Index: The Book of Statistical ProofsProbability Distributions ▷ Multivariate continuous distributions ▷ Normal-gamma distribution ▷ Kullback-Leibler divergence

Theorem: Let $x$ be an $n \times 1$ random vector and let $y$ be a positive random variable. Assume two normal-gamma distributions $P$ and $Q$ specifying the joint distribution of $x$ and $y$ as

\[\label{eq:NGs} \begin{split} P: \; (x,y) &\sim \mathrm{NG}(\mu_1, \Lambda_1^{-1}, a_1, b_1) \\ Q: \; (x,y) &\sim \mathrm{NG}(\mu_2, \Lambda_2^{-1}, a_2, b_2) \; . \\ \end{split}\]

Then, the Kullback-Leibler divergence of $P$ from $Q$ is given by

\[\label{eq:NG-KL} \begin{split} \mathrm{KL}[P\,||\,Q] &= \frac{1}{2} \frac{a_1}{b_1} \left[ (\mu_2 - \mu_1)^\mathrm{T} \Lambda_2 (\mu_2 - \mu_1) \right] + \frac{1}{2} \, \mathrm{tr}(\Lambda_2 \Lambda_1^{-1}) - \frac{1}{2} \ln \frac{|\Lambda_2|}{|\Lambda_1|} - \frac{n}{2} \\ &+ a_2 \, \ln \frac{b_1}{b_2} - \ln \frac{\Gamma(a_1)}{\Gamma(a_2)} + (a_1 - a_2) \, \psi(a_1) - (b_1 - b_2) \, \frac{a_1}{b_1} \; . \end{split}\]

Proof: The probabibility density function of the normal-gamma distribution is

\[\label{eq:NG-pdf} p(x,y) = p(x|y) \cdot p(y) = \mathcal{N}(x; \mu, (y \Lambda)^{-1}) \cdot \mathrm{Gam}(y; a, b) \; .\]

The Kullback-Leibler divergence of the multivariate normal distribution is

\[\label{eq:mvn-KL} \mathrm{KL}[P\,||\,Q] = \frac{1}{2} \left[ (\mu_2 - \mu_1)^\mathrm{T} \Sigma_2^{-1} (\mu_2 - \mu_1) + \mathrm{tr}(\Sigma_2^{-1} \Sigma_1) - \ln \frac{|\Sigma_1|}{|\Sigma_2|} - n \right]\]

and the Kullback-Leibler divergence of the univariate gamma distribution is

\[\label{eq:gam-KL} \mathrm{KL}[P\,||\,Q] = a_2 \, \ln \frac{b_1}{b_2} - \ln \frac{\Gamma(a_1)}{\Gamma(a_2)} + (a_1 - a_2) \, \psi(a_1) - (b_1 - b_2) \, \frac{a_1}{b_1}\]

where $\Gamma(x)$ is the gamma function and $\psi(x)$ is the digamma function.


The KL divergence for a continuous random variable is given by

\[\label{eq:KL-cont} \mathrm{KL}[P\,||\,Q] = \int_{\mathcal{Z}} p(z) \, \ln \frac{p(z)}{q(z)} \, \mathrm{d}z\]

which, applied to the normal-gamma distribution over $x$ and $y$, yields

\[\label{eq:NG-KL0} \mathrm{KL}[P\,||\,Q] = \int_{0}^{\infty} \int_{\mathbb{R}^n} p(x,y) \, \ln \frac{p(x,y)}{q(x,y)} \, \mathrm{d}x \, \mathrm{d}y \; .\]

Using the law of conditional probability, this can be evaluated as follows:

\[\label{eq:NG-KL1} \begin{split} \mathrm{KL}[P\,||\,Q] &= \int_{0}^{\infty} \int_{\mathbb{R}^n} p(x|y) \, p(y) \, \ln \frac{p(x|y) \, p(y)}{q(x|y) \, q(y)} \, \mathrm{d}x \, \mathrm{d}y \\ &= \int_{0}^{\infty} \int_{\mathbb{R}^n} p(x|y)\, p(y) \, \ln \frac{p(x|y)}{q(x|y)} \, \mathrm{d}x \, \mathrm{d}y + \int_{0}^{\infty} \int_{\mathbb{R}^n} p(x|y)\, p(y) \, \ln \frac{p(y)}{q(y)} \, \mathrm{d}x \, \mathrm{d}y \\ &= \int_{0}^{\infty} p(y) \int_{\mathbb{R}^n} p(x|y) \, \ln \frac{p(x|y)}{q(x|y)} \, \mathrm{d}x \, \mathrm{d}y + \int_{0}^{\infty} p(y) \, \ln \frac{p(y)}{q(y)} \int_{\mathbb{R}^n} p(x|y) \, \mathrm{d}x \, \mathrm{d}y \\ &= \left\langle \mathrm{KL}[p(x|y)\,||\,q(x|y)] \right\rangle_{p(y)} + \mathrm{KL}[p(y)\,||\,q(y)] \; . \end{split}\]

In other words, the KL divergence between two normal-gamma distributions over $x$ and $y$ is equal to the sum of a multivariate normal KL divergence regarding $x$ conditional on $y$, expected over $y$, and a univariate gamma KL divergence regarding $y$.


From equations \eqref{eq:NG-pdf} and \eqref{eq:mvn-KL}, the first term becomes

\[\label{eq:exp-mvn-KL-s1} \begin{split} &\left\langle \mathrm{KL}[p(x|y)\,||\,q(x|y)] \right\rangle_{p(y)} \\ &= \left\langle \frac{1}{2} \left[ (\mu_2 - \mu_1)^\mathrm{T} (y \Lambda_2) (\mu_2 - \mu_1) + \mathrm{tr}\left( (y \Lambda_2) (y \Lambda_1)^{-1} \right) - \ln \frac{|(y \Lambda_1)^{-1}|}{|(y \Lambda_2)^{-1}|} - n \right] \right\rangle_{p(y)} \\ &= \left\langle \frac{y}{2} (\mu_2 - \mu_1)^\mathrm{T} \Lambda_2 (\mu_2 - \mu_1) + \frac{1}{2} \, \mathrm{tr}(\Lambda_2 \Lambda_1^{-1}) - \frac{1}{2} \ln \frac{|\Lambda_2|}{|\Lambda_1|} - \frac{n}{2} \right\rangle_{p(y)} \\ \end{split}\]

and using the relation $y \sim \mathrm{Gam}(a,b) \Rightarrow \left\langle y \right\rangle = a/b$, we have

\[\label{eq:exp-mvn-KL-s2} \begin{split} \left\langle \mathrm{KL}[p(x|y)\,||\,q(x|y)] \right\rangle_{p(y)} = \frac{1}{2} \frac{a_1}{b_1} (\mu_2 - \mu_1)^\mathrm{T} \Lambda_2 (\mu_2 - \mu_1) + \frac{1}{2} \, \mathrm{tr}(\Lambda_2 \Lambda_1^{-1}) - \frac{1}{2} \ln \frac{|\Lambda_2|}{|\Lambda_1|} - \frac{n}{2} \; . \end{split}\]

By plugging \eqref{eq:exp-mvn-KL-s2} and \eqref{eq:gam-KL} into \eqref{eq:NG-KL1}, one arrives at the KL divergence given by \eqref{eq:NG-KL}.

Sources:
  • Soch J, Allefeld A (2016): "Kullback-Leibler Divergence for the Normal-Gamma Distribution" ; in: arXiv math.ST , 1611.01437 ; URL: https://arxiv.org/abs/1611.01437 .

Metadata: ID: P6 | shortcut: ng-kl | author: JoramSoch | date: 2019-12-06, 09:35.