0

I understand that $\theta$ is the true distribution parameter (great explanation here). I also know that $\hat\theta$ is an estimator of the true $\theta$ (so for example, MLE is an example of $\hat\theta$).

But sometimes, I also see $\theta^*$. What does it denote, and what is its relationship to $\theta$ and $\hat\theta$?

  • During my search, I came across this MIT lecture, which seems to attach different meanings to the three types of theta, but I couldn't really get the relationship. An example slide from that lecture (can be seen on 25:50):

    enter image description here

HeyJude
  • 350
  • 3
  • 8
  • 2
    Fair warning: notation is not globally consistent across all statistics or machine learning contexts, so there can't really be an answer. It would perhaps be better to show a context in which you see these symbols occurring. – Arya McCarthy Jan 04 '24 at 01:04
  • @AryaMcCarthy, that's funny, because I didn't want to limit the question to a certain case. But I guess you're right, so example added. – HeyJude Jan 04 '24 at 01:12
  • 1
    $\theta^$ is defined right on the slide in your question -- "the true* parameter". i.e. at least for that slide, it's the thing we're trying to estimate. In the video, the definition is different; he explains what $\theta^*$ is just before the 19 minute mark. – Glen_b Jan 04 '24 at 01:14
  • @Glen_b but then what is plain $\theta$? – HeyJude Jan 04 '24 at 01:20
  • 1
    In the slide, clearly not the true parameter. Where $\theta$ is used in the slide, it seems to be treated like a variable in algebra, seemingly representing a potential value that the parameter might take; more context for it might arise from the rest of the lecture (or indeed earlier ones). I'm not watching more of that long video to try to guess exactly how he's using it there; it might well have been defined in an earlier lecture. Indeed, from what little I watched it's possible the meaning changes slightly from the first 17 odd minutes to the part he's discussing when he defines $\theta^*$ – Glen_b Jan 04 '24 at 01:23
  • Sometimes variables are explicitly defined, either in text or words, sometimes you will need to infer it from how it's being used. – Glen_b Jan 04 '24 at 02:03
  • @Glen_b, you're great, thanks. I really appreciate it. – HeyJude Jan 04 '24 at 02:08

0 Answers0