normal-gamma
Parameters	${\displaystyle \mu }$location (real) ${\displaystyle \lambda >0}$ (real) ${\displaystyle \alpha >0}$ (real) ${\displaystyle \beta >0}$ (real)
Support	${\displaystyle x\in (-\mathrm{\infty },\mathrm{\infty }),\tau \in (0,\mathrm{\infty })}$
PDF	${\displaystyle f(x,\tau \mid \mu ,\lambda ,\alpha ,\beta )=\frac{{\beta }^{\alpha }\sqrt{\lambda }}{\mathrm{\Gamma }(\alpha )\sqrt{2\pi }}{\tau }^{\alpha -\frac{1}{2}}{e}^{-\beta \tau }{e}^{-\frac{\lambda \tau (x-\mu {)}^2}{2}}}$
Mean	[1]${\displaystyle \mathrm{E}(X)=\mu ,\mathrm{E}(\mathrm{T})=\alpha {\beta }^{-1}}$
Mode	${\displaystyle \left(\mu ,\frac{\alpha -\frac{1}{2}}{\beta }\right)}$
Variance	[1]${\displaystyle \mathrm{var}(X)=(\frac{\beta }{\lambda (\alpha -1)}),\mathrm{var}(\mathrm{T})=\alpha {\beta }^{-2}}$

Inprobability theory andstatistics, thenormal-gamma distribution (orGaussian-gamma distribution) is a bivariate four-parameter family of continuousprobability distributions. It is theconjugate prior of anormal distribution with unknownmean andprecision.^[2]

For a pair ofrandom variables, (X,T), suppose that theconditional distribution ofX givenT is given by

${\displaystyle X\mid T\sim N(\mu ,1/(\lambda T)),}$

meaning that the conditional distribution is anormal distribution withmean${\displaystyle \mu }$ andprecision${\displaystyle \lambda T}$ — equivalently, withvariance${\displaystyle 1/(\lambda T).}$

Suppose also that the marginal distribution ofT is given by

${\displaystyle T\mid \alpha ,\beta \sim \mathrm{Gamma}(\alpha ,\beta ),}$

where this means thatT has agamma distribution. Hereλ,α andβ are parameters of the joint distribution.

Then (X,T) has a normal-gamma distribution, and this is denoted by

${\displaystyle (X,T)\sim \mathrm{NormalGamma}(\mu ,\lambda ,\alpha ,\beta ).}$

The jointprobability density function of (X,T) is

$${\displaystyle f(x,\tau \mid \mu ,\lambda ,\alpha ,\beta )=\frac{{\beta }^{\alpha }\sqrt{\lambda }}{\mathrm{\Gamma }(\alpha )\sqrt{2\pi }}{\tau }^{\alpha -\frac{1}{2}}{e}^{-\beta \tau }\exp \left(-\frac{\lambda \tau (x-\mu {)}^2}{2}\right),}$$

where theconditional probability for$${\displaystyle f(x,\tau \mid \mu ,\lambda ,\alpha ,\beta )=f(x\mid \tau ,\mu ,\lambda ,\alpha ,\beta )f(\tau \mid \mu ,\lambda ,\alpha ,\beta )}$$was used.

By construction, themarginal distribution of${\displaystyle \tau }$ is agamma distribution, and theconditional distribution of${\displaystyle x}$ given${\displaystyle \tau }$ is aGaussian distribution. Themarginal distribution of${\displaystyle x}$ is a three-parameter non-standardizedStudent's t-distribution with parameters${\displaystyle (\nu ,\mu ,{\sigma }^2)=(2\alpha ,\mu ,\beta /(\lambda \alpha ))}$.^{[citation needed]}

The normal-gamma distribution is a four-parameterexponential family withnatural parameters${\displaystyle \alpha -1/2,-\beta -\lambda {\mu }^2/2,\lambda \mu ,-\lambda /2}$ andnatural statistics${\displaystyle \ln \tau ,\tau ,\tau x,\tau x^2}$.^{[citation needed]}

The following moments can be easily computed using themoment generating function of the sufficient statistic:^[3]

${\displaystyle \mathrm{E}(\ln T)=\psi \left(\alpha \right)-\ln \beta ,}$

where${\displaystyle \psi \left(\alpha \right)}$ is thedigamma function,

If${\displaystyle (X,T)\sim \mathrm{N}\mathrm{o}\mathrm{r}\mathrm{m}\mathrm{a}\mathrm{l}\mathrm{G}\mathrm{a}\mathrm{m}\mathrm{m}\mathrm{a}(\mu ,\lambda ,\alpha ,\beta ),}$ then for any${\displaystyle b>0,(bX,bT)}$ is distributed as^{[citation needed]}${\displaystyle \mathrm{N}\mathrm{o}\mathrm{r}\mathrm{m}\mathrm{a}\mathrm{l}\mathrm{G}\mathrm{a}\mathrm{m}\mathrm{m}\mathrm{a}(b\mu ,\lambda /b^3,\alpha ,\beta /b).}$

Assume thatx is distributed according to a normal distribution with unknown mean${\displaystyle \mu }$ and precision${\displaystyle \tau }$.

${\displaystyle x\sim \mathcal{N}(\mu ,{\tau }^{-1})}$

and that the prior distribution on${\displaystyle \mu }$ and${\displaystyle \tau }$,${\displaystyle (\mu ,\tau )}$, has a normal-gamma distribution

${\displaystyle (\mu ,\tau )\sim \text{NormalGamma}({\mu }_0,{\lambda }_0,{\alpha }_0,{\beta }_0),}$

for which the densityπ satisfies

${\displaystyle \pi (\mu ,\tau )\propto {\tau }^{{\alpha }_0-\frac{1}{2}}\exp [-{\beta }_0\tau ]\exp \left[-\frac{{\lambda }_0\tau (\mu -{\mu }_0{)}^2}{2}\right].}$

Suppose

${\displaystyle x_1,\dots ,x_n\mid \mu ,\tau \sim \mathrm{i}.\mathrm{i}.\mathrm{d}.\mathrm{N}\left(\mu ,{\tau }^{-1}\right),}$

i.e. the components of${\displaystyle \mathbf{X}=(x_1,\dots ,x_n)}$ are conditionally independent given${\displaystyle \mu ,\tau }$ and the conditional distribution of each of them given${\displaystyle \mu ,\tau }$ is normal with expected value${\displaystyle \mu }$ and variance${\displaystyle 1/\tau .}$ The posterior distribution of${\displaystyle \mu }$ and${\displaystyle \tau }$ given this dataset${\displaystyle \mathbb{X}}$ can be analytically determined byBayes' theorem^[4] explicitly,

${\displaystyle \mathbf{P}(\tau ,\mu \mid \mathbf{X})\propto \mathbf{L}(\mathbf{X}\mid \tau ,\mu )\pi (\tau ,\mu ),}$

where${\displaystyle \mathbf{L}}$ is the likelihood of the parameters given the data.

Since the data are i.i.d, the likelihood of the entire dataset is equal to the product of the likelihoods of the individual data samples:

${\displaystyle \mathbf{L}(\mathbf{X}\mid \tau ,\mu )=\prod _{i=1}^n\mathbf{L}(x_i\mid \tau ,\mu ).}$

This expression can be simplified as follows:

where${\displaystyle \overline{x}=\frac{1}{n}\sum _{i=1}^n{x}_i}$, the mean of the data samples, and${\displaystyle s=\frac{1}{n}\sum _{i=1}^n(x_i-\overline{x}{)}^2}$, the sample variance.

The posterior distribution of the parameters is proportional to the prior times the likelihood.

The final exponential term is simplified by completing the square.

On inserting this back into the expression above,

This final expression is in exactly the same form as a Normal-Gamma distribution, i.e.,

${\displaystyle \mathbf{P}(\tau ,\mu \mid \mathbf{X})=\text{NormalGamma}\left(\frac{{\lambda }_0{\mu }_0+n\overline{x}}{{\lambda }_0+n},{\lambda }_0+n,{\alpha }_0+\frac{n}{2},{\beta }_0+\frac{1}{2}\left(ns+\frac{{\lambda }_{0}n(\overline{x}-{\mu }_0{)}^2}{{\lambda }_0+n}\right)\right)}$

The interpretation of parameters in terms of pseudo-observations is as follows:

• The new mean takes a weighted average of the old pseudo-mean and the observed mean, weighted by the number of associated (pseudo-)observations.
• The precision was estimated from${\displaystyle 2\alpha }$ pseudo-observations (i.e. possibly a different number of pseudo-observations, to allow the variance of the mean and precision to be controlled separately) with sample mean${\displaystyle \mu }$ and sample variance${\displaystyle \frac{\beta }{\alpha }}$ (i.e. with sum ofsquared deviations${\displaystyle 2\beta }$).
• The posterior updates the number of pseudo-observations (${\displaystyle {\lambda }_0}$) simply by adding the corresponding number of new observations (${\displaystyle n}$).
• The new sum of squared deviations is computed by adding the previous respective sums of squared deviations. However, a third "interaction term" is needed because the two sets of squared deviations were computed with respect to different means, and hence the sum of the two underestimates the actual total squared deviation.

As a consequence, if one has a prior mean of${\displaystyle {\mu }_0}$ from${\displaystyle n_{\mu }}$ samples and a prior precision of${\displaystyle {\tau }_0}$ from${\displaystyle n_{\tau }}$ samples, the prior distribution over${\displaystyle \mu }$ and${\displaystyle \tau }$ is

${\displaystyle \mathbf{P}(\tau ,\mu \mid \mathbf{X})=\mathrm{NormalGamma}\left({\mu }_0,n_{\mu },\frac{n_{\tau }}{2},\frac{n_{\tau }}{2{\tau }_0}\right)}$

and after observing${\displaystyle n}$ samples with mean${\displaystyle \mu }$ and variance${\displaystyle s}$, the posterior probability is

${\displaystyle \mathbf{P}(\tau ,\mu \mid \mathbf{X})=\text{NormalGamma}\left(\frac{n_{\mu }{\mu }_0+n\mu }{n_{\mu }+n},n_{\mu }+n,\frac{1}{2}(n_{\tau }+n),\frac{1}{2}\left(\frac{n_{\tau }}{{\tau }_0}+ns+\frac{n_{\mu }n(\mu -{\mu }_0{)}^2}{n_{\mu }+n}\right)\right)}$

Note that in some programming languages, such asMatlab, the gamma distribution is implemented with the inverse definition of${\displaystyle \beta }$, so the fourth argument of the Normal-Gamma distribution is${\displaystyle 2{\tau }_0/n_{\tau }}$.

Generation of random variates is straightforward:

1. Sample${\displaystyle \tau }$ from a gamma distribution with parameters${\displaystyle \alpha }$ and${\displaystyle \beta }$
2. Sample${\displaystyle x}$ from a normal distribution with mean${\displaystyle \mu }$ and variance${\displaystyle 1/(\lambda \tau )}$

• Thenormal-inverse-gamma distribution is the same distribution parameterized by variance rather than precision
• Thenormal-exponential-gamma distribution

• Bernardo, J.M.; Smith, A.F.M. (1993)Bayesian Theory, Wiley.ISBN 0-471-49464-X
• Dearden et al."Bayesian Q-learning",Proceedings of the Fifteenth National Conference on Artificial Intelligence (AAAI-98), July 26–30, 1998, Madison, Wisconsin, USA.

• Multivariate continuous distributions
• Conjugate prior distributions
• Normal distribution

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