Index: The Book of Statistical ProofsProbability Distributions ▷ Matrix-variate continuous distributions ▷ Wishart distribution ▷ Kullback-Leibler divergence

Theorem: Let $S$ be a $p \times p$ random matrix. Assume two Wishart distributions $P$ and $Q$ specifying the probability distribution of $S$ as

\[\label{eq:wishs} \begin{split} P: \; S &\sim \mathcal{W}(V_1, n_1) \\ Q: \; S &\sim \mathcal{W}(V_2, n_2) \; . \end{split}\]

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

\[\label{eq:wish-KL} \mathrm{KL}[P\,||\,Q] = \frac{1}{2} \left[ n_2 \left( \ln |V_2| - \ln |V_1| \right) + n_1 \mathrm{tr}(V_2^{-1} V_1) + 2 \ln \frac{\Gamma_p\left(\frac{n_2}{2}\right)}{\Gamma_p\left(\frac{n_1}{2}\right)} + (n_1-n_2) \psi_p\left(\frac{n_1}{2}\right) - n_1 p \right]\]

where $\Gamma_p(x)$ is the multivariate gamma function

\[\label{eq:mult-gam-fct} \Gamma_p(x) = \pi^{p(p-1)/4} \, \prod_{j=1}^k \Gamma\left(x - \frac{j-1}{2}\right)\]

and $\psi_p(x)$ is the multivariate digamma function

\[\label{eq:mult-psi-fct} \psi_p(x) = \frac{\mathrm{d}\ln \Gamma_p(x)}{\mathrm{d}x} = \sum_{j=1}^k \psi\left(x - \frac{j-1}{2}\right) \; .\]

Proof: The KL divergence for a continuous random variable is given by

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

which, applied to the Wishart distributions in \eqref{eq:wishs}, yields

\[\label{eq:wish-KL-s1} \begin{split} \mathrm{KL}[P\,||\,Q] &= \int_{\mathcal{S}^p} \mathcal{W}(S; V_1, n_1) \, \ln \frac{\mathcal{W}(S; V_1, n_1)}{\mathcal{W}(S; V_2, n_2)} \, \mathrm{d}S \\ &= \left\langle \ln \frac{\mathcal{W}(S; \alpha_1)}{\mathcal{W}(S; \alpha_1)} \right\rangle_{p(S)} \end{split}\]

where $\mathcal{S}^p$ is the set of all positive-definite symmetric $p \times p$ matrices.

Using the probability density function of the Wishart distribution, this becomes:

\[\label{eq:wish-KL-s2} \begin{split} \mathrm{KL}[P\,||\,Q] &= \left\langle \ln \frac{\frac{1}{\sqrt{2^{n_1 p} |V_1|^{n_1}} \Gamma_p \left( \frac{n_1}{2} \right)} \cdot |S|^{(n_1-p-1)/2} \cdot \exp\left[ -\frac{1}{2} \mathrm{tr}\left( V_1^{-1} S \right) \right]}{\frac{1}{\sqrt{2^{n_2 p} |V_2|^{n_2}} \Gamma_p \left( \frac{n_2}{2} \right)} \cdot |S|^{(n_2-p-1)/2} \cdot \exp\left[ -\frac{1}{2} \mathrm{tr}\left( V_2^{-1} S \right) \right]} \right\rangle_{p(S)} \\ &= \left\langle \ln \left( \sqrt{2^{(n_2-n_1)p} \cdot \frac{|V_2|^{n_2}}{|V_1|^{n_1}}} \cdot \frac{\Gamma_p\left( \frac{n_2}{2} \right)}{\Gamma_p\left( \frac{n_1}{2} \right)} \cdot |S|^{(n_1-n_2)/2} \cdot \exp\left[ -\frac{1}{2} \mathrm{tr}\left( V_1^{-1} S \right) +\frac{1}{2} \mathrm{tr}\left( V_2^{-1} S \right) \right] \right) \right\rangle_{p(S)} \\ &= \left\langle \frac{(n_2-n_1)p}{2} \ln 2 + \frac{n_2}{2} \ln |V_2| - \frac{n_1}{2} \ln |V_1| + \ln \frac{\Gamma_p\left( \frac{n_2}{2} \right)}{\Gamma_p\left( \frac{n_1}{2} \right)} \right. \\ &+ \left. \quad \frac{n_1-n_2}{2} \ln |S| - \frac{1}{2} \mathrm{tr}\left( V_1^{-1} S \right) + \frac{1}{2} \mathrm{tr}\left( V_2^{-1} S \right) \right\rangle_{p(S)} \\ &= \frac{(n_2-n_1)p}{2} \ln 2 + \frac{n_2}{2} \ln |V_2| - \frac{n_1}{2} \ln |V_1| + \ln \frac{\Gamma_p\left( \frac{n_2}{2} \right)}{\Gamma_p\left( \frac{n_1}{2} \right)} \\ &+ \frac{n_1-n_2}{2} \left\langle \ln |S| \right\rangle_{p(S)} - \frac{1}{2} \left\langle \mathrm{tr}\left( V_1^{-1} S \right) \right\rangle_{p(S)} + \frac{1}{2} \left\langle \mathrm{tr}\left( V_2^{-1} S \right) \right\rangle_{p(S)} \; . \end{split}\]

Using the expected value of a Wishart random matrix

\[\label{eq:wish-mean} S \sim \mathcal{W}(V,n) \quad \Rightarrow \quad \left\langle S \right\rangle = n V \; ,\]

such that the expected value of the matrix trace becomes

\[\label{eq:wish-trmean} \left\langle \mathrm{tr}(AS) \right\rangle = \mathrm{tr}\left( \left\langle AS \right\rangle \right) = \mathrm{tr}\left( A \left\langle S \right\rangle \right) = \mathrm{tr}\left( A \cdot (nV) \right) = n \cdot \mathrm{tr}(AV) \; ,\]

and the expected value of a Wishart log-determinant

\[\label{eq:wish-logdetmean} S \sim \mathcal{W}(V,n) \quad \Rightarrow \quad \left\langle \ln |S| \right\rangle = \psi_p\left(\frac{n}{2}\right) + p \cdot \ln 2 + \ln |V| \; ,\]

the Kullback-Leibler divergence from \eqref{eq:wish-KL-s2} becomes:

\[\label{eq:wish-KL-s3} \begin{split} \mathrm{KL}[P\,||\,Q] &= \frac{(n_2-n_1)p}{2} \ln 2 + \frac{n_2}{2} \ln |V_2| - \frac{n_1}{2} \ln |V_1| + \ln \frac{\Gamma_p\left( \frac{n_2}{2} \right)}{\Gamma_p\left( \frac{n_1}{2} \right)} \\ &+ \frac{n_1-n_2}{2} \left[ \psi_p\left(\frac{n_1}{2}\right) + p \cdot \ln 2 + \ln |V_1| \right] - \frac{n_1}{2} \mathrm{tr}\left( V_1^{-1} V_1 \right) + \frac{n_1}{2} \mathrm{tr}\left( V_2^{-1} V_1 \right) \\ &= \frac{n_2}{2} \left( \ln |V_2| - \ln |V_1| \right) + \ln \frac{\Gamma_p\left( \frac{n_2}{2} \right)}{\Gamma_p\left( \frac{n_1}{2} \right)} + \frac{n_1-n_2}{2} \psi_p\left(\frac{n_1}{2}\right) - \frac{n_1}{2} \mathrm{tr}\left( I_p \right) + \frac{n_1}{2} \mathrm{tr}\left( V_2^{-1} V_1 \right) \\ & = \frac{1}{2} \left[ n_2 \left( \ln |V_2| - \ln |V_1| \right) + n_1 \mathrm{tr}(V_2^{-1} V_1) + 2 \ln \frac{\Gamma_p\left(\frac{n_2}{2}\right)}{\Gamma_p\left(\frac{n_1}{2}\right)} + (n_1-n_2) \psi_p\left(\frac{n_1}{2}\right) - n_1 p \right] \; . \end{split}\]

Metadata: ID: P295 | shortcut: wish-kl | author: JoramSoch | date: 2021-12-02, 15:33.