When using the "non-informative" prior $\pi(\mu,\sigma)\propto\frac{1}{\sigma^2}$ where $\pi(\mu)\propto1$ and $\pi(\sigma^2)\propto\frac{1}{\sigma^2}$

Where is the no information for the parameter?

and if would be informative prior, Where one would see the 'information' given?

I saw the definition on Wikipedia https://en.wikipedia.org/wiki/Prior_probability and there mentions 'An informative prior expresses specific, definite $\color{blue}{\text{information}}$ about a variable.'

My question is where is this $\color{blue}{\text{information}}$ given in my particular example for instance?

Could you help please?

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    $\begingroup$ I don't think this question is a duplicate. The proposed duplicate is asking about the rationale for using informative vs. uninformative priors, whereas this question seems to be asking about how to identify informative vs. uninformative priors. $\endgroup$
    – user20160
    Feb 11, 2019 at 0:02
  • $\begingroup$ @user20160 I don't think this is a duplicate neither. Though I didn't mean to ask how to identify informative vs. uninformative priors. I meant the current questions actually :) . I saw the definitions on Wikipedia en.wikipedia.org/wiki/Prior_probability and there mentions An informative prior expresses specific, definite information about a variable. where is this information given? $\endgroup$
    – user208618
    Feb 11, 2019 at 1:17
  • $\begingroup$ @user20160 Your question 'how to identify informative vs. uninformative priors' is an interesting one btw. $\endgroup$
    – user208618
    Feb 11, 2019 at 1:20
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    $\begingroup$ There are several entries on X validated that discuss why there is not such thing as a no-information prior: What is the point of non-informative priors? and History of uninformative prior theory and Why are Jeffreys priors considered noninformative? and What is an “uninformative prior”? Can we ever have one with truly no information?. $\endgroup$
    – Xi'an
    Feb 11, 2019 at 6:09
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    $\begingroup$ I still think this large collection of detailed and pertinent answers all address the most elusive notion of information that your question seeks. Rather than asking where the information is, you should consider what information means in this context. $\endgroup$
    – Xi'an
    Feb 11, 2019 at 8:35