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The authors have introduced an interesting and mathematically intricate method for Markov chain Monte Carlo (MC) simulation on an embedded manifold. The geodesic MC method provides large proposals as part of the scheme, which are devised by careful study of the Riemannian geometry of the space and the geodesics in particular. The aim of the resulting algorithm is to produce a chain with low autocorrelation and high acceptance probabilities. As displayed by the authors, the method is well geared up for simulating from unimodal distributions on a manifold via the gradient of the log-density and the geodesic flow. They also demonstrate its effective use in multimodal scenarios via parallel tempering. Given that there are always many choices of embedding, should one choose as low-dimensional embedding as possible?

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Figure 1. Simulated values of x5 for the Fisher–Bingham example with c = 0. There are 6588 simulated values from two billion proposals.

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There are various levels of approximation in the algorithm, and so it is worth exploring in any specific application if simpler algorithms can end up providing more efficient or more accurate simulations. Consider the Fisher–Bingham example, and recall that the Fisher–Bingham (c,A) distribution can be defined as

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with inline image, a is chosen such that (A + aIp) is negative definite (see Mardia & Jupp, 2000, p.175), and Ip is the p × p identity matrix. Because the Fisher–Bingham density is unchanged by adding aIp to A, we can, for example, choose a such that trace(Σ) = 1. The integrating constant of the Fisher–Bingham can be expressed in terms of the density of a linear combination of non-central inline image random variables (Kume & Wood, 2005), which can be evaluated using a saddlepoint approximation. Hence, simulation via rejection methods is feasible.

An even simpler approach when c is small could be to simulate from Y ∼ Np(μ,Σ), and then keep only the observations that fall within | ∥ Y ∥ − 1 | < ν, for small ν > 0. This naive conditioning method might appear rather inefficient, but the accepted observations are independent draws. Note that if the dimension p is large and X Bingham distributed with trace(Σ) = 1,trace(Σ2) ࣈ 1, c = 0 then from Dryden (2005) we have the approximation X ࣈ Np(0,Σ). Hence, even for large p, this can still be a practical method for certain Σ. In Fig. 1, we show the results of this algorithm in the example from Section 5.1 of the paper, with c = 0 and with two billion proposals and ν = 2 × 10 − 6. Here, a = − 23.06176, and the acceptance rate is 0.00033%.

There is always a trade-off with any simulation method, and one needs to compromise between the level of approximation (through ν here), the efficiency in run time, the independence of observations and the amount of coding involved in the implementation. For this Bingham example, the naive conditional method seems reasonable here, giving independent, near exact realizations and very minimal effort in coding. However, the beauty of the geodesic MC method of the paper is that the algorithm is quite general, and so can be tried out in a range of scenarios where there may be no reasonable alternative.

References

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  2. References
  • Dryden, I. L. (2005). Statistical analysis on high-dimensional spheres and shape spaces. Ann. Statist. 33, (4), 16431665.
  • Kume, A. & Wood, A. T. A. (2005). Saddlepoint approximations for the Bingham and Fisher-Bingham normalising constants. Biometrika 92, (2), 465476.
  • Mardia, K. V. & Jupp, P. E. (2000). Directional statistics, Wiley Series in Probability and Statistics, John Wiley & Sons Ltd., Chichester.