von Mises  Fisher (vMF)
 The von Mises  Fisher (vMF) distribution is a probability distribution on directions in R^{D}. It is natural to think of it as a distribution on the (D1)sphere of unit radius, that is on the surface of the Dball of unit radius.
 The von Mises  Fisher's probability density function is
 pdf(v  μ, κ) = C_{D} e^{κμ.v}
 where datum v is a normalised Dvector, equivalently a point on the (D1)sphere,
 mu, μ, is the mean (a normalised vector), and
 kappa, κ ≥ 0, is the concentration parameter (a scalar).
 The distribution's normalising constant
 C_{D}(κ) = κ^{D/21} / {(2π)^{D/2} I_{D/21}(κ)}
 where I_{order}(.) is the "modified Bessel function of the first kind"!
 In the special case that D = 3,
 C_{3}(κ) = κ / {2π (e^{κ}  e^{κ})}
 The negative log pdf is
  log pdf(v  μ, κ) =  log C_{D}  κ μ . v,
 and

C_{D}  Given data
N
 R = ∑_{i=0..N1} v_{i},
R  and
 Rbar = R / N.

 logLH   logLH =  N log C_{D}  κ μ . R.
 It is obvious that the maximum likelihood estimate of μ is R normalised,
 μ_{ML} = R / R,

μ_{MML}  μ_{MML} = μ_{ML} = R / R,
 the most general prior for μ being the uniform distribution.
 For given μ and κ, the expected value of Rbar equals

A_{D}(κ)  and the (less obvious) maximum likelihood estimate of κ is
 κ_{ML} = A^{1}(Rbar).
 This is because
 ^{∂}/_{∂κ}  logLH =  N {^{∂}/_{∂κ} log C_{D}(κ)}  μ . R
 which is zero if
  ^{∂}/_{∂κ} log C_{D}(κ) = μ . R / N,
 where
 ^{∂}/_{∂κ} log C_{D}(κ)
 = ω / κ  I'_{ω}(κ) / I_{ω}(κ), where ω = D/2  1
 = ω {I_{ω}(κ)  ^{κ}/_{ω} I'_{ω}(κ)} / (κ I_{ω}(κ))
 = ω {^{κ}/_{2ω} {I_{ω1}(κ)  I_{ω+1}(κ)}  ^{κ}/_{2ω} {I_{ω1}(κ) + I_{ω+1}(κ)}} / (κ I_{ω}(κ))
 =  I_{D/2}(κ) / I_{D/21}(κ),
 using the "well known" relations,
 I_{ν}(z) = ^{z}/_{2ν} {I_{ν1}(z)  I_{ν+1}(z)},
 and
 I'_{ν}(z) = ^{1}/_{2} {I_{ν1}(z) + I_{ν+1}(z)}, (I'_{0}(z) = I_{1}(z)).
 The MML estimate, κ_{MML},
κ_{MML}  The Fisher information of the vMF distribution.
 The expected second derivative of  logLH w.r.t. κ is
 ^{∂2}/_{∂κ2}  logLH = N A'_{D}(κ).
 The vMF distribution is symmetric about μ on the (D1)sphere; there is no preferred orientation around μ. A direction, such as μ, has D  1 degrees of freedom. The expected 2nd derivative of  logLH w.r.t. any one of μ's degrees of freedom is
 N κ A_{D}(κ).
 This is for the following reason:
 Without loss of generality, let μ = (1, 0, ...), and then μ → (cos δ, sin δ, 0, ...), say, where δ is small,
 ^{∂}/_{∂δ}  logLH = N κ R sin δ,
 ^{∂2}/_{∂δ2}  logLH = N κ R cos δ ≈ N κ R, as δ is small
 which is
 N κ A_{D}(κ) in expectation.
 Symmetry implies that the offdiagonal elements for μ are zero. And, μ is a position parameter and κ a scale parameter, so the offdiagonal elements between μ and κ are also zero.
 F, the Fisher information of the vMF is therefore,

F  Sources
 Search for [vonMises direction] in the [Bib], and
 see section 6.5, p.266 of Wallace's book (2005).
 P. Kasarapu & L. Allison, Minimum message length estimation of mixtures of multivariate Gaussian and von MisesFisher distributions, Machine Learning (Springer Verlag), March 2015 [click].
 The special case of the probability distribution where D = 2 is known as the von Mises distribution for directions in R^{2}, that is for angles and periodic quantitites such as annual events.