Calculating the normalising constant of the Bingham distribution on the sphere using the holonomic gradient method

Abstract

In this paper we implement the holonomic gradient method to exactly compute the normalising constant of Bingham distributions. This idea is originally applied for general Fisher–Bingham distributions in Nakayama et al. (Adv. Appl. Math. 47:639–658, 2011). In this paper we explicitly apply this algorithm to show the exact calculation of the normalising constant; derive explicitly the Pfaffian system for this parametric case; implement the general approach for the maximum likelihood solution search and finally adjust the method for degenerate cases, namely when the parameter values have multiplicities.

Appendices

Appendix A: One-dimensional representation

First we briefly describe derivation of the one-dimensional representation (3) according to Kume and Wood (2005). Note that the parameter λ _i in their paper is our −θ _i . Consider p independent normal random variables x _i ∼N(0,(−2θ _i )⁻¹), where θ _i <0 for all i. Then the marginal density of $r=\sum_{i=1}^{p}x_{i}^{2}$ is directly calculated as

$$\begin{aligned} f(r) = \frac{r^{p/2-1}\prod_k\sqrt{-\theta_k}}{\varGamma (p/2)}\frac{c(r\boldsymbol{\theta})}{c(\boldsymbol{0})}. \end{aligned}$$

(26)

On the other hand, the characteristic function of r is

$$\begin{aligned} \phi(s) = E\bigl[e^{is\sum_ix_i^2}\bigr] = \prod_k \sqrt{\frac{-\theta_k}{-\theta_k-is}}. \end{aligned}$$

(27)

In general, the density function is represented by its characteristic function as

$$\begin{aligned} f(r) = \frac{1}{2\pi}\lim_{\epsilon\to0}\int_{-\infty}^{\infty} \phi(s)e^{-is r-\epsilon s^2/2}ds \end{aligned}$$

(28)

at arbitrary continuous point of f (see e.g. Feller 1971). By combining Eqs. (26) to (28), we have

$$\begin{aligned} c(\boldsymbol{\theta}) &= \frac{\varGamma(p/2)c(\boldsymbol{0})}{\prod_k\sqrt{-\theta_k}}f(1) \\ &= \frac{c(\boldsymbol{0})}{2\pi}\lim_{\epsilon\to0}\int_{-\infty}^{\infty} \frac{1}{\prod_k\sqrt{-\theta_k-is}}e^{-is-\epsilon s^2/2}ds. \end{aligned}$$

Note that this expression holds for any p≥2. If p≥3, then

$$\begin{aligned} c(\boldsymbol{\theta}) &= \frac{c(\boldsymbol{0})}{2\pi}\int_{-\infty}^{\infty} \frac{1}{\prod_k\sqrt{-\theta_k-is}}e^{-is}ds \end{aligned}$$

since $|\prod_{k=1}^{p}(-\theta_{k}-is)^{-1/2}|$ is integrable over (−∞,∞). By analytic continuation with respect to s, we obtain

$$\begin{aligned} c(\boldsymbol{\theta}) &= \frac{c(\boldsymbol{0})}{2\pi}\int_{-\infty}^{\infty} \frac{1}{\prod_k\sqrt{-\theta_k-t_0-is}}e^{-t_0-is}ds \end{aligned}$$

(29)

for any real number t ₀ less than min _k (−θ _k ). Even if some θ _i ’s are not negative, Eq. (29) still holds due to analytic continuation with respect to θ, as long as t ₀<min _k (−θ _k ). Hence we obtain (3).

Appendix B: Truncation error of the power series

We derive the power series expansion of c(θ) and evaluate the truncation error according to Koyama et al. (2012a). Let θ(ϕ,d)=(ϕ ₁,…,ϕ ₁,…,ϕ _q ,…,ϕ _q ) be a parameter vector with multiplicities d=(d ₁,…,d _q ). By Kume and Wood (2007), we have

$$\begin{aligned} c\bigl(\boldsymbol{\theta}(\boldsymbol{\phi},\boldsymbol{d})\bigr) =& \int _{S^{p-1}} e^{\sum_{i=1}^p \theta_i(\boldsymbol{\phi },\boldsymbol{d}) x_i^2} d\boldsymbol{x} = c(\boldsymbol{0})\sum _{k_1=0}^{\infty}\cdots \sum _{k_q=0}^{\infty} \frac{\phi_1^{k_1}\cdots\phi_q^{k_q}}{k_1!\cdots k_q!} \frac{\prod_i \varGamma(k_i+d_i/2)}{\varGamma(\sum_i (k_i+d_i/2))} \frac{\varGamma(\sum_i d_i/2)}{\prod_i\varGamma(d_i/2)}. \end{aligned}$$

Let

$$c_N\bigl(\boldsymbol{\theta}(\boldsymbol{\phi},\boldsymbol{d})\bigr) := c(\boldsymbol{0}) \sum_{k_1+\cdots+k_q < N} \frac{\phi_1^{k_1}\cdots\phi_q^{k_q}}{k_1!\cdots k_q!} \frac{\prod_i \varGamma(k_i+d_i/2)}{\varGamma(\sum_i (k_i+d_i/2))} \frac{\varGamma(\sum_i d_i/2)}{\prod_i\varGamma(d_i/2)}. $$

Then the truncation error is evaluated as

$$\begin{aligned} \frac{|c(\boldsymbol{\theta}(\boldsymbol{\phi},\boldsymbol{d})) - c_N(\boldsymbol{\theta}(\boldsymbol{\phi},\boldsymbol {d}))|}{c(\boldsymbol{0})} \leq&\sum_{k_1+\cdots+k_q\geq N} \frac{|\phi_1|^{k_1}\cdots|\phi_q|^{k_q}}{k_1!\cdots k_q!} \frac{\prod_i\varGamma(k_i+d_i/2)}{\varGamma(\sum_i(k_i+d_i/2))} \frac{\varGamma(\sum_i d_i/2)}{\prod_i\varGamma(d_i/2)} \\ \leq&\sum_{k_1+\cdots+k_q\geq N} \frac{|\phi_1|^{k_1}\cdots|\phi_q|^{k_q}}{k_1!\cdots k_q!} \\ =& \sum_{n=N}^{\infty} \frac{1}{n!}\sum _{k_1+\cdots+k_q=n} \frac{n!}{k_1!\cdots k_q!} |\phi_1|^{k_1} \cdots|\phi_q|^{k_q} = \sum_{n=N}^{\infty} \frac{\|\boldsymbol{\phi}\|_1^n}{n!} \\ \leq&\frac{\|\boldsymbol{\phi}\|_1^N}{N!} \sum_{n=N}^{\infty} \frac{\|\boldsymbol{\phi}\|_1^{n-N}}{(N+1)^{n-N}} = \frac{\|\boldsymbol{\phi}\|_1^N}{N!} \frac{N+1}{N+1-\|\boldsymbol{\phi}\|_1}. \end{aligned}$$