Talk:Normal-inverse-gamma distribution
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The CDF does not look right, if sigma^2 to Infty, the CDF goes to zero if I am not mistaken, which does not make sense, no? DoubleMatchPoint (talk) 19:57, 8 November 2017 (UTC)
We probably ought to make the parameters match with those listed in the Conjugate prior article or vice-versa. --Rhaertel80 (talk) 16:56, 29 May 2008 (UTC)
Notation is inconsistent with regard to the distribution of . In the definition, mean is and variance is , but in the section about generating values from the distribution, mean is and variance is . I don't have a reference handy so I'm not sure which one is correct. Ksimek (talk) 23:40, 12 June 2009 (UTC)
User:rewtnode: I'm trying to verify the formula for the Expectation of \sigma^2 E[sigma^2] which is given now as \beta/(\alpha-1/2), but was just recently changed. Just a few days ago it was \beta/(2 (\alpha-1)). But I tried to carry out the integral myself and come to the old result. The issue is whether we assume this is a density function of (x, \sigma^2) or of (x, \sigma). If I assume it's a function of (x,\sigma) I get the old result E[\sigma^2] = \beta/(2(\alpha-1)). However if I assume that it is a function of (x,\sigma^2) I need to do a different variable substitution in the integral and get the result E[\sigma^2] = \beta^{1/2} \Gamma(\alpha-3/2)/\Gamma(\alpha). Very confusing. So I wonder if this distribution should be indeed seen as function of (x, \sigma), that is, a joint density over the domain (x, \sigma) — Preceding unsigned comment added by Rewtnode (talk • contribs) 00:40, 8 June 2018 (UTC)
Derivation of expected value of
editThere seems to be some disagreement about whether is (correct) or (incorrect). To get this right, you have to remember to integrate with respect to and not since the support of the distribution is defined in terms of and .
To make the above clear, all instances of below are written as to indicate that is our variable and not .
From the definition of :
From the definition of the normal-inverse-gamma distribution:
Rearrange:
Integrate out , which appears as a squared exponential function (proportional to the pdf of the normal distribution):
Simplify:
Integrate out , which appears in the same form as the pdf of an inverse-gamma distribution with argument :
We can further confirm this result by generating samples from the normal-inverse-gamma distribution (the sampling procedure is described in the article) and estimating the expected value of the argument empirically. It converges to .
On the marginal distribution of x in the univariate case
editThe formula for the marginal distribution of is wrong (which can be readily observed by comparison with the result of the proof reported for the case below) and is taken from the posterior predictive formula that one can find in the table of Conjugate prior.
I think the correct formula is . A derivation with a different notation for the parameters can be found in [1]https://bookdown.org/aramir21/IntroductionBayesianEconometricsGuidedTour/sec42.html#sec42:~:text=%2C%20respectively.-,The%20marginal%20posterior%20of,is,-%CF%80 Br1 Ursino (talk) 19:05, 21 September 2023 (UTC)