. And do you know how to read the second part? $\mathbb{E}_{z \sim \mathcal{D}}[f(z)]$ – Clay Sep 10 '17 at 01:04
  • Honestly this is not the usual mathematical notation and I don't have the book in front of me to contextualize it. If I had to guess, I'd say that $z\sim \mathcal D$ is expressing that $z$ (an element from the sample space) is drawn from some distribution (in the standard mathematical framework, $\mathcal D$ would correspond to the probability measure on the sample space). I would guess that you would expand this to something like $$ \mathbb E_{z\sim \mathcal D}[f(z)] = \int f(z)p_{\mathcal D}(z), dz$$ where $p_{\mathcal D}(z)$ is the probability distribution on the sample space. – spaceisdarkgreen Sep 10 '17 at 01:23
  • Thank you so much! I added image links if that helps. https://i.stack.imgur.com/skD5a.png https://i.stack.imgur.com/LtG3i.png – Clay Sep 10 '17 at 01:36
  • Yeah, I think that validates what I said above. $E_{z\sim \mathcal D}(f(z))$ is just the expectation value of the RV $f$ and they say they often sensibly abbreviate the notation to $E(f).$ I think they're being precise here to get you used to their formalism. – spaceisdarkgreen Sep 10 '17 at 02:44