Hartigan’s Method for $k$-MLE: Mixture Modeling with Wishart Distributions and Its Application to Motion Retrieval

Abstract

We describe a novel algorithm called $k$-Maximum Likelihood Estimator ($k$-MLE) for learning finite statistical mixtures of exponential families relying on Hartigan’s $k$-means swap clustering method. To illustrate this versatile Hartigan $k$-MLE technique, we consider the exponential family of Wishart distributions and show how to learn their mixtures. First, given a set of symmetric positive definite observation matrices, we provide an iterative algorithm to estimate the parameters of the underlying Wishart distribution which is guaranteed to converge to the MLE. Second, two initialization methods for $k$-MLE are proposed and compared. Finally, we propose to use the Cauchy-Schwartz statistical divergence as a dissimilarity measure between two Wishart mixture models and sketch a general methodology for building a motion retrieval system.

Appendix A

This Appendix details some calculations for distributions $\mathcal {W}_{d}$, $\mathcal {W}_{d,\underline{n}}$, $\mathcal {W}_{d,\underline{S}}$.

11.1.1 Wishart Distribution $\mathcal {W}_{d}$

$$\begin{aligned} \mathcal {W}_{d}(X;n,S)&= \frac{|X|^{\frac{n-d-1}{2}}\exp \{-\frac{1}{2} {\mathrm {tr}}(S^{-1} X)\}}{2^{\frac{nd}{2}} |S|^{\frac{n}{2}} \varGamma _{d}(\frac{n}{2})}\\&= \exp \left\{ \frac{n-d-1}{2} \log |X|-\frac{1}{2} {\mathrm {tr}}(S^{-1} X)- \frac{nd}{2}\log (2) - \frac{n}{2} \log |S| - \log \varGamma _{d}\left( \frac{n}{2}\right) \right\} \end{aligned}$$

Letting $(\theta _{n},\theta _{S})=(\frac{n-d-1}{2},S^{-1}) \longleftrightarrow (n,S) = (2\theta _{n}+d+1,\theta _{S}^{-1})$

$$\begin{aligned} \mathcal {W}_{d}(X;\theta _{n},\theta _{S}) =&\exp \left\{ \frac{2\theta _{n} + d + 1-d-1}{2} \log |X|-\frac{1}{2} {\mathrm {tr}}(\theta _{S} X)- \frac{(2\theta _{n} + d + 1)d}{2}\log (2) \right. \\&- \left. \frac{(2\theta _{n} + d + 1)}{2} \log |\theta _{S}^{-1}| - \log \varGamma _{d}\left( \frac{2\theta _{n} + d + 1}{2}\right) \right\} \\ =&\exp \left\{ \theta _{n}\log |X|-\frac{1}{2} {\mathrm {tr}}(\theta _{S} X)- \left( \theta _{n} + \frac{(d + 1)}{2}\right) \left( d\log (2) - \log |\theta _{S}|\right) \right. \\&\left. - \log \varGamma _{d}\left( \theta _{n} + \frac{(d + 1)}{2}\right) \right\} \\ =&\exp \left\{ <\theta _{n},\log |X|>_{\mathbb {R}} + <\theta _{S},- \frac{1}{2}X>_{HS} - F(\varTheta )\right\} \\&\text{ with } F(\varTheta ) = \left( \theta _{n} + \frac{(d + 1)}{2}\right) \left( d\log (2) - \log |\theta _{S}|\right) + \log \varGamma _{d}\left( \theta _{n} + \frac{(d + 1)}{2}\right) \\ =&\exp \left\{ <\varTheta ,t(X)> - F(\varTheta ) + k(X)\right\} \\&\text{ with } t(X) =(\log |X|,- \frac{1}{2}X) \text{ and } k(X)=0 \end{aligned}$$

$$F(\varTheta ) = \left( \theta _{n} + \frac{(d + 1)}{2}\right) \left( d\log (2) - \log |\theta _{S}|\right) + \log \varGamma _{d}\left( \theta _{n} + \frac{(d + 1)}{2}\right) $$

$$\begin{aligned} \frac{\partial F}{\partial \theta _{n}}(\theta _{n},\theta _{S}) = d\log (2) - \log |\theta _{S}|+ \varPsi _{d}\left( \theta _{n} + \frac{(d + 1)}{2}\right) \end{aligned}$$

(11.43)

where $\varPsi _{d}$ is the multivariate Digamma function (or multivariate polygamma of order 0).

$$\begin{aligned} \frac{\partial F}{\partial \theta _{S}}(\theta _{n},\theta _{S}) = - \left( \theta _{n} + \frac{(d + 1)}{2}\right) \theta _{S}^{-1} \end{aligned}$$

(11.44)

Dissimilarity $\varDelta (\theta ,\theta ')$ between natural parameters $\theta =(\theta _{n},\theta _{S})$ and $\theta '=(\theta '_{n},\theta '_{S})$ is

$$\begin{aligned} \varDelta (\theta ,\theta ') =&F(\theta +\theta ') - (F(\theta ) + F(\theta ')) = \left( \theta _{n} + \theta _{n}'+ \frac{(d + 1)}{2}\right) \left( d\log (2) - \log |\theta _{S}+\theta '_{S}|\right) \nonumber \\&-\left( \theta _{n} + \frac{(d + 1)}{2}\right) \left( d\log (2) - \log |\theta _{S}|\right) - \left( \theta _{n}'+ \frac{(d + 1)}{2}\right) \left( d\log (2) - \log |\theta '_{S}|\right) \nonumber \\&+ \log \varGamma _{d}\left( \theta _{n} + \theta '_{n} + \frac{(d + 1)}{2}\right) - \log \varGamma _{d}\left( \theta _{n}+ \frac{(d + 1)}{2}\right) - \log \varGamma _{d}\left( \theta '_{n} + \frac{(d + 1)}{2}\right) \nonumber \\ =&- \frac{(d + 1)}{2}d\log (2) + \left( \theta _{n}+ \frac{(d + 1)}{2}\right) \log |\theta _{S}| + \left( \theta _{n}'+ \frac{(d + 1)}{2}\right) \log |\theta _{S}'|\nonumber \\&- \left( \theta _{n} + \theta _{n}'+ \frac{(d + 1)}{2}\right) \log |\theta _{S}+\theta _{S}'| + \log \left( \frac{\varGamma _{d}\left( \theta _{n} + \theta '_{n} + \frac{(d + 1)}{2}\right) }{\varGamma _{d}\left( \theta _{n} + \frac{(d \,+\, 1)}{2}\right) \varGamma _{d}\left( \theta '_{n} + \frac{(d\, +\, 1)}{2}\right) }\right) \end{aligned}$$

(11.45)

Remark $\varDelta (\theta ,\theta ) \ne 0$. Same quantity with source parameters $\lambda =(n,S)$ and $\lambda '=(n',S')$ is

$$\begin{aligned} \varDelta (\lambda ,\lambda ') =\,&- \frac{(d + 1)}{2}d\log (2) -\frac{n}{2} \log |S| - \frac{n'}{2} \log |S'| - \frac{n+n'-d-1}{2} \log |S^{-1}\nonumber \\&+ S'^{-1}| + \log \left( \frac{\varGamma _{d}\left( \frac{n+n'-d-1}{2}\right) }{\varGamma _{d}\left( \frac{n}{2}\right) \varGamma _{d}\left( \frac{n'}{2}\right) }\right) \end{aligned}$$

(11.46)

11.1.2 Distribution $\mathcal {W}_{d,\underline{n}}$

$$\begin{aligned} \mathcal {W}_{d}(X;\underline{n},S)&= \frac{|X|^{\frac{\underline{n}-d-1}{2}}\exp \{-\frac{1}{2} {\mathrm {tr}}(S^{-1} X)\}}{2^{\frac{\underline{n}d}{2}} |S|^{\frac{\underline{n}}{2}} \varGamma _{d}(\frac{\underline{n}}{2})}\\&= \exp \left\{ \frac{\underline{n}-d-1}{2} \log |X|-\frac{1}{2} {\mathrm {tr}}(S^{-1} X)- \frac{\underline{n}d}{2}\log (2) - \frac{\underline{n}}{2} \log |S| - \log \varGamma _{d}\left( \frac{\underline{n}}{2}\right) \right\} \end{aligned}$$

Letting $\theta _{S}=S^{-1}$,

$$\begin{aligned} \mathcal {W}_{d}(X;\underline{n},\theta _{S})&= \exp \left\{ -\frac{1}{2} {\mathrm {tr}}(\theta _{S} X) + \frac{\underline{n}-d-1}{2} \log |X| - \frac{\underline{n}d}{2}\log (2) - \frac{\underline{n}}{2} \log |\theta _{S}^{-1}| - \log \varGamma _{d}\left( \frac{\underline{n}}{2}\right) \right\} \\&= \exp \left\{ <\theta _{S},- \frac{1}{2}X>_{HS} + k(X) - F_{\underline{n}}(\theta _{S})\right\} \\&\quad \quad \text{ with } F_{\underline{n}}(\theta _{S}) = \frac{\underline{n}d}{2}\log (2) - \frac{\underline{n}}{2} \log |\theta _{S}| + \log \varGamma _{d}\left( \frac{\underline{n}}{2}\right) \\&\quad \quad \text{ with } k_{\underline{n}}(X) = \frac{\underline{n}-d-1}{2}\log |X| \end{aligned}$$

Using the rule $\frac{\partial log |X|}{\partial X} = ^{t}(X^{-1})$ [33] and the symmetry of $\theta _{S}$, we get

$$\nabla _{\theta _{S}} F_{\underline{n}} (\theta _{S})= - \frac{\underline{n}}{2}\theta _{S}^{-1}$$

The correspondence between natural parameter $\theta _{S}$ and expectation parameter $\eta _{S}$ is

$$\eta _{S} = \nabla _{\theta _{S}} F_{\underline{n}} (\theta _{S}) = - \frac{\underline{n}}{2}\theta _{S}^{-1} \longleftrightarrow \theta _{S} = \nabla _{\eta _{S}} F_{\underline{n}}^{*}(\eta _{S}) = (\nabla _{\theta _{S}} F_{\underline{n}})^{-1}(\eta _{S}) = - \frac{\underline{n}}{2}\eta _{S}^{-1}$$

Finally, we obtain the MLE for $\theta _{S}$ in this sub family:

$$\hat{\theta }_{S} = - \frac{\underline{n}}{2} \left( \frac{1}{N}\sum _{i=1}^{N} - \frac{1}{2}X_{i}\right) ^{-1} = \underline{n} N \left( \sum _{i=1}^{N} X_{i}\right) ^{-1}$$

Same formulation with source parameter $S$:

$$\hat{S} = \hat{\theta }_{S}^{-1} = \left( \underline{n} N\left( \sum _{i=1}^{N} X_{i}\right) ^{-1}\right) ^{-1} = \frac{\sum _{i=1}^{N} X_{i}}{\underline{n}N}$$

Dual log-normalizer $F_{\underline{n}}^{*}$ for $\mathcal {W}_{d,\underline{n}}$ is

$$\begin{aligned} F_{\underline{n}}^{*} (\eta _{S})&= \langle (\nabla F_{\underline{n}})^{-1} (\eta _{S}), \eta _{S}\rangle - F_{\underline{n}}((\nabla F_{\underline{n}})^{-1} (\eta _{S}))\\&= \langle - \frac{\underline{n}}{2}\eta _{S}^{-1}, \eta _{S}\rangle - F_{\underline{n}}(- \frac{\underline{n}}{2}\eta _{S}^{-1})\\&= - \frac{\underline{n}}{2} {\mathrm {tr}}(\eta _{S}^{-1} \eta _{S}) - \frac{\underline{n}d}{2}\log (2) + \frac{\underline{n}}{2} \log \left[ (\frac{\underline{n}}{2})^{d} |-\eta _{S}^{-1} | \right] - \log \varGamma _{d}\left( \frac{\underline{n}}{2}\right) \\&= - \frac{\underline{n}d}{2} (1+ \log (2) - \log \underline{n} + \log 2) + \frac{\underline{n}}{2} \log |-\eta _{S}^{-1} | - \log \varGamma _{d}\left( \frac{\underline{n}}{2}\right) \\&= \frac{\underline{n}d}{2} \log \left( \frac{\underline{n}}{4e}\right) + \frac{\underline{n}}{2} \log |-\eta _{S}^{-1}| - \log \varGamma _{d}\left( \frac{\underline{n}}{2}\right) \\[-28pt] \end{aligned}$$

$$\begin{aligned} {\mathrm {KL}}(\mathcal {W}_{d,\underline{n}}^{1} || \mathcal {W}_{d,\underline{n}}^{2})&= B_{F_{\underline{n}}}(\theta _{S_{2}}:\theta _{S_{1}})\\&= F_{\underline{n}}(\theta _{S_{2}}) - F_{\underline{n}}(\theta _{S_{1}}) - {<}\theta _{S_{2}}-\theta _{S_{1}}, \nabla _{\theta _{S}} F_{\underline{n}}(\theta _{S_{1}}){>}\\&= \frac{\underline{n}}{2} \left( \log |\theta _{S_{1}}| - \log |\theta _{S_{2}}| \right) + \frac{\underline{n}}{2} {\mathrm {tr}}((\theta _{S_{2}}-\theta _{S_{1}})\theta _{S_{1}}^{-1})\\&= \frac{\underline{n}}{2} \left( \log \frac{|\theta _{S_{1}}|}{|\theta _{S_{2}}|} + {\mathrm {tr}}(\theta _{S_{2}}\theta _{S_{1}}^{-1}) - d\right) \\&= \frac{\underline{n}}{2} \left( - \log \frac{|\theta _{S_{2}}|}{|\theta _{S_{1}}|} + {\mathrm {tr}}(\theta _{S_{2}}\theta _{S_{1}}^{-1}) - d\right) \end{aligned}$$

also with source parameter

$${\mathrm {KL}}(\mathcal {W}_{d,\underline{n}}^{1} || \mathcal {W}_{d,\underline{n}}^{2}) = \frac{\underline{n}}{2} \left( - \log \frac{|S_{1}|}{|S_{2}|} + {\mathrm {tr}}(S_{2}^{-1}S_{1}) - d\right) $$

Let’s remark that KL divergence depends now on $\underline{n}$.

$$\begin{aligned} B_{F_{\underline{n}}^*} (\eta _{S_{1}} : \eta _{S_{2}})&= {F_{\underline{n}}^*}(\eta _{S_{1}}) - {F_{\underline{n}}^*}(\eta _{S_{2}}) - {<}\eta _{S_{1}} - \eta _{S_{2}}, \nabla F_{\underline{n}}^{*}(\eta _{S_{2}}){>}_{HS}\\&= \frac{\underline{n}}{2} \left( \log |-\eta _{S_{1}}^{-1}| - \log |-\eta _{S_{2}}^{-1}|\right) - {<}\eta _{S_{1}} - \eta _{S_{2}}, -\frac{\underline{n}}{2} \eta _{S_{2}}^{-1} {>}_{HS}\\&= \frac{\underline{n}}{2} \left( \log \frac{|-\eta _{S_{1}}^{-1}|}{|-\eta _{S_{2}}^{-1}|} + {\mathrm {tr}}(\eta _{S_{1}}\eta _{S_{2}}^{-1}) - d\right) \end{aligned}$$

11.1.3 Distribution $\mathcal {W}_{d,\underline{S}}$

For fixed $\underline{S}$, the p.d.f of $\mathcal {W}_{d,\underline{S}}$ can be rewritten⁴ as

$$\begin{aligned} \mathcal {W}_{d}(X;n,\underline{S})&= \frac{|X|^{\frac{n-d-1}{2}}\exp \{-\frac{1}{2} {\mathrm {tr}}(\underline{S}^{-1} X)\}}{|2\underline{S}|^{\frac{n}{2}} \varGamma _{d}(\frac{n}{2})}\\&= \exp \left\{ \frac{n-d-1}{2} \log |X|-\frac{1}{2} {\mathrm {tr}}(\underline{S}^{-1} X)- \frac{n}{2}\log |2\underline{S}| - \log \varGamma _{d}\left( \frac{n}{2}\right) \right\} \end{aligned}$$

Letting $\theta _{n}=\frac{n-d-1}{2}$ ($n=2\theta _{n}+d+1$)

$$\begin{aligned} \mathcal {W}_{d}(X;\theta _{n},\underline{S})&= \exp \left\{ \theta _{n} \log |X| -\frac{1}{2} {\mathrm {tr}}(\underline{S}^{-1} X) - \left( \theta _{n} + \frac{d+1}{2}\right) \log \left| 2\underline{S}\right| - \log \varGamma _{d}\left( \theta _{n} + \frac{d+1}{2}\right) \right\} \\&= \exp \left\{ <\theta _{n},\log |X|> + k_{\underline{S}}(X) - F_{\underline{S}}(\theta _{n})\right\} \\&\quad \quad \text{ with } F_{\underline{S}}(\theta _{n}) = \left( \theta _{n} + \frac{d+1}{2}\right) \log \left| 2\underline{S}\right| + \log \varGamma _{d}\left( \theta _{n} + \frac{d+1}{2}\right) \\&\quad \quad \text{ with } k_{\underline{S}}(X) = -\frac{1}{2} {\mathrm {tr}}(\underline{S}^{-1} X) \end{aligned}$$

The correspondence between natural parameter $\theta _{n}$ and expectation parameter $\eta _{n}$ is

$$\eta _{n} = \nabla _{\theta _{n}} F_{\underline{S}} (\theta _{n}) = \log \left| 2\underline{S}\right| + \varPsi _{d}\left( \theta _{n} + \frac{(d + 1)}{2}\right) $$

$$\begin{aligned} \Leftrightarrow&\qquad \qquad \varPsi _{d}\left( \theta _{n} + \frac{(d + 1)}{2}\right) = \eta _{n} - \log \left| 2\underline{S}\right| \\ \Leftrightarrow&\qquad \qquad \theta _{n} + \frac{(d + 1)}{2} = \varPsi _{d}^{-1}\left( \eta _{n} - \log \left| 2\underline{S}\right| \right) \\ \Leftrightarrow&\theta _{n} = \varPsi _{d}^{-1}\left( \eta _{n} - \log \left| 2\underline{S}\right| \right) - \frac{(d + 1)}{2} = (\nabla F_{\underline{S}})^{-1} (\eta _{n}) = \nabla F_{\underline{S}}^* (\eta _{n})\\ \end{aligned}$$

Finally, we obtain the MLE for $\theta _{n}$ in this sub family:

$$\hat{\theta }_{n} = \varPsi _{d}^{-1}\left( \left[ \frac{1}{N} \sum _{i=1}^{N} \log \left| X\right| \right] - \log \left| 2\underline{S}\right| \right) - \frac{(d + 1)}{2}$$

Same formulation with source parameter $n$:

$$\begin{aligned} \frac{\hat{n} - d -1}{2}&= \varPsi _{d}^{-1}\left( \left[ \frac{1}{N} \sum _{i=1}^{N} \log \left| X\right| \right] - \log \left| 2\underline{S}\right| \right) - \frac{(d + 1)}{2}\\ \hat{n}&= 2 \varPsi _{d}^{-1}\left( \left[ \frac{1}{N} \sum _{i=1}^{N} \log \left| X\right| \right] - \log \left| 2\underline{S}\right| \right) \end{aligned}$$

Dual log-normalizer $F_{\underline{S}}^{*}$ for $\mathcal {W}_{d,\underline{S}}$ is

$$\begin{aligned} F_{\underline{S}}^{*} (\eta _{n}) = \,&\langle (\nabla F_{\underline{S}})^{-1} (\eta _{n}), \eta _{n}\rangle - F_{\underline{S}}((\nabla F_{\underline{S}})^{-1} (\eta _{n}))\\ = \,&\langle \varPsi _{d}^{-1}\left( \eta _{n} - \log \left| 2\underline{S}\right| \right) - \frac{(d + 1)}{2}, \eta _{n}\rangle \\&-\varPsi _{d}^{-1}\left( \eta _{n} - \log \left| 2\underline{S}\right| \right) \log \left| 2\underline{S}\right| - \log \varGamma _{d}\left( \varPsi _{d}^{-1}\left( \eta _{n} - \log \left| 2\underline{S}\right| \right) \right) \\ =\,&\varPsi _{d}^{-1}\left( \eta _{n} - \log \left| 2\underline{S}\right| \right) \left( \eta _{n} - \log \left| 2\underline{S}\right| \right) - \frac{(d + 1)}{2} \eta _{n} - \log \varGamma _{d}\left( \varPsi _{d}^{-1}\left( \eta _{n} - \log \left| 2\underline{S}\right| \right) \right) \\[-38pt] \end{aligned}$$

$$\begin{aligned} {\mathrm {KL}}(\mathcal {W}_{d,{\underline{S}}}^{1} || \mathcal {W}_{d,{\underline{S}}}^{2}) =\,&B_{F_{\underline{S}}} (\theta _{n_{2}} : \theta _{n_{1}}) = F_{\underline{S}}(\theta _{n_{2}}) - F_{\underline{S}}(\theta _{n_{1}}) - \langle \theta _{n_{2}} - \theta _{n_{1}} , \nabla F_{\underline{S}} (\theta _{n_{1}})\rangle \\ = \,&\left( \theta _{n_{2}} + \frac{d+1}{2}\right) \log \left| 2\underline{S}\right| + \log \varGamma _{d}\left( \theta _{n_{2}} + \frac{d+1}{2}\right) \\&-\left( \theta _{n_{1}} + \frac{d+1}{2}\right) \log \left| 2\underline{S}\right| - \log \varGamma _{d}\left( \theta _{n_{1}} + \frac{d+1}{2}\right) \\&-\langle \theta _{n_{2}} - \theta _{n_{1}}, \log \left| 2\underline{S}\right| + \varPsi _{d}\left( \theta _{n_{1}} + \frac{(d + 1)}{2}\right) \rangle \\ {\mathrm {KL}}(\mathcal {W}_{d,{\underline{S}}}^{1}|| \mathcal {W}_{d,{\underline{S}}}^{2}) =\,&\log \frac{\varGamma _{d}\left( \theta _{n_{2}} + \frac{d+1}{2}\right) }{\varGamma _{d}\left( \theta _{n_{1}} + \frac{d+1}{2}\right) } - (\theta _{n_{2}} - \theta _{n_{1}}) \varPsi _{d}\left( \theta _{n_{1}} + \frac{(d + 1)}{2}\right) \\ =\,&- \log \frac{\varGamma _{d}\left( \theta _{n_{1}} + \frac{d+1}{2}\right) }{\varGamma _{d}\left( \theta _{n_{2}} + \frac{d+1}{2}\right) } + (\theta _{n_{1}} - \theta _{n_{2}}) \varPsi _{d}\left( \theta _{n_{1}} + \frac{(d + 1)}{2}\right) \end{aligned}$$

also with source parameter

$${\mathrm {KL}}(\mathcal {W}_{d,{\underline{S}}}^{1}|| \mathcal {W}_{d,{\underline{S}}}^{2}) = - \log \left( \frac{\varGamma _{d}\left( \frac{n_{1}}{2}\right) }{\varGamma _{d}\left( \frac{n_{2}}{2}\right) }\right) + \left( \frac{n_{1} - n_{2}}{2}\right) \varPsi _{d}\left( \frac{n_{1}}{2}\right) $$

Let us remark that this quantity does not depend on $\underline{S}$.

$$\begin{aligned} B_{F_{\underline{S}}^*} (\eta _{n_{1}} : \eta _{n_{2}}) =\,&{F_{\underline{S}}^*}(\eta _{n_{1}}) - {F_{\underline{S}}^*}(\eta _{n_{2}}) - {<}\eta _{n_{1}} - \eta _{n_{2}}, \nabla F_{\underline{S}}^{*}(\eta _{n_{2}}){>}_{HS}\\ =\,&\varPsi _{d}^{-1}\left( \eta _{n_{1}} - \log \left| 2\underline{S}\right| \right) \left( \eta _{n_{1}} - \log \left| 2\underline{S}\right| \right) - \frac{(d + 1)}{2} \eta _{n_{1}} \\&\quad - \log \varGamma _{d}\left( \varPsi _{d}^{-1}\left( \eta _{n_{1}} - \log \left| 2\underline{S}\right| \right) \right) \\&- \varPsi _{d}^{-1}\left( \eta _{n_{2}} - \log \left| 2\underline{S}\right| \right) \left( \eta _{n_{2}} - \log \left| 2\underline{S}\right| \right) + \frac{(d + 1)}{2} \eta _{n_{2}} \\&\quad + \log \varGamma _{d}\left( \varPsi _{d}^{-1}\left( \eta _{n_{2}} - \log \left| 2\underline{S}\right| \right) \right) \\&- \langle \eta _{n_{1}} - \eta _{n_{2}}, \varPsi _{d}^{-1}\left( \eta _{n_{2}} - \log \left| 2\underline{S}\right| \right) - \frac{(d + 1)}{2}\rangle _{HS}\\ B_{F_{\underline{S}}^*} (\eta _{n_{1}} : \eta _{n_{2}}) = \,&\log \frac{\varGamma _{d}\left( \varPsi _{d}^{-1}\left( \eta _{n_{2}} - \log \left| 2\underline{S}\right| \right) \right) }{\varGamma _{d}\left( \varPsi _{d}^{-1}\left( \eta _{n_{1}} - \log \left| 2\underline{S}\right| \right) \right) }\\&-\left[ \varPsi _{d}^{-1}\left( \eta _{n_{2}} - \log \left| 2\underline{S}\right| \right) - \varPsi _{d}^{-1}\left( \eta _{n_{1}} - \log \left| 2\underline{S}\right| \right) \right] \left( \eta _{n_{1}} - \log \left| 2\underline{S}\right| \right) \end{aligned}$$