I am watching this video on maximum likelyhood estimation. I'm confused by the notation when the presenter says "assume a set of distributions P-theta." (the URL links to the relevant part of the video:
What is the presenter saying here? I am confused by a bunch of things. I have seen set builder notation, but not this notation with a colon. Are they the same? Also he says that little theta "ranges over" big theta. What does it mean to "range over" something?
Basically there are enough unfamiliar things going on here that I can't totally follow it from context. Is the idea that this statement builds a set of all of the distributions that are possible with all of the different parameters (thetas)? Is this the idea?
"$\theta$ ranges over $\Theta$" means that the value of $\theta$ could be any member of the set $\Theta$ but not anything else.
The distribution $P_\theta$ depends on the value of $\theta$. Change the value of $\theta$ and you change the probability distribution $P_\theta$.
The set $\{P_\theta : \theta\in\Theta\}$, also denoted $\{P_\theta \mid \theta\in\Theta\}$, is simply the set of all probability distributions $P_\theta$ corresponding to the value of $\theta$ belonging to the set $\Theta$.