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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?

kjetil b halvorsen
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user208618
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    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. – user20160 Feb 11 '19 at 00:02
  • @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 https://en.wikipedia.org/wiki/Prior_probability and there mentions An informative prior expresses specific, definite information about a variable. where is this information given? – user208618 Feb 11 '19 at 01:17
  • @user20160 Your question 'how to identify informative vs. uninformative priors' is an interesting one btw. – user208618 Feb 11 '19 at 01:20
  • @Isa I see. You might consider editing your question to include this. This could help clear up the duplicate issue, and help people understand exactly what you're asking. – user20160 Feb 11 '19 at 01:42
  • @user20160 I've edited – user208618 Feb 11 '19 at 02:03
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    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?. – Xi'an Feb 11 '19 at 06:09
  • @Xi'an notice that I didn't ask 'why there is not such thing as a no-information prior' .My question is where is this $\color\blue{information}$ given in my particular example for instance? – user208618 Feb 11 '19 at 06:16
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    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. – Xi'an Feb 11 '19 at 08:35
  • @Xi'an Ok. I think that question: What information means in this context ? implicitly answer my original question. I'll read the 5 linked questions with their many many answers then.. – user208618 Feb 11 '19 at 19:44
  • @Xi'an In your answer this question https://stats.stackexchange.com/questions/27813/what-is-the-point-of-non-informative-priors you say 'they represent an input from the statistician, hence are informative about something!' and then 'Those priors indeed give a reference against which one can compute either the reference estimator/test/prediction' . So with informative about something do you mean the arbitrary distribution given to the prior by the statistician? – user208618 Feb 12 '19 at 23:52
  • When choosing a prior distribution, whether or not proper, one sets a measure over the parameter space. This measure $\mu$ defines the volumes $\mu(A)$ of measurable sets and as such is informative about which measurable sets are more weighted than others, &tc. – Xi'an Feb 13 '19 at 08:35
  • @Xi'an Is there another way to explain this without measures? I don't understand what you mean with your comment. Also, I read the other 4 links that you mentioned in your previous comment and I couldn't find the answer to my question. – user208618 Feb 13 '19 at 18:39
  • The shortest possible answer is that information is not a mathematically well-defined concept and that many versions have been proposed in the literature, differing from one field to the next and from one era to the next. – Xi'an Feb 13 '19 at 18:55
  • @Xi'an I am not seeking for a mathematically well-defined concept, just an understandable answer . You said that I was going to find an answer in the links and there's nothing. It was not fair from you nor the others to mark my question as [duplicate] then. – user208618 Feb 13 '19 at 19:08

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