normal-gamma

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

In probability theory and statistics, the normal-gamma distribution (or Gaussian-gamma distribution) is a bivariate four-parameter family of continuous probability distributions. It is the conjugate prior of a normal distribution with unknown mean and precision.^[2]

Definition

For a pair of random variables, (X,T), suppose that the conditional distribution of X given T is given by

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

meaning that the conditional distribution is a normal distribution with mean ${\displaystyle \mu }$ and precision ${\displaystyle \lambda T}$ — equivalently, with variance ${\displaystyle 1/(\lambda T).}$

Suppose also that the marginal distribution of T is given by

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

where this means that T has a gamma distribution. Here λ, α and β are parameters of the joint distribution.

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

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

Properties

Posterior distribution of the parameters

Assume that x 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 density π 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

${\displaystyle x_{1},\ldots ,x_{n}\mid \mu ,\tau \sim \operatorname {{i.}{i.}{d.}} \operatorname {N} \left(\mu ,\tau ^{-1}\right),}$

i.e. the components of ${\displaystyle \mathbf {X} =(x_{1},\ldots ,x_{n})}$ 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 by Bayes' 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:

${\displaystyle {\begin{aligned}\mathbf {L} (\mathbf {X} \mid \tau ,\mu )&\propto \prod _{i=1}^{n}\tau ^{1/2}\exp \left[{\frac {-\tau }{2}}(x_{i}-\mu )^{2}\right]\\[5pt]&\propto \tau ^{n/2}\exp \left[{\frac {-\tau }{2}}\sum _{i=1}^{n}(x_{i}-\mu )^{2}\right]\\[5pt]&\propto \tau ^{n/2}\exp \left[{\frac {-\tau }{2}}\sum _{i=1}^{n}(x_{i}-{\bar {x}}+{\bar {x}}-\mu )^{2}\right]\\[5pt]&\propto \tau ^{n/2}\exp \left[{\frac {-\tau }{2}}\sum _{i=1}^{n}\left((x_{i}-{\bar {x}})^{2}+({\bar {x}}-\mu )^{2}\right)\right]\\[5pt]&\propto \tau ^{n/2}\exp \left[{\frac {-\tau }{2}}\left(ns+n({\bar {x}}-\mu )^{2}\right)\right],\end{aligned}}}$

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

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

${\displaystyle {\begin{aligned}\mathbf {P} (\tau ,\mu \mid \mathbf {X} )&\propto \mathbf {L} (\mathbf {X} \mid \tau ,\mu )\pi (\tau ,\mu )\\&\propto \tau ^{n/2}\exp \left[{\frac {-\tau }{2}}\left(ns+n({\bar {x}}-\mu )^{2}\right)\right]\tau ^{\alpha _{0}-{\frac {1}{2}}}\,\exp[{-\beta _{0}\tau }]\,\exp \left[-{\frac {\lambda _{0}\tau (\mu -\mu _{0})^{2}}{2}}\right]\\&\propto \tau ^{{\frac {n}{2}}+\alpha _{0}-{\frac {1}{2}}}\exp \left[-\tau \left({\frac {1}{2}}ns+\beta _{0}\right)\right]\exp \left[-{\frac {\tau }{2}}\left(\lambda _{0}(\mu -\mu _{0})^{2}+n({\bar {x}}-\mu )^{2}\right)\right]\end{aligned}}}$

The final exponential term is simplified by completing the square.

${\displaystyle {\begin{aligned}\lambda _{0}(\mu -\mu _{0})^{2}+n({\bar {x}}-\mu )^{2}&=\lambda _{0}\mu ^{2}-2\lambda _{0}\mu \mu _{0}+\lambda _{0}\mu _{0}^{2}+n\mu ^{2}-2n{\bar {x}}\mu +n{\bar {x}}^{2}\\&=(\lambda _{0}+n)\mu ^{2}-2(\lambda _{0}\mu _{0}+n{\bar {x}})\mu +\lambda _{0}\mu _{0}^{2}+n{\bar {x}}^{2}\\&=(\lambda _{0}+n)(\mu ^{2}-2{\frac {\lambda _{0}\mu _{0}+n{\bar {x}}}{\lambda _{0}+n}}\mu )+\lambda _{0}\mu _{0}^{2}+n{\bar {x}}^{2}\\&=(\lambda _{0}+n)\left(\mu -{\frac {\lambda _{0}\mu _{0}+n{\bar {x}}}{\lambda _{0}+n}}\right)^{2}+\lambda _{0}\mu _{0}^{2}+n{\bar {x}}^{2}-{\frac {\left(\lambda _{0}\mu _{0}+n{\bar {x}}\right)^{2}}{\lambda _{0}+n}}\\&=(\lambda _{0}+n)\left(\mu -{\frac {\lambda _{0}\mu _{0}+n{\bar {x}}}{\lambda _{0}+n}}\right)^{2}+{\frac {\lambda _{0}n({\bar {x}}-\mu _{0})^{2}}{\lambda _{0}+n}}\end{aligned}}}$

On inserting this back into the expression above,

${\displaystyle {\begin{aligned}\mathbf {P} (\tau ,\mu \mid \mathbf {X} )&\propto \tau ^{{\frac {n}{2}}+\alpha _{0}-{\frac {1}{2}}}\exp \left[-\tau \left({\frac {1}{2}}ns+\beta _{0}\right)\right]\exp \left[-{\frac {\tau }{2}}\left(\left(\lambda _{0}+n\right)\left(\mu -{\frac {\lambda _{0}\mu _{0}+n{\bar {x}}}{\lambda _{0}+n}}\right)^{2}+{\frac {\lambda _{0}n({\bar {x}}-\mu _{0})^{2}}{\lambda _{0}+n}}\right)\right]\\&\propto \tau ^{{\frac {n}{2}}+\alpha _{0}-{\frac {1}{2}}}\exp \left[-\tau \left({\frac {1}{2}}ns+\beta _{0}+{\frac {\lambda _{0}n({\bar {x}}-\mu _{0})^{2}}{2(\lambda _{0}+n)}}\right)\right]\exp \left[-{\frac {\tau }{2}}\left(\lambda _{0}+n\right)\left(\mu -{\frac {\lambda _{0}\mu _{0}+n{\bar {x}}}{\lambda _{0}+n}}\right)^{2}\right]\end{aligned}}}$

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

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

Generating normal-gamma random variates

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 ${\displaystyle 1/(\lambda \tau )}$

Related distributions

• The normal-inverse-gamma distribution is essentially the same distribution parameterized by variance rather than precision
• The normal-exponential-gamma distribution

Notes

References

• 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.