In the context of the Bingham probability distribution the ${ }_1F_1$ hypergeometric function of matrix argument naturally arises as a normalization constant of the probability distribution function. Thus, it is of interest to evaluate this function effectively. For the more general class of hypergeometric functions $ _pF_q$ of Matrix argument, an algorithm was given by Koev. However, this algorithm still has some open problems discussed in Koevs paper. My question is, whether there is a way to compute the normalization constant of a bingham distribution for arbitrary dimensions effectively.
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2 Answers
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It would be useful, if you'd point out the precise "open questions" regarding these evaluations that are of interest to you.
But since you mentioned Plamen Koev's work, I am sure you have tried out his matlab code for computing Hypergeometric functions of matrix argument. For the particular case of the normalization constant of the Bingham distribution, I am sure methods for approximating high-dimensional integrals numerically will prove to be effective, because, although scary looking, the normalization constant still has a very nice form:
\begin{equation*} {}_1F_1(\frac12,\frac p2, A) := \int_{S^{p-1}} e^{x^TAx}dx, \end{equation*} where the integration is wrt to the uniform distribution on the unit hypersphere $S^{p-1}$.
(PS: This integral has previously been discussed on MO, e.g., here in this question of L. Nicolaescu)
Edit: Regarding numerical approximation. Here is what seems to be the latest in this direction. Have a look at the geodesic monte carlo sampling method of Byrne and Girolami; (they discuss sampling from the Bingham distribution). Once you have that, then it should be "easy" to estimate the normalization constant. But I guess the bad thing might again be a lack of guarantees on how long it takes to get a given accuracy approximation---but for now, seems like the abovecited approach might be the most promising.
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$\begingroup$ Thank you for your post. I was aware of the code and the contribution in Koevs paper is very interesting. Unfortunately, there is no error analysis for his proposed algorithm yet and thus it is not possible to know the number of terms needed in his algorithm to obtain a pre-defined accuracy. Numerical integration would work. Still, it would be necessary to handle the curse of dimensionality for large p $\endgroup$ Commented Mar 21, 2013 at 10:30
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$\begingroup$ Thanks for your further remarks. I have come around a paper by Kume & Walker (link.springer.com/article/10.1007%2Fs11222-008-9081-z), which gives a way to compute the normalization constant with a rigorous error analysis. Unfortunately their method might become computationally quite burdensome for some interesting cases. $\endgroup$ Commented Mar 28, 2013 at 9:54
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1$\begingroup$ I guess, that getting rigorous error bounds on approximating this integral will be tedious----however, the route I would take is to do some "asymptotic approximations" or "Pade approximations" so that one has a numerically fast method.... $\endgroup$– SuvritCommented Mar 28, 2013 at 18:36
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$\begingroup$ I assume the question to be answered by your reply and our discussion below. One more thing I'll be trying is the use of saddlepoint approximations, which were also derived by Kume. $\endgroup$ Commented Apr 1, 2013 at 12:54
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$\begingroup$ if you are really interested in solving this question, please get in touch with me by email---I have previously (though for a much simpler case) discussed such problems with a colleague of mine. $\endgroup$– SuvritCommented Apr 2, 2013 at 0:47
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I have used Kume's method of Fisher Bingham sampling in my master's thesis. Using 100-500 samples were enough. Please have a look on the paper; I might have sampling code of my implementation in Matlab somewhere on my machine http://link.springer.com/chapter/10.1007/978-3-642-24085-0_69