# Talk:Normal-gamma distribution

Can anyone write this article? (Specifically, the pdf of the normal-gamma distribution would be really useful to me. It might be in your probability textbook, perhaps Bernardo and Smith?) MisterSheik 19:20, 25 April 2007 (UTC)

## Definition

According to both Christopher Bishop's "Pattern Recognition and Machine Learning" and Bernardo and Smith's "Bayesian Theory" the Normal gamma distribution only has three parameters: $\mu$, $\alpha$, and $\beta$. The $\lambda$ parameter is simply $(2 \alpha - 1)^{-1}$. Should we change the definition or at least mention this common convention for the value of $\lambda$?

## Maximum Likelihood Estimation

This section has some problems. First, the notation does not agree with the notation used in the definition section.

Second, conventions used in this section disagree with those in the Definition section. In particular, this section seems to use $\theta$ as the second parameter of the gamma function, whereas the Definition section uses the $\beta = \theta^{-1}$ convention.

Third, the result for the last parameter of the normal-gaussian posterior seems to disagree with those in Bishop's "Pattern Recognition and Machine Learning" and Bernardo and Smith's "Bayesian Theory." This may be due to the different parameterization used here, but I couldn't get the results to agree after re-parameterizing to their forms. Incidentally, Bishop and Bernardo disagree with each other, but both give solutions with much simpler form than the solution provided here. Bishop's approach is an elegant two-line derivation that's easy to follow; I found the six-line derivation provided here to be rather verbose. I might suggest a re-write using bishop's parameterization and results.

Finally, it seems like the title of this section is misleading, since a prior is involved. Maybe "Maximum A-Posteriori Estimate" would be more accurate?

Ksimek (talk) 06:18, 23 August 2009 (UTC)

## naming conventions

I think it should be $\tau$ instead of $\sigma$ since $\sigma$ is usually used for standard deviations but here it is used as precision, which is usually labeled $\tau$. 129.26.160.2 11:40, 14 September 2007 (UTC)

## Infobox

I think that the infobox is misleading. This is a multivariate distribution. So are the reported means, variances, etc. for $x$ or $\tau$? In this case these values are vectors of dimension 2. —Preceding unsigned comment added by Rhaertel80 (talkcontribs) 06:29, 29 May 2008 (UTC)

The infobox is right. You could draw a mean from the normal distribution and a precision from a gamma distribution, and then defining a normal distribution using that mean and precision, you could draw a value. These final drawn values have the given variance and mean (in fact, they're distributed according to a student's t distribution.) MisterSheik (talk) 23:00, 22 September 2008 (UTC)
The infobox is right for the $x$ variable. $\tau$ mean and variance should also be added though. 82.41.241.70 (talk) 15:04, 2 November 2008 (UTC)

## Problem with the formula

Something's wrong with the definition of normal-gamma. \gamma should vary inversely with variance, but it does not. MisterSheik (talk) 23:00, 22 September 2008 (UTC)

I looked into the references, and the paper cited does indeed define it this way as a prior. That paper cites a book which has a method for deriving priors--but does not have this distribution defined. There are however, other papers with a different definition based on the sum (or convolution) of the variables which has no known analytical distribution. In particular Plancade S., Rozenholc Y. and Lund E., Improving background correction for Illumina BeadArrays: the normal-gamma model (http://arxiv.org/abs/1112.4180)., and they also have an R package that solves a "NormalGamma" as a sum of a normal and gamma variable. I think this article should reference both models, since there doesn't seem to be consensus about the use of this name in defining the distribution.

Shawn@garbett.org (talk) 17:20, 16 August 2012 (UTC)

I found this: http://onlinelibrary.wiley.com/doi/10.1002/0471667196.ess1827.pub2/full. It uses the nomenclature Normal-gamma prior Distribution. This makes more sense in describing this distribution. It's obviously useful, as I've found several more articles on it.

Shawn@garbett.org (talk) 15:16, 17 August 2012 (UTC)

## Expert-subject tag

I have added the expert-subject tag because the article needs a thorough overhaul, including giving better context and applications, and using a consistent notation throughout. There are also many section headings that will probally never have any content. Melcombe (talk) 09:44, 27 May 2010 (UTC)

## Generating normal-gamma random variates

I got really confused in this section.Assume I perform a gibbs sampler on the gamma-normal distribution. Then, the way gibbs sampler sample random numbers is

0. i = 0.set a random number to x_0

1. Given x_i, sample random number y_i from gamma-normal

2. Given y_i, sample random number x_{i+1} from gamma-normal

3. i = i+1 and go 1.

Now, compared the gibbs sampler with the proposed random number generation in the article, 1 is different because of the surviving tau term in an exponential term stemmed from the normal distribution. However, the gibbs sampler is supposed to sample true random numbers given sufficient burn-in period and enough thinning. Now, is the proposed random number generation correct in the article? I do not think so. — Preceding unsigned comment added by Ssoga (talkcontribs) 04:39, 13 March 2014 (UTC)

## Big edit

I did a lot of editing to the page in the 'Posterior distribution of the parameters' and the 'Interpretation of parameters' sections. I re-did the notation to line up more with how it would be used, fixed some of the errors in the application of Bayes rule, and made it more clear how the parameters of the gamma-normal distribution are related to pseudo-observations. I am not sure what to do about notation on this page. Normally, this distribution would be used as the conjugate distribution for a gaussian. In that case, having $\mu$ be one of the parameters of the gamma-normal is confusing. On the other hand, changing it to something else in the intro section makes the connections between the gamma and the normal definition less clear. As a result, I just left the upper notation as-is, and introduced new notation when talking about pseudo-observations. Future tweaks for clarity would be helpful. Howeman (talk) 08:16, 25 August 2012 (UTC)