Abstract
Maximum likelihood estimation of the concentration parameter of von Mises–Fisher distributions involves inverting the ratio R_nu = I_{nu +1} / I_nu of modified Bessel functions and computational methods are required to invert these functions using approximative or iterative algorithms. In this paper we use Amos-type bounds for R_nu to deduce sharper bounds for the inverse function, determine the approximation error of these bounds, and use these to propose a new approximation for which the error tends to zero when the inverse of R_nu is evaluated at values tending to 1 (from the left). We show that previously introduced rational bounds for R_nu which are invertible using quadratic equations cannot be used to improve these bounds.
Highlights
A random unit length vector in Rd has a von Mises–Fisher distribution with parameter θ ∈ Rd if its density with respect to the uniform distribution on the unit hypersphere Sd−1 = {x ∈ Rd : x = 1} is given by B
We show that the error of the suggested new approximation tends to zero for ρ → 1−, whereas the error tends to −1/2 for the Dhillon–Sra approximation, which is too large for large ρ
In the following we compare the number of iterations required to reach convergence by two different algorithms based on nested intervals for finding roots using (1) the Tanabe et al (2007) and (2) the newly established bounds for initialization
Summary
Tanabe et al (2007) suggest to use the “mid-point” approximation with c = 1, i.e., Rν−1(ρ) ≈ (2ν + 1)ρ/(1 − ρ2) as the starting value for iterative schemes for solving Rν(t) = ρ, such as the fixed-point iteration tn+1 = tnρ/Rν(tn). We use a family of bounds for Rν first introduced in Amos (1974) to provide substantially sharper bounds for Rν−1, which have approximation error at most 3ρ/2, and use these results to suggest a new approximation. We establish that these improved bounds hold for the Dhillon–Sra approximation, which has the same maximal approximation error. We investigate whether the rational bounds for Rν developed by Nåsell (1978) can be used to obtain improved explicit bounds for Rν−1, and show that for the rational bounds which can be inverted by solving quadratic equations, no such improvement is possible
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