Index: The Book of Statistical ProofsProbability DistributionsMultivariate continuous distributions ▷ Normal-gamma distribution ▷ Probability density function

Theorem: Let $x$ and $y$ follow a normal-gamma distribution:

$\label{eq:ng} x,y \sim \mathrm{NG}(\mu, \Lambda, a, b) \; .$

Then, the joint probability density function of $x$ and $y$ is

$\label{eq:ng-pdf} p(x,y) = \sqrt{\frac{|\Lambda|}{(2 \pi)^n}} \frac{b^a}{\Gamma(a)} \cdot y^{a+\frac{n}{2}-1} \exp \left[ -\frac{y}{2} \left( (x-\mu)^\mathrm{T} \Lambda (x-\mu) + 2b \right) \right] \; .$

Proof: The normal-gamma distribution is defined as $X$ conditional on $Y$ following a multivariate distribution and $Y$ following a gamma distribution:

$\label{eq:mvn-gam} \begin{split} X \vert Y &\sim \mathcal{N}(\mu, (Y \Lambda)^{-1}) \\ Y &\sim \mathrm{Gam}(a, b) \; . \end{split}$

Thus, using the probability density function of the multivariate normal distribution and the probability density function of the gamma distribution, we have the following probabilities:

$\label{eq:mvn-gam-pdf} \begin{split} p(x \vert y) &= \mathcal{N}(x; \mu, (y \Lambda)^{-1}) \\ &= \sqrt{\frac{|y \Lambda|}{(2 \pi)^n}} \exp \left[ -\frac{1}{2} (x-\mu)^\mathrm{T} (y \Lambda) (x-\mu) \right] \\ p(y) &= \mathrm{Gam}(y; a, b) \\ &= \frac{b^a}{\Gamma(a)} y^{a-1} \exp\left[-by\right] \; . \end{split}$

The law of conditional probability implies that

$\label{eq:prob-cond} p(x,y) = p(x \vert y) \, p(y) \; ,$

such that the normal-gamma density function becomes:

$\label{eq:ng-pdf-prod} p(x,y) = \sqrt{\frac{|y \Lambda|}{(2 \pi)^n}} \exp \left[ -\frac{1}{2} (x-\mu)^\mathrm{T} (y \Lambda) (x-\mu) \right] \cdot \frac{b^a}{\Gamma(a)} y^{a-1} \exp\left[-by\right] \; .$

Using the relation $\lvert y A \rvert = y^n \lvert A \rvert$ for an $n \times n$ matrix $A$ and rearranging the terms, we have:

$\label{eq:ng-pdf-qed} p(x,y) = \sqrt{\frac{|\Lambda|}{(2 \pi)^n}} \frac{b^a}{\Gamma(a)} \cdot y^{a+\frac{n}{2}-1} \exp \left[ -\frac{y}{2} \left( (x-\mu)^\mathrm{T} \Lambda (x-\mu) + 2b \right) \right] \; .$
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Metadata: ID: P44 | shortcut: ng-pdf | author: JoramSoch | date: 2020-02-07, 20:44.